> <\body> <\hide-preamble> >> >> >> >> >> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> |CEREMADE & CNRS UMR 7534Université Paris-Dauphine>>>|Institut für MathematikHumboldt-Universität zu Berlin>>>|CEREMADE & CNRS UMR 7534Université Paris-Dauphine>>>|>|>> We introduce an approach to study certain singular PDEs which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough paths. We illustrate its applicability on some model problems like differential equations driven by fractional Brownian motion, a fractional Burgers type SPDE driven by space-time white noise, and a non-linear version of the parabolic Anderson model with a white noise potential. |> In this paper we introduce the notion of and show how to use it to give a meaning to and solve partial differential equations involving non-linear operations on generalized functions. More precisely, we combine the idea of , introduced in, with the introduced by Bony and the related paradifferential calculus, in order to develop a non-linear theory for a certain class of distributions. The approach presented here works for generalized functions defined on an index set of arbitrary dimension and constitutes a flexible and lightweight generalization of Lyons' rough path theory. In particular it allows to handle problems involving singular stochastic PDEs which were substantially out of reach with previously known methods. In order to set the stage for our analysis let us list some of the problems which are amenable to be analyzed in the paracontrolled framework: <\enumerate> The differential equation () driven by a --dimensional Gaussian process : <\equation*> \*u=F|)>*\*X, where \,|)>> is a smooth vector-field . Typically, will be a Brownian motion or a fractional Brownian motion with Hurst exponent >. The paracontrolled analysis works up to 1/3>. While we do not have any substantial new results for this problem, it is a useful pedagogical example on which we can easily describe our approach. Generalizations of Hairer's Burgers-like SPDE (): <\equation*> L*u=G*\*u+\. Here \\>, where =/2*\|)>> denotes the torus, +(-\)>>, where )>> is the fractional Laplacian with periodic boundary conditions and we will take \5/6>, and > is a space-time white noise with values in >. Moreover, \,|)>> is a smooth field of linear transformations. A non-linear generalization of the parabolic Anderson model (): <\equation*> L*u=F*\*\, where \\>, -\> is the parabolic operator corresponding to the heat equation, and where > is a random potential which is sampled according to the law of the white noise on > and is therefore independent of the time variable. We allow for a general smooth function \>, the linear case =u> corresponding to the standard parabolic Anderson model. The symbol > stands for a renormalized product which is necessary to have a well defined problem. The one-dimensional periodic Kardar--Parisi--Zhang equation (): <\equation*> L*h=`*`**h|)>*"+\, where \\>, -\>, and where > is a space-time white noise. Here *h|)>*"> denotes the necessity of an additive renormalization in the definition of the square of the distribution *h>. The three-dimensional, periodic, stochastic quantization equation for the |)>> euclidean quantum field (): <\equation*> L*\=`*`|4!>|)>*"+\, where :\\>, -\>, > is a space-time white noise, and where |)>*"> denotes a suitable renormalization of a cubic polynomial of > and > is the coupling constant of the scalar theory. In this paper we will consider in detail the three cases, , . In all cases we will exhibit a space of paracontrolled distributions where the equations are well posed (in a suitable sense), and admit a global solution which is unique. The three-dimensional stochastic quantization equation is studied by R.Catellier and K.Chouk in by applying the paracontrolled technique. The paracontrolled analysis of will be presented elsewhere. The kind of results which will be obtained below can be exemplified by the following statement for s. Below >=B>,\>> stands for the Hölder-Besov space of index > on >. Given two distributions >> and >> with +\\0> we can always consider a certain distribution g> which is obtained via a bilinear operation of and which belongs to +\>>. <\theorem> Let :\> be a continuous function and \,|)>> be a family of smooth vector-fields. Let \> be a solution of the Cauchy problem <\equation*> \*u=F|)>*\,u=u, where \>. Let > be a solution to *\=\> and let =,\\\|)>>. Then for all \> there exists a continuous map :\-1>\-1>\>> such that *,R*\|)>> for all \C;|)>>. In particular, this theorem provides a natural way of extending the solution map to data > which are merely distributions in -1>>. It suffices to approximate > by a sequence of smooth functions |)>> converging to > in -1>>, and to prove that the ``lifted" sequence |)>> converges to some limit in -1>\-1>>. The uniqueness of this limit is not guaranteed however, and each possible limit will give rise to a different notion of solution to the , just like in standard rough path theory. The space > obtained by taking the closure in -1>\-1>> of the set of all elements of the form > for smooth > replaces the space of (geometric) rough paths, and the above theorem is a partial restatement of Lyons' continuity result: namely that the (Itô) solution map >, going from data to solution of the differential equation, is a continuous map from the rough path space > to >>. The space > is fibered over -1>>. It allows us to equip the driving distribution with enough information to control the continuity of the solution map to our problem -- and as we will see below, also the continuity of the solution maps to suitable PDEs. In various contexts the space > can take different forms, and in general it does not seem to have the rich geometrical and algebraic structure of standard rough paths. The verification that suitable approximations |)>> are such that their lifts |)>> converge in -1>\-1>> depends on the particular form of >. In the case of > being a Gaussian stochastic process (like in all our examples above), this verification is the result of almost sure convergence results for elements in a fixed chaos of an underlying Gaussian process, and the proofs rely on elementary arguments on Gaussian random variables. Even in the case of s, the paracontrolled analysis leads to some interesting insights. For example, we have that a more general equation of the form <\equation*> \*u=F|)>*\+F|)>*F|)>*\,u=u, where \C*;\|)>>, has a solution map which depends continuously on ,\\\+\|)>\-1>\-1>>. The remarkable fact here is that the solution map depends only on the combination \\+\> and not on each term separately. Such structural features of the solution map, which can be easily seen using the paracontrolled analysis, are very important in situations where renomalizations are needed, as for example in the model. In the context we can simply remark that setting =-\\\>, the solution map becomes a continuous function of \-1>>, without any further requirement on the bilinear object \\>. Thus, the equation <\equation*> \*u=F|)>*\-F|)>*F|)>*\\|)>,u=u, can be readily extended to any \-1>> by continuity. In that sense, this equation should be interpreted as a deterministic version of an Itô stochastic differential equation where, at the price of a modification of standard rules of calculus, we are able to solve more general problems than in the Stratonovich setting. We remark that, even if only quite implicitly, paraproducts have been already exploited in the rough path context in the work of Unterberger on the renormalization of rough paths, where it is referred to as ``Fourier normal-ordering'', and in the related work of Nualart and Tindel. In this paper we construct weak solutions for the SPDEs under consideration. For an approach using mild solutions see. See also, where we use the decomposition of continuous functions in a certain Fourier series and similar ideas as developed below, in order to give a new and relatively elementary approach to rough path integration. Before going into the details, let us describe the context of our study. Consider for example the problem above. Schwartz' theory of distributions gives a robust framework for defining linear operations on irregular generalized functions. But when trying to handle non-linear operations, we quickly run into problems. For example, in Schwartz' theory, it is not possible to define the product *\*X> in the case where is the sample path of a Brownian motion. The standard analysis of this difficulty goes as follows: is an >--Hölder continuous process for any \1/2>, but not better. The solution has to have the same regularity, which is transferred to > if is smooth. In this situation, the product *\*X> corresponds to the product of an >--Hölder continuous function with the distribution *X> which is of order -1>. A well known result of analysis (see Section below) tells us that a necessary condition for this product to be well defined is that the sum of the orders is positive, that is -1\0>, which is barely violated in the Brownian setting. This is the classical problem which motivated Itô's theory of stochastic integrals. Itô's integral has however quite stringent structural requirements: an ``arrow of time'' (i.e. a filtration and adapted integrands), a probability measure (it is defined as >--limit), and >--orthogonal increments of the integrator (the integrator needs to be a (semi-) martingale). If one or several of these assumptions are violated, then Lyons' rough path integral can be an effective alternative. For example, it allows to construct pathwise integrals for, among other processes, fractional Brownian motion, which is not a semimartingale. In the last years, several other works applied rough path techniques to SPDEs. But they all relied on special features of the problem at hand in order to apply the integration theory provided by the rough path machinery. A first series of works attempts to deal with ``time"-like irregularities by adapting the standard rough path approach: <\itemize> Deya, Gubinelli, Lejay, and Tindel deal with SPDEs of the form =\|)>*\,>>>>> where >, -\>, the noise > is a space-time Gaussian distribution (for example white in time and colored in space), and > is some non-linear coefficient. They interpret this as an evolution equation (in time), taking values in a space of functions (with respect to the space variable). They extend the rough path machinery to handle the convolution integrals that appear when applying the heat flow to the noise. Friz, Caruana, Diehl, and Oberhauser deal with fully non-linear stochastic PDEs with a special structure. Among others, of the form *u=F**u,\*u|)>+\*\*u*\,>>>>> where the spatial index can be multidimensional, but the noise > only depends on time. Such an SPDE can be reinterpreted as a standard PDE with random coefficients via a change of variables involving the flow of the stochastic characteristics associated to >. This flow is handled using usual rough path results for s. Teichmann studies semilinear SPDEs of the form -A|)>*u=\*\,>>>>> where is a suitable linear operator, in general unbounded, and > is a general non-linear operation on the unknown which however should satisfy some restrictive conditions. The SPDE is transformed into an SDE with bounded coefficients by applying a transformation based on the group generated by on a suitable space. The ``arrow of time'' condition of Itô's integral is typically violated if the index is a spatial variable and not a temporal variable. Another series of works applied rough path integrals to deal with situations involving irregularities in the ``space" directions: <\itemize> Bessaih, Gubinelli, and Russo and Brzezniak, Gubinelli, and Neklyudov consider the vortex filament equation which describes the (approximate) motion of a closed vortex line |)>\C,|)>> in an incompressible three-dimensional fluid: <\equation*> \*x|)>=u|)>>|)>|)>,u|)>>=>K*|)>|)>*\>*x|)>*\*\, where \,|)>> is a smooth antisymmetric field of linear transformations of >. In the modeling of turbulence it is interesting to study this equation with initial condition |)>> sampled according to the law of the three-dimensional Brownian bridge. In this case, the regularity of |)>> with respect to > is no better than Brownian for any positive time, and thus the integral in the definition of the velocity field |)>>> is not well defined. Rough path theory allows to make sense of this integral and then of the equation. Hairer, Maas, and Weber build on the insight of Hairer that rough path theory allows to make sense of SPDEs which are ill-defined in standard function spaces due to spatial irregularities. Hairer and Weber extend the type SPDE that we presented above to the case of multiplicative noise. Hairer, Maas, and Weber study approximations to this equation, where they discretize the spatial derivative as *u\1/\*|)>-u|)>>. They show that in the limit \0>, the approximation may introduce a Stratonovich type correction term to the equation. Finally, Hairer uses this approach to define and solve for the first time the Kardar--Parisi--Zhang (KPZ) equation, an SPDE of one spatial index variable that describes the random growth of an interface. The KPZ equation was introduced by Kardar, Parisi, and Zhang, and prior to Hairer's work it could only be solved by applying a spatial transform (the Cole-Hopf transform) which had the effect of linearizing the equation. In all the papers cited above, the intrinsic one-dimensional nature of rough path theory severely limits possible improvements or applications to other contexts. To the best of our knowledge, the first attempt to remove these limitations is the still unpublished work by Chouk and Gubinelli, extending rough path theory to handle (fractional) Brownian sheets (Gaussian two-parameter stochastic processes akin to (fractional) Brownian motion). In the recent paper, Hairer has introduced a theory of with the aim of giving a more general and versatile notion of regularity. Hairer's theory is also inspired by the theory of controlled rough paths, and it can also be considered a generalization of it to functions of a multidimensional index variable. The crucial insight is that the regularity of the solution to an equation driven by -- say -- Gaussian space-time white noise should not be described in the classical way. Usually we say that a function is smooth if it can be approximated around every point by a polynomial of a given degree (the Taylor polynomial). Since the solution to an SPDE does not look like a polynomial at all, this is not the correct way of describing its regularity. We rather expect that the solution locally looks like the driving noise (more precisely like the noise convoluted with the Green kernel of the linear part of the equation; so in the case of s the time integral of the white noise, i.e. the Brownian motion). Therefore, in Hairer's theory a function is called smooth if it can locally be well approximated by this convolution (and higher order terms depending on the noise). Hairer's notion of smoothness induces a natural topology in which the solutions to semilinear SPDEs depend continuously on the driving signal. This approach is very general, and allows to handle more complicated problems than the ones we are currently able to treat in the paracontrolled approach. If there is a merit in our approach, then its relative simplicity, the fact that it seems to be very adaptable so that it can be easily modified to treat problems with a different structure, and that we make the connection between harmonic analysis and rough paths. Section develops the calculus of paracontrolled distributions. In Section we solve ordinary differential equations driven by suitable Gaussian processes such as the fractional Brownian motion with Hurst index 1/3>. In Section we solve a fractional Burgers type equation driven by white noise, and in Section we study a non-linear version of the parabolic Anderson model. In Appendix we recall the main concepts of Littlewood-Paley theory and of Bony's paraproduct, and Appendix contains a commutator estimate between paraproduct and time integral. We stress the fact that this paper is mostly self-contained, and in particular we will not need any results from rough path theory and just basic elements of the theory of Besov spaces. This research has been partly supported by the ANR Project ECRU (ANR-09-BLAN-0114-01), by a Junior fellowship of the Institut Universitaire de France (IUF), and by a Postdoctoral grant of the Fondation Sciences Mathématiques de Paris. The main part of the research was carried out while N.P. was employed by Humboldt-Universität zu Berlin. During an Oberwolfach workshop in the summer of 2012, M. Hairer discussed with one of us (M.G.) his approach to extend rough path theory and M.G. would like to thank M. Hairer for suggesting the application to the two-dimensional non-linear parabolic Anderson model discussed in this paper. Throughout the paper, we use the notation b> if there exists a constant 0>, independent of the variables under consideration, such that c\b>, and we write b> if b> and a>. If we want to emphasize the dependence of on the variable , then we write \b>. For index variables and of Littlewood-Paley decompositions (see below) we write j> if \2>, so in other words if there exists >, independent of , such that j+N>, and we write j> if j> and i>. An is a set of the form =:a\\b|}>> for some a\b>. A is a set of the form =:\b|}>>. =/|)>> denotes the torus. The Hölder-Besov space >,\>,|)>> for \> will be denoted by >>, equipped with the norm |\|\>>=|\|\>>,\>>>. The local space >>l*o*c> consists of all which satisfy *u\>> for every infinitely differentiable > of compact support. Given two Banach spaces we denote by > the Banach space of linear maps from to , endowed with the operator norm |\|\>>>. More generally, given > and Banach spaces ,\,X>, we write \\\X,Y|)>> for the space of -linear maps from \\\X> to , and |\|\>\\\X,Y|)>>> for the operator norm. We denote by > the Banach space of continuous maps from to , endowed with the supremum norm |\|\>>>. We write *Y=C,Y|)>> for the space of continuous maps from > to , equipped with the supremum norm |\|\>*Y>>. If \>, then we also define >*Y> as the space of >-Hölder continuous functions from > to , endowed with the seminorm <\equation*> |f|\<\|\|\>>>*Y>=sups\t\T> |f-f|\<\|\|\>>|>>. If is a map from > to the linear space , then we write =f-f>, so that |f|\<\|\|\>>>*Y>=sups\t\T>|f|\<\|\|\>>/>>. For L|)>> we write |f|\<\|\|\>>|)>>=>|\|>*\*x>. For a multi-index =,\,\|)>\> we write |\|>=\+\+\> and >=\|\|>>/\>>*\*\>>>. *F> or > denote the total derivative of . For > we denote by *F> the -th order derivative of . For \0>, >=C>,|)>> is the space of |\|\>> times continuously differentiable functions, bounded with bounded partial derivatives, and with -|\|\>|)>>--Hölder continuous partial derivatives of order |\|\>>, equipped with its usual norm |\|\>>>>. We also write > for the partial derivative in direction , and if \\>, then we write *F> for its spatial derivative in the point \\>. The space of real valued infinitely differentiable functions of compact support is denoted by |)>> or >. The space of Schwartz functions is denoted by |)>> or >. Its dual, the space of tempered distributions, is |)>> or >. If is a vector of tempered distributions on >, then we write ,|)>>. The Fourier transform is defined with the normalization <\equation*> u==>e|z,x|\>>*u*\*x, so that the inverse Fourier transform is given by u=|)>u(-z)>. If > is a smooth function, such that > and all its partial derivatives are at most of polynomial growth at infinity, then we define the Fourier multiplier |)>> by |)>*u=u|)>> for any >. More generally, we define |)>*u> by this formula whenever the right hand side makes sense. The scaling operator > on > is defined for \0> by >*u=u*\|)>>. Throughout the paper, ,\|)>> will denote a dyadic partition of unity, and |)>-1>> will denote the Littlewood-Paley blocks associated to this partition of unity, i.e. =\|)>> and =\**\|)>> for 0>. We will often write >, by which we mean > if , and we mean *\|)>> if 0>. We also use the notation =j>\>. Paraproducts are bilinear operations introduced by Bony in order to linearize a class of non-linear PDE problems. In this section we will introduce paraproducts to the extent of our needs. We will be using the Littlewood-Paley theory of Besov spaces. The reader can peruse Appendix, where we summarize the basic elements of Besov space theory and Littlewood-Paley decompositions which will be needed in the remainder of the paper. One of the simplest situations where paraproducts appear naturally is in the analysis of the product of two Besov distributions. In general, the product of two distributions >> and >> is not well defined unless +\\0>. In terms of Littlewood--Paley blocks, the product can be (at least formally) decomposed as <\equation*> f*g=-1>-1>\*f*\*g=f\g+f\g+f\g. Here g> is the part of the double sum with j-1>, and g> is the part with j+1>, and g> is the ``diagonal'' part, where \1>. More precisely, we define <\equation*> f\g=g\f=-1>\*f*\*g*f\g=\1>\*f*\*g. We also introduce the notation <\equation*> f\g=f\g+f\g. This decomposition behaves nicely with respect to Littlewood--Paley theory. Of course, it depends on the dyadic partition of unity used to define the blocks >, and also on the particular choice of the pairs > in the diagonal part. Our choice of taking all > with \1> into the diagonal part corresponds to property . in the definition of dyadic partition of unity in Appendix, where we assumed that **\|)>|)>\supp**\|)>|)>=\> for \1>. This means that every term in the series <\equation*> f\g=-1>\*f*\*g=-1>S*f*\*g has a Fourier transform which is supported in a suitable annulus, and of course the same holds true for g>. On the other side, every term in the diagonal part g> has a Fourier transform that is supported in a ball. We call g> and g> , and g> the term. Bony's crucial observation is that g> (and thus g>) is always a well-defined distribution. In particular, if \0> and \>, then \f\g> is a bounded bilinear operator from >\>> to >>. Heuristically, g> behaves at large frequencies like (and thus retains the same regularity), and provides only a modulation of at larger scales. The only difficulty in defining for arbitrary distributions lies in handling the diagonal term g>. The basic result about these bilinear operations is given by the following estimates. <\lemma> >For any \> we have <\equation> |f\g|\<\|\|\>>>\>|f|\<\|\|\>>>>|g|\<\|\|\>>>, and for \0> furthermore <\equation> |f\g|\<\|\|\>>+\>\,\>|f|\<\|\|\>>>|g|\<\|\|\>>>. For +\\0> we have <\equation> |f\g|\<\|\|\>>+\>\,\>|f|\<\|\|\>>>|g|\<\|\|\>>>. <\proof> Observe that there exists an annulus > such that *f*\*g> has Fourier transform supported in >, and that for L>> we have <\equation*> |S*f*\*g|\<\|\|\>>>>\|S*f|\<\|\|\>>>>*|\*g|\<\|\|\>>>>\|f|\<\|\|\>>>>*2>|g|\<\|\|\>>>. On the other side, if \0> and >>, then <\equation*> |S*f*\*g|\<\|\|\>>>>\j-2>|\*f|\<\|\|\>>>>*|\*g|\<\|\|\>>>>\|f|\<\|\|\>>>|g|\<\|\|\>>>*j-2>2-j*\>\|f|\<\|\|\>>>|g|\<\|\|\>>>*2+\|)>>. By Lemma, we thus obtain and. To estimate g>, observe that the term =\*f*\1>\*g> has Fourier transform supported in a ball >, and that <\equation*> |u|\<\|\|\>>>>\|\*f|\<\|\|\>>>>*\1>|\*g|\<\|\|\>>>>\|f|\<\|\|\>>>|g|\<\|\|\>>>*2+\|)>*j>. So if +\\0>, then we can apply the second part of Lemma to obtain that g=-1>u> is an element of +\>> and that equation holds. A natural corollary is that the product of two elements >> and >> is well defined as soon as +\\0>, and that it belongs to >>, where =min ,\,\+\|}>>. Consider the <\equation> \*u=F*\,u=u, where \>, \> is a continuous vector valued function, > is the time derivative, :\> is a vector valued distribution with values in -1>> for some \>, and \,|)>> is a family of vector fields on >. A natural approach is to understand this equation as limit of the classical ODEs <\equation> \*u>=F>|)>*\>,u>=u, for a family of smooth approximations >|)>> of > such that >\\> in -1>> as \0>. In order to pass to the limit, we are looking for a priori estimates on >> which require only a control on the -1>> norm of >. To avoid cumbersome notation, we will work at the level of equation for smooth >, where it should be understood that our aim is to obtain a priori estimates for the solution, in order to safely pass to the limit and extend the solution map to a larger class of data. The natural regularity of is >>, since should gain one derivative with respect to *\>, which will not behave better than >, and will therefore be in -1>>. We use the paraproduct decomposition to write the right hand side of as a sum of the three terms <\equation> \\|\>-1>+\\|\>-1>+\\|\>-1> (where the quantity indicated by the underbrace corresponds to the expected regularity of each term). Note however that unless -1\0>, the resonant term \\> cannot be controlled using only the >>--norm of and the -1>>--norm of >. If is at least in >, we can use a paralinearization result (see Lemma below) to rewrite this term as <\equation> F\\=F*\|)>+\|)>, where the remainder |)>> is well defined under the condition -1\0>, provided that >> and \-1>>. In this case it belongs to -1>>. The difficulty is now localized in the linearized resonant product \>. In order to control this term, we would like to exploit the fact that the function is not a generic element of >> but that it has a specific structure, since its derivative *u> has to match the paraproduct decomposition given in. Thus, we postulate that the solution is given by the following <\equation*> u=u>\\+u>, where >,\\>> and the remainder >> is in >>. This decomposition allows for a finer analysis of the resonant term \>. Indeed, we have <\equation> u\\=>\\|)>\\+u>\\=u>*\\|)>+C>,\,\|)>+u>\\, where the commutator is defined by >,\,\|)>=>\\|)>\\-u>*\\|)>>. Observe now that the term >\\> does not pose any further problem, as it is bounded in -1>>. Moreover, we will show that the commutator is a bounded multilinear function of its arguments as long as the sum of their regularities is strictly positive, see Lemma below. By assumption, we have -1\0>, and therefore >,\,\|)>\-1>>. The only problematic term which remains to be handled is thus \\>. Here we need to make the assumption that \\\-1>> in order for the product >*\\|)>> to be well defined. That assumption is not guaranteed by the analytical estimates at hand, and it has to be added as a further requirement. Granting this, we have obtained that the right hand side of equation is well defined and a continuous function of >,u>,\,\,\\\|)>>. The paracontrolled ansatz and the Leibniz rule for the paraproduct now imply that can be rewritten as <\equation*> \*u=\*>\\+u>|)>=\*u>\\+u>\\*\+\*u>=F\\+F\\+F\\. If we choose > such that *\=\> and we set >=F>, then we can use and to obtain the following equation for the remainder >>: *u>>|*F*\\|)>+F\\-*F\\|)>>>||+F*C,\,\|)>+F*>\\|)>+\|)>.>>>>> Together with the equation \\+u>>, this completely describes the solution and allows us to obtain an a priori estimate on in terms of ,|\|\<\|\|\>>-1>,|\\\|\<\|\|\>>-1>|)>>. With this estimate at hand, it is now easy to show that if C>, then depends continuously on the data ,\,\\\|)>>, so that we can pass to the limit in and make sense of the solution to also for irregular \-1>> as long as \1/3>. In this section we prove some lemmas which will allow us to perform algebraic computations with the paraproduct and the resonant term, thus justifying the analysis of the previous section. <\lemma> >Let >> for \>, and let L>>. For any -1> we have <\equation*> |,f|]>*g|\<\|\|\>>>>=|\*-f*\*g|\<\|\|\>>>>\2*j>|f|\<\|\|\>>>|g|\<\|\|\>>>>. This commutator lemma is easily proven by writing =\|)>> as a convolution operator, and using the embedding of >> in the space of Hölder continuous functions. <\lemma> Assume that \> and \>, and let >> and >>. Then <\equation*> \*g|)>=f*\*g+R, for all -1>, with a remainder > which satisfies |R|\<\|\|\>>>>\2+\|)>>|f|\<\|\|\>>>|g|\<\|\|\>>>>. <\proof> Note that g=f\\*g>, and that the Fourier transform of \*g> is supported in an annulus of the form >. Hence, we have *\*g|)>\0> only if i>, which leads to *g|)>>|j>\*\*g|)>=j>\**g|)>-j>\*\*g|)>>>||j>f*\*\*g-j>,f|]>*\*g-j>\*\*g|)>,>>>>> where we recall that ,f|]>*\*g=\**g|)>-f*\*\*g> denotes the commutator. The sum over with j> can be chosen to encompass enough terms so that *g=j>\*\*g>, and therefore we conclude that <\equation*> |\*g|)>-f*\*g|\<\|\|\>>>>\j>|,f|]>*\*g|\<\|\|\>>>>-j>|\*\*g|)>|\<\|\|\>>>>. We apply Lemma to each term of the first sum, and the paraproduct estimates to each term of the second sum, to obtain <\equation*> |\*g|)>-f*\*g|\<\|\|\>>>>\2+\|)>>|f|\<\|\|\>>>|g|\<\|\|\>>>. Using this result, it is easy to prove our basic commutator lemma. <\lemma> Assume that \> and ,\\> are such that +\+\\0> and +\\0>. Then for smooth , the trilinear operator <\equation*> C=g|)>\h|)>-f*h|)> allows for the bound <\equation*> |C|\<\|\|\>>+\+\>\|f|\<\|\|\>>>|g|\<\|\|\>>>|h|\<\|\|\>>>. Thus, can be uniquely extended to a bounded trilinear operator in >\>\>,+\+\>|)>>. <\proof> Let be smooth functions and write <\equation*> C=g|)>\h|)>-f*h|)>=-1>\1>**f\g|)>*\*h-\*f*\*g*\*h|]>. Observe that for fixed , the term *f\g> has a Fourier transform supported outside of a ball >. Thus, we have **f\g|)>=k>*\**f\g|)>>. We can therefore apply Lemma to obtain *C>|-1>\1>k>**f*\*g+R**f,g|)>|)>*\*h-\*f*\*g*\*h|]>>>||-1>\1>k>*R**f,g|)>*\*h-k-N>*\*f*\*g*\*h|]>>>>>> for some fixed >. We treat the two sums separately. First observe that for fixed , the term -1>\1>k-N>*\*f*\*g*\*h> has a Fourier transform which is supported in a ball >. Moreover, -1>\1>k-N>*\*f*\*g*\*h>>>|2>|f|\<\|\|\>>>*2+\|)>>|g|\<\|\|\>>>|h|\<\|\|\>>>>>||2+\+\|)>>|f|\<\|\|\>>>|g|\<\|\|\>>>|h|\<\|\|\>>>,>>>>> where in the second step we used that +\\0>. Since +\+\\0>, the estimate for the second series in follows from Lemma. For the first series, recall that **f,g|)>=\**f\g|)>-\*f*\*g>. So for fixed , the Fourier transform of -1>\1>k>*R**f,g|)>*\*h> is supported in ball >. Furthermore, Lemma yields |-1>\1>k>*R**f,g|)>*\*h>>=\1>Ri>\*f,g|)>*\*h>>>>||\\1>2+\|)>>i>\*f>|g|\<\|\|\>>>*2>|h|\<\|\|\>>>\2+\+\|)>>|f|\<\|\|\>>>|g|\<\|\|\>>>|h|\<\|\|\>>>,>>>>> so that the claimed bound for |C|\<\|\|\>>+\+\>> follows from another application of Lemma. Now we can uniquely extend to a bounded trilinear operator on the closure of the smooth functions in >\>\>>. Unfortunately, this is a strict subset of >\>\>>. But we obtain similar bounds for acting on >\>\>> for \> and ,\\>, such that \\>, \\>, \\>, and +\+\\0>. Since >\>\>> is contained in the closure of the smooth functions in >\>\>>, the extension of to >\>\>> is unique. <\remark> The restriction +\\0> is not problematic. If +\\0>, then g|)>\h> can be treated with the usual paraproduct estimates, without the need of introducing the commutator. If +\=0>, then we can apply the commutator estimate with \\> sufficiently close to > such that +\+\\0>. Our next result is a simple paralinearization lemma for non-linear operators. <\lemma> , Theorem 2.92>Let \>, \|]>>, and let C/\>>. There exists a locally bounded map :>\+\>> such that <\equation> F=F\f+R for all >>. More precisely, we have <\equation*> |R|\<\|\|\>>+\>\|F|\<\|\|\>>/\>>*|f|\<\|\|\>>>/\>|)>. If C/\>>, then > is locally Lipschitz continuous: <\equation*> |R-R|\<\|\|\>>+\>\|F|\<\|\|\>>/\>>*|f|\<\|\|\>>>+|g|\<\|\|\>>>|)>/\>*|f-g|\<\|\|\>>>. <\proof> The difference -F\f> is given by <\equation*> R=F-F\f=-1>*F-S*F*\*f|]>=-1>u, and every > is spectrally supported in a ball >. For 1>, we simply estimate |u|\<\|\|\>>>>\|F|\<\|\|\>>>*|f|\<\|\|\>>>|)>>. For 1> we use the fact that is a bounded function to write the Littlewood-Paley projections as convolutions and obtain >|K**Ki-1>**|)>-F|)>*f|]>*\*y*\*z>>||K**Ki-1>**|)>-F|)>-F|)>*-f|)>|]>*\*y*\*z,>>>>> where =\>, i-1>=i-1>K>, and where we used that K*\*y=\=0> for 0> and Ki-1>*\*z=1> for 1>. Now we can apply a first order Taylor expansion to and use the /\>--Hölder continuity of > in combination with the >--Hölder continuity of , to deduce |\|>>||F|\<\|\|\>>/\>>|f|\<\|\|\>>>/\>***Ki-1>*|\|>\+\>*\*y*\*z>>|||F|\<\|\|\>>/\>>|f|\<\|\|\>>>/\>*2+\|)>>.>>>>> Therefore, the estimate for > follows from Lemma. The estimate for -R> is shown in the same way. Let be a distribution belonging to >> for some \0>. Then the map f\g> behaves, modulo smoother correction terms, like a derivative operator: <\lemma> Let \>, \|]>>, \> be such that +\+\\0> but +\\0>. Let C/\>>. Then there exists a locally bounded map :>\>\+\+\>> such that <\equation> F\g=F*g|)>+\ for all >> and all smooth . More precisely, we have <\equation*> |\|\<\|\|\>>+\+\>\|F|\<\|\|\>>/\>>*|f|\<\|\|\>>>/\>|)>|g|\<\|\|\>>>. If C/\>>, then > is locally Lipschitz continuous: <\equation*> |\-\|\<\|\|\>>+\+\>\|F|\<\|\|\>>/\>>*|f|\<\|\|\>>>+|u|\<\|\|\>>>|)>/\>+|v|\<\|\|\>>>|)>*|f-u|\<\|\|\>>>+|g-v|\<\|\|\>>>|)>. <\proof> Just use the paralinearization and commutator lemmas above to deduce that >|\g-F*g|)>=R\g+\f|)>\g-F*g|)>>>||\g+C,f,g|)>,>>>>> so that the claimed bounds easily follow from Lemma and Lemma. Besides this sort of chain rule, we also have a Leibniz rule for f\g>: <\lemma> Let \> and \0> be such that +\\0> but +\\0> Then there exists a bounded trilinear operator >:>\>\>\+\>>, such that <\equation*> \g=f*g|)>+u*g|)>+\> for all >|)>> and all smooth . <\proof> It suffices to note that u+f\u+f\u>, which leads to <\equation*> \>=\g-f*g|)>+u*g|)>=C+C+u|)>\g. Let us now resume the analysis of Section. We want to study the <\equation> \*u=F*\,u=u, where \>, \> is a continuous vector valued function, :\> is a vector valued distribution with values in -1>> for some \>, and \,|)>> is a family of vector fields on >. In order to obtain concrete estimates, we have to localize the equation. Therefore, we introduce a smooth cut-off function > with support on , which is equal to 1 on and modify the equation as <\equation*> \*u=\*F*\,u=u. In the regular setting, if is solution to this equation, it is also a solution of the original equation on , and thus it is sufficient to study the last equation for local bounds. To avoid problems with the fact that the paraproduct is a (mildly) non-local operation, we modify the paracontrolled ansatz as follows: <\equation> u=\*\\|)>+u>. If C>, an easy computation gives *u>>|*F*\-*\|)>*\\|)>-\**F\\|)>-\*\\|)>>>||*\\|)>+F*|)>\\|)>+\>>*,\|)>-*F\\|)>|]>>>||-*\|)>*\\|)>,>>>>> where we set >=F*+x|)>> and used that >|)>*|)>=F> for all >. We subtract the contribution of the initial condition, because this will eventually allow us to solve the equation on a small interval whose length does not depend on >. If we plug in the modified paracontrolled ansatz for , then *|)>\\|)>> becomes <\equation*> F*|)>\\|)>=F**\\|)>|)>\\|)>+F*>-u|)>\\|)>. For the first term on the right hand side we can further use that <\equation*> *\\|)>|)>\\=\*\\|)>\\|)>+\\|)>*\\|)>+\>*,F\\,\|)>, where we recall that >> was defined in Lemma. So finally, an application of our commutator lemma yields \*u>>|*\\|)>+\>>*,\|)>+F*>-u|)>\\|)>+\\|)>*\\|)>|\>>>|>||+\>*,F\\,\|)>+\*C,\,\|)>+F*\*F*\\|)>-*F\\|)>|]>>>|>|-*\|)>*\\|)>>>||*\>-*\|)>*\\|)>,>>>>> where >> is defined to be the term in the square large brackets. Let us summarize our observations so far. <\lemma> Let > be a smooth path, let > be such that *\=\>, and let C>. Then solves the ODE <\equation*> \*u=\*F*\,u=u, if and only if *\\|)>+u>>, where >> solves <\equation*> \*u>=\*\>-*\|)>*\\|)>,u>=u-\*\\|)>, and where >> is defined in. Moreover, for \> we have the estimate <\equation*> |\>|\<\|\|\>>-1>\C*C>*|u-u|\<\|\|\>>>+|u-u|\<\|\|\>>>+|u>-u|\<\|\|\>>>|)>, where <\equation*> C>=|\|\<\|\|\>>-1>+|\|\<\|\|\>>>+|\\\|\<\|\|\>>-1>+|\|\<\|\|\>>>|\|\<\|\|\>>-1>*andC=|F|\<\|\|\>>>+|F|\<\|\|\>>>. The estimate for >> follows from a somewhat lengthy but elementary calculation based on the decomposition, where we estimate the >> norm rather than the -1>> norm for each term where this is possible. Plugging in the correct initial condition for >> leads to >>|-\\|)>+\*u>*\*s>>||-\\|)>+*\>|)>*\*s-*\|)>*\\|)>*\*s.>>>>> Now > is compactly supported, and therefore Lemma gives estimates for the integrals appearing on the right hand side in terms of distributional norms of the integrands, and we obtain the bound <\equation*> |u>-u|\<\|\|\>>>\|F\\|\<\|\|\>>-1>+|\>|\<\|\|\>>-1>\C*C>*|u-u|\<\|\|\>>>+|u-u|\<\|\|\>>>+|u>-u|\<\|\|\>>>|)>. Using that *\\|)>+u>>, we moreover have <\equation*> |u-u|\<\|\|\>>>\|F|\<\|\|\>>>>|\|\<\|\|\>>>+|u>-u|\<\|\|\>>>. From these two estimates we deduce that if > is small enough (depending only on >> but not on |\|>>), then |u>|\<\|\|\>>>\|\|>+1>. This is the required uniform estimate on the problem. Similarly we can show that if C> and if |F|\<\|\|\>>>> is small enough, then the map <\equation*> ,\,\,\\\|)>\>|)> is locally Lipschitz continuous from -1>\>\-1>\> to >\-1>>. To summarize: <\lemma> Let 0> and let |F|\<\|\|\>>>> be small enough (depending on ). Let >, >, and > be smooth functions with =\*\> and such that > has compact support. If \1/3> and <\equation> max |\|\<\|\|\>>-1>,|\|\<\|\|\>>>,|\\\|\<\|\|\>>-1>,|\|\<\|\|\>>>|}>\a, then for every \> there exists a unique global solution to <\equation*> \*u=\*F*\,u=u. For fixed > and , the solution depends in a Lipschitz continuous way on ,\,\,\\\|)>> satisfying. In order to ensure that |F|\<\|\|\>>>> is small enough we can use a dilation argument. Recall that the scaling operator >> is defined for \0> by >*u=u*\|)>>. If we let >=\>*u> and >=\>*\>*\> for \0>, then >> solves <\equation*> \*u>=\>*F>|)>*\>,u>=u. The rescaling of >> is chosen so that its >> norm is uniformly bounded by that of > as \0>. Indeed, Lemma yields <\equation*> |\>|\<\|\|\>>-1>=\>*|\>*\|\<\|\|\>>-1>\>|)>|\|\<\|\|\>>-1>\|\|\<\|\|\>>-1> for \1>. If moreover we let >=\>*\>*\>, then |\>\\>|\<\|\|\>>-1>\|\\\|\<\|\|\>>-1>+|\|\<\|\|\>>>|\|\<\|\|\>>-1>> by Lemma in Appendix below. Thus, we deduce from Lemma that for every > of compact support there exists \0>, such that for all \> we have a unique global solution >> to <\equation*> \*u>=\*\>*F>|)>*\>,u>=u. The rescaled problem is equivalent to the original one upon the change \>*F>, \\>> and \\\\>\\>>. So if we set >*u>>, then is the unique global solution to <\equation*> \*u=\>*F*\,u=u, where we set >=\*|)>>. In particular, if \1> on , then is the unique solution to the original in the interval ,\]>. Since > can be chosen independently of >, we can now iterate on intervals of length >, and obtain a global solution >>. This analysis can be summarized in the following statement. <\theorem> Let \1/3>. Assume that >|)>\0>> is a family of smooth functions with values in >, >|)>> is a family of initial conditions in >, and is a family of > vector fields on >. Suppose that there exist \>, \-1>> and \-1>> such that >,\>,\>,>\\>|)>|)>> converges to ,\,\,\|)>> in -1>\>\-1>>, where >> and > are solutions to *\>=\>> and *\=\>, respectively. Let for \0> the function >> be the unique global solution to the Cauchy problem <\equation*> \*u>=F>|)>*\>,u>=u>. Then there exists >>l*o*c> such that >\u> in >> as \0>. The limit depends only on ,\,\,\|)>>, and not on the approximating family >,\>,\>,>\\>|)>|)>>. <\proof> The only point which remains to be shown is the convergence of >|)>> to in >>. A priori, we only know that for sufficiently small \0>, the solutions >> to *>=\>*F>|)>*\>> with >=u> converge, as \0>, in >> to a unique limit >. But since >\1> on ,\]>, we have >\|,\]>=u>\|,\]>>. So if we define ,\]>=\|,\]>>, then ,\]>> does not depend on >>. Moreover, for every \> with support contained in ,\]>, we also have that |\*>-u|)>|\<\|\|\>>>> converges to zero as \0>. Now we can iterate this construction of on intervals of length >. We end up with a distribution >, which only depends on ,F,\,\,\|)>>, but not on >> or on the approximating sequence >,\>,\>,\>\\>|)>\0>>. If \>, then it can be written as a finite sum of smooth functions with support contained in intervals of length >, and therefore *u=lim\> \*u>>, where convergence takes places in >>. <\remark> By Lemma, it suffices if C/\>> for some \0> with +\\1> to obtain existence and uniqueness solutions. If we only suppose C/\>> and not that and its derivatives are bounded, we still obtain local existence and uniqueness of solutions. In that case we may consider a function C/\>> that coincides with on \a|}>> for some |\|>>. The Cauchy problem <\equation*> \*v=G*\,v=u, then has a unique global solution in the sense of Theorem. If we stop upon leaving the set \a|}>>, we obtain a local solution to the with vector field . So far we showed that under the assumptions of Theorem there exists a unique limit of the solutions to the regularized equations, which does not depend on the particular approximating sequence. In that sense, one may formally call the unique solution to <\equation*> \*u=F*\,u=u. But is actually a weak solution to the equation if we interpret the product *\> appropriately. Below we will introduce a map which extends the pointwise product *\> from smooth > to \-1>> by a continuity argument. But first we present an auxiliary result which shows that the considered topologies and operators do not depend on the particular dyadic partition of unity that we use to describe them. <\lemma> Let ,\\>. Let ,\|)>> and |~>,|~>|)>> be two dyadic partitions of unity and let ,\,\)> and |~>,|~>,|~>|)>> denote paraproducts and resonant term defined in terms of ,\|)>> and |~>,|~>|)>>, respectively. Then <\equation*> \v-u*|~>*v,u\v-u*|~>*v,u\v-u*|~>*v|)> is a bounded bilinear operator from >\>> to +\>|)>>. <\proof> The statement for \v-u*|~>*v|)>> (and thus for \v-u*|~>*v|)>>) is shown in Bony, Theorem 2.1. But for smooth functions and we have v=u*v-u\v-u\v>, and similarly for |~>*v>. Thus, the bound on v-u*|~>*v> follows from the bounds on v-u*|~>*v> and on v-u*|~>*v> in combination with a continuity argument. Our commutator lemma states that if the product h> is given, then we can unambiguously make sense of the product g|)>\h> for suitable . This leads us to the following definition. <\definition> Let \>, \0>, and let >>. A pair of distributions |)>\>\>> is called by if <\equation*> u>=u-u\v\+\>. In that case we abuse notation and write >=>>, and we define the norm <\equation*> |u|\<\|\|\>>>>=|u|\<\|\|\>>>+|u>|\<\|\|\>>+\>. According to Lemma, the space >> does not depend on the specific partition of unity used to define it. To construct the product *\>, we could now show that smooth preserve the paracontrolled structure of . This can be achieved by combining Lemma with another commutator lemma (Theorem2.3 in). But we do not need the full strength of that result, let us just show that if is paracontrolled by > and is smooth enough, then *\> is well defined. <\theorem> Let \>, \|]>>, \0> be such that +\+\\0>. Let C/\>> and let >>, >>, \+\>> be such that there exist sequences of smooth functions |)>>, |)>>, converging to and respectively, such that \w|)>> converges to >. Then >\u\F*w>|\w+F\w+\+F*>\w|)>>>|>|+F*C,v,w|)>+F*u*\\>>>>>> defines a locally Lipschitz continuous function. If is a smooth function and =v\w>, then *w> is simply the pointwise product. The product *w> does not depend on the specific dyadic partition used to construct it: if |~>,|~>,|~>|)>> denote paraproducts and resonant term defined in terms of another partition unity, if <\equation*> |~>=\+v\w+v\w-v|~>*w-v|~>*w, and >=u*|~>*v>, then *w> is equal to the right hand side of if we replace every operator by the corresponding operator defined in terms of |~>,|~>,|~>|)>>, and we replace >> by >> and > by |~>>. <\proof> The local Lipschitz continuity of the product follows from its definition in combination with Lemma, Lemma, and the paraproduct estimates Lemma. If is smooth and =v\w>, then <\equation*> F*C,v,w|)>+F*u*\=F*\v|)>\w|)>, and therefore <\equation*> \+F*>\w|)>+F*C,v,w|)>+F*u*\=\+F*w|)>=F\w, which shows that we recover \w+F\w+F\w>, i.e. the pointwise product. It remains to show that *w> does not depend on the specific dyadic partition of unity. By continuity of the operators involved, we have *w>|\>\w+F\w+\|)>+F*>\w|)>|\>>>|||+F*C,v,w|)>+F*u*\w|)>|]>>>||\>*w+F*\-v|)>|)>\w|)>|]>.>>>>> Assume now that we defined \w> in terms of another partition of unity, as described above. Then Lemma implies the convergence of *|~>*w|)>> to |~>> in +\>>, and therefore <\equation*> F\w=lim\>*w+F**|~>*-v|)>|)>|~>*w|)>|]>. Another application of Lemma then yields *w=F\w>. <\remark> If in the setting of Theorem we let =v+f> for some +\>>, then we have >=>|)>>, and it is easy to see that if we set |~>=\+f\w>, >=u-u\>, and define *w|~>> like *w>, with >, >>, |~>> replacing >,v,\>, then *w|~>=F*w>. With this product operator at hand, it is relatively straightforward to show that if > has compact support (which in general is necessary to have >> and not just in >>>), then the solution that we constructed in Theorem is the unique element of >> which solves *u=F*\>, =u>, in the weak sense. Remark explains why we did not fix the initial condition > in Theorem: it is of no importance whatsoever. We briefly describe an alternative approach to s which avoids the paracontrolled ansatz. The idea is to control \> directly by exploiting that solves the differential equation *u=F*\>. Indeed, let as above > be a solution to *\=\> and observe that the Leibniz rule yields <\equation*> u\\=u\\*\=\*\|)>-\*u\\=\*\|)>-*\|)>\\. Now the second term on the right hand side can be rewritten as *\|)>\\>|\\|)>\\+\\|)>\\+\\|)>\\>>||*\\|)>+C,\,\|)>+*\|)>|)>\\+\|)>\\+\\|)>\\.>>>>> Combining these two equations, we see that \=\-*\|)>|)>\\,>>|=\*\|)>-F*\\|)>-C,\,\|)>-\|)>\\-\\|)>\\.>>>>> This is an implict equation for \> which can be solved by fixed point methods. For example, it is easy to obtain the estimate <\equation*> |u\\|\<\|\|\>>-1>\|\|\<\|\|\>>-1>+C*|u\\|\<\|\|\>>-1>|\|\<\|\|\>>>, and if > is small enough this leads to |u\\|\<\|\|\>>-1>\|\|\<\|\|\>>-1>>. Moreover, we have |\|\<\|\|\>>-1>\C>*|u|\<\|\|\>>>+C*|u|\<\|\|\>>>|)>|]>>. These estimates can be reinjected into the equation <\equation*> \*u=F*\=F\\+F*\|)>+F\\+R,u,\|)> to obtain a local estimate for . In this section, we would like to discuss how the combination of analytic and algebraic requirements generate very interesting phenomena in the context of irregular PDEs. We will discuss only the simple case of s but similar considerations apply also to the other models. These remarks are intentionally sketchy and have only a heuristic purpose, we plan to come back to them more systematically in a further publication. As we have seen, the solution to the for smooth > can be understood as a regular function of ,\,\\\|)>\\-1>,|)>\-1>,\|)>>, where > is a solution to *\=\>. Let us denote this function by *,\,\\\|)>>. The structure of the solution which we derived above shows that for \-1>>, the <\equation> \*v=F*\+F*F*\,v=u, has a solution which can be obtained from the same map by replacing \\> with \\+\>, i.e. <\equation*> v=\*,\,\\\+\|)>. Moreover, the solution depends continuously on the data \\+\\-1>>. First note that this regularity hypothesis concerns only the combination \\+\>, and second note that many equations share a similar structure and one can pass from solutions of one equation to solutions of other equations via a transformation of the (extended) data of the problem. For many reasons, all these differential equations should be considered to be the same object, especially when dealing with data of low regularity. To understand this point of view, consider two different families >|)>\0>> and |~>>|)>\0>> of smooth functions, such that both converge to > in -1>>. The corresponding solutions >|)>> and >|)>> to equation are given by >=\*,\>,\>\\>|)>> and >=\*,|~>>,|~>>\|~>>|)>>. Let us take the limit as \0> and assume that >\\>\\> in -1>> and also that |~>>\|~>>\|~>> in -1>>, where \|~>>. This could happen in principle, and it is not difficult to find specific and relevant examples of this multiplicity of limits. Of course, we have <\equation*> \>\\>-|~>>\|~>>=\>*\>-|~>>*|~>>-\>\\>-\>\\>+|~>>\|~>>+|~>>\|~>>. Since >|)>> and |~>>|)>> both converge to >, where *\=\>, we deduce from the continuity of the paraproduct that all the terms on the right hand side cancel, except the first two, and we remain with <\equation*> \=\-\=lim\>>*\>-|~>>*|~>>|)>. Incidentally, this line of reasoning also shows that if the limit exists, it does not depend on the particular Littlewood-Paley decomposition we use to compute it. From the continuity of > it moreover follows that >\u> and >\> where ,\,\|)>> and =\*,\,\+\|)>>. That is, different approximations of the same equation could lead to different equations in the limit. In particular, if > is smooth enough (but >|)>> does not converge to > in a space of sufficiently high regularity), we can interpret as a classical solution to a differential equation, and in this case > will solve a modified equation. We can therefore identify -1>\-1>> with a transformation group |)>-1>\-1>>>, which acts on solutions to ODEs via <\equation*> T*\,\,\|)>=\*,\+f,\+\\f+\\\+\\f+g|)>. where > solves *\=f>. In particular, ,\,\|)>=T,\-\\\>*\,0,0|)>>. The neutral element is > and the group operation is *T,g>=T,g+g>>, so that the group is abelian. A simple distance is given by <\equation*> d,T,g>|)>=d,g-g>,T|)>=|f-f|\<\|\|\>>-1>+|g-g+-\|)>\|)>|\<\|\|\>>-1>, where > and > are the definite integrals with =\=0> of and > respectively. Note that if > and > denote respectively the projections onto symmetric and antisymmetric tensors, then for smooth ,\> we have <\equation*> J*\\|)>=J*\\*\|)>=*\*J*\\|)>, where the right hand side is now well defined for all \>> and defines a distribution in -1>>. So for any family >\\> in -1>> we have that <\equation*> J*>\\>|)>=*\*J*>\\>|)>\*\*J*\\|)> as \0>, and only the antisymmetric part *>\\>|)>> could feature multiple accumulation points. To highlight the geometric meaning of the symmetric and antisymmetric parts of \\> we can submit the to a nontrivial transformation given by the application of a smooth diffeomorphism :\>. If is a solution to the *u=F*\>, then > solves <\equation*> \*v=\*\*u=\*F*\, i.e. an with vector field *F|)>\\>. We have <\equation*> \*\**u|)>=\**u|)>*\**u|)>=\**u|)>**u|)>*+f|)>+F**u|)>*F**u|)>*g|)>. On the other hand, we obtain for *v=T*\> that <\equation*> \*T*v=*F|)>\\|)>**v|)>*+f|)>+\**F|)>\\|)>**v|)>**F|)>\\|)>**v|)>*g, and we have **F|)>\\|)>=*F+\*F|)>\\|)>|)>\\>, where we slightly abuse notation by writing > for the inverse function of > and |)>> for the inverse matrix of >. Thus, we obtain <\equation*> \*T*v=*F|)>\\|)>**v|)>*+f|)>+*F*F+\*F*F|)>\\|)>**v|)>*g, so that **u|)>=T|)>> only if *F*F|)>***v|)>|)>*g=0>. If this holds for all and >, then the symmetric part *g> of must be 0. In this sense, > acts geometrically only if *g=0>. We saw in the previous section that the solution to an of the form *u=F*\> depends on the driving signal in a continuous way, provided that we not only keep track of > but also of \\>. From the theory of rough paths it is well known that the same holds true if we keep track of > and its iterated integrals >\*\|)>*\*\|)>>. But in fact the convergence of >\\>|)>> is equivalent to the convergence of the iterated integrals \*\>*\*\>>: <\corollary> Let >,v>|)>\0>> be a family of smooth functions on >. Define for every \0> the ``area'' <\equation*> A>=>\*u>|)>*\*v>|)>,s\t\. Let ,\\> with +\\1> and let >,v\>,\\+\-1>>. Then >,v>,u>\\*v>|)>> converges to |)>> in >\>\+\-1>> if and only if >,v>|)>> converges to > in >\>>, and if moreover <\equation> lim\0>t\,\1> -A>|\|>|+\>>|)>=0, where we set =+\*v|)>+\*v|)>|)>*\*r-u*-v|)>> for >. <\proof> First suppose that >,v>,u>\\*v>|)>> converges to |)>> in >\>\+\-1>>, and let > with \1>. We have *A-A>>|+u\\*v-u>\\*v>-u>\\*v>|)>*\*r>>|>|+>-u|)>\\*v>|)>*\*r->-u|)>*>-v>|)>>>||+\*>-v|)>|)>*\*r-*>-v|)>->-v|)>|)>.>>>>> The first term on the right hand side can be estimated with the help of Lemma, which allows us to bound increments of the integral in terms of Besov norms of the integrand. We get |+u\\*v-u>\\*v>-u>\\*v>|)>*\*r|\|>>>||\|\-u>\\*v>|\<\|\|\>>+\-1>+|u-u>|\<\|\|\>>>*|\*v|\<\|\|\>>-1>+|u>|\<\|\|\>>>*|\*>-v|)>|\<\|\|\>>-1>|)>*+\>,>>>>> using also the paraproduct estimates. Since |\*>-v|)>|\<\|\|\>>-1>\|v>-v|\<\|\|\>>>>, the right hand side goes to zero if we divide it by +\>> and let \0>. The second term on the right hand side of can be estimated using Lemma, which roughly states that time integral and paraproduct commute with each other, at the price of introducing a smoother remainder term: <\equation*> >-u|)>\\*v>|)>*\*r->-u|)>*>-v>|)>|\|>\+\>*|u>-u|\<\|\|\>>>|v>|\<\|\|\>>>, The third term on the right hand side of is of the same type as the second term, and therefore the convergence in follows. Conversely, assume that >,v>|)>> converges to > in >\>>, and that the convergence in holds. It follows from the representation and the convergence of >,v>|)>> to > in >\>>, that also <\equation*> lim\0>t\,\1> -u>\\*v>|)>*\*r|+\>>|)>=0. Due to the restriction \1>, it is not entirely obvious that this implies the convergence of >\\*v>> to > in +\-1>>. However, here we can use an alternative characterization of Besov spaces in terms of local means. Let > and be infinitely differentiable functions on > with support in , such that *k\0>, and such that there exists \0> with *k\0> for all \\>. Then an equivalent norm on +\-1>|)>> is given by <\equation*> |w|\<\|\|\>>+\-1>\max |k\w|\<\|\|\>>>>,sup0> 2+\-1|)>>*|2*k**\|)>\w|\<\|\|\>>>>|}>, see, Theorem 1.10. Let us write >-u>\\*v>|)>*\*r> and let > and 0>. Then *k**\|)>\*f|)>|\|>=2>*k|)>**|)>*-f|)>*\*s|\|>>>|2*>*k|)>**|)>*t-s|\|>+\>*\*s*sup\1> -f|\|>|+\>>\2+\-1|)>>*sup\1> -f|\|>|+\>>,>>>>> where we used that >\*k*\*t=0>, and that is supported in . Similarly, we obtain <\equation*> \*f|)>|\|>\>*k**t-s|\|>+\>*\*s*sup\1> -f|\|>|+\>>\sup\1> -f|\|>|+\>>, from where the convergence of >\\*v>> to > in +\-1>> follows. <\corollary> Let be an --dimensional centered Gaussian process with independent components and measurable trajectories, whose covariance function satisfies for some > the inequalities -X|\|>|]>\*and>>|-X|)>*-X|)>|]>|\|>\*r>>>>> for all > and all |)>>. Then *X\>> for all \H> and all \>, and there exists \-1>> such that for every \> with \*\*t=1> and for every \0> we have <\equation*> lim\0> *|\>\*X|)>-*X|)>|\<\|\|\>>>+|>\*X|)>|)>\\*>\*X|)>|)>-\|\<\|\|\>>-1>>\\|)>=0, where we define >=\*\**\|)>>. <\proof> Since > is smooth and of compact support, it is easy to see that also the Gaussian process *X> satisfies the covariance condition, and using Gaussian hypercontractivity we obtain *X-\*X|\|>|]>\> for all 1>. Using the fact that has measurable trajectories, we can apply this estimate to show that |\*X|\<\|\|\>>>>|]>\\> for all 1>, \H>. Now it suffices to apply Besov embedding, Lemma, to obtain that *X\>>. Moreover, *X> has compact support. So by Theorem 15.45 of, for every 1>, the iterated integrals >\*\>\*X|)>|)>*\*\>\*X|)>|)>> converge in > in the sense of. The statement then follows from Corollary. <\remark> The proof of Corollary actually shows more than the equivalence of the convergence of >> and of >\\*v>>: it shows that the norm of >\\*v>-\|)>> can be controlled by a polynomial of the norms of >-A|)>>, >-u|)>>, and >-v|)>>. So in fact we have >--convergence in Corollary, and not just convergence in probability. Alternatively, the >--convergence is obtained from the convergence in probability because we are considering random variables living in a fixed Gaussian chaos, see Theorem 3.50 of. Combining Corollary with Theorem, we obtain the following corollary: <\corollary> Let be a --dimensional centered Gaussian process satisfying the conditions of Corollary for some 1/3>, and let \> and C>. Then there exists a unique solution to <\equation*> \*u=F*\**X|)>,u=u, in the following sense: If \> with \*\*t=1> and if for \0> the function >> solves <\equation*> \*u>=F>|)>*\**X|)>>,u=u, where *X|)>>=\*\*\|)>\*X|)>>, then >> converges to in probability in >> for all \H>. Fix now \5/6> and consider the following PDE on \> for some fixed 0>: <\equation> L*u=G*\*u+\,u=u, where +(-\)>> and \>> for a suitable >. We would like to consider solutions in the case of a distributional >, and in particular we want to allow > to be a typical realization of a space-time white noise. We will see below that in this case the solution > to the linear equation =\>, =0>, belongs (locally in time) to >|)>> for any \\-1/2>, but it is not better than that. This is also the regularity to be expected from the solution of the non-linear problem, and so for \1> the term |)>*\*u> is not well defined since |)>\>|)>> and *u\-1>|)>>, and the sum of their regularities fails to be positive. For =1>, this equation has been solved by Hairer, who used rough path integrals to define the product *\*u>. In the following, we will show how to solve the equation using paracontrolled distributions. While in general it is possible to set up the equation in a space-time Besov space, the fact that the distribution > (which is a genuine space-time distribution) enters the problem linearly allows for a small simplification. Indeed, if we let >, then solves the PDE <\equation> L*w=G*+w|)>*\*+w|)>, which can be studied as an evolution equation for a continuous function of time with values in a suitable Hölder-Besov space: Recall that for 0> and \> we defined the spaces >=C,>,|)>|)>> with norm |u|\<\|\|\>>>>=sups\T>|u|\<\|\|\>>>>. By the regularity theory for we expect C-1+2*\>> whenever +w|)>*\*+w|)>\C-1>> (at least in the sense of uniform estimates as the regularization goes to zero). The paraproduct allows us to decompose the right hand side of as <\equation*> G*+w|)>*\*+w|)>=G*+w|)>\\*\+G*+w|)>\\*\+G*+w|)>\\*\+G*+w|)>*\*w, where we have expanded only the term containing *\> since the one linear in *w> is well defined under the hypothesis that C-1+2*\>>. Note that here we only let the paraproduct act on the spatial variables, i.e. +w|)>\\*\> should really be understood as <\equation*> t\G*+w|)>\\*\, an element of -1>>. A simple modification of the proof of Lemma shows that, for \>, we have <\equation*> |G*+w|)>-G*+w|)>\\|\<\|\|\>>>\|G|\<\|\|\>>>*|\|\<\|\|\>>>|)>*|w|\<\|\|\>>>|)>\|G|\<\|\|\>>>*|\|\<\|\|\>>>|)>*|w|\<\|\|\>>-1+2*\>|)>, where we used that -1+2*\\2*\>, which holds because \\-1/2\2*\-1>. The linear dependence on the norm of will be crucial for obtaining global solutions. We can now rewrite <\equation*> G*+w|)>\\*\=+w|)>-G*+w|)>\\|)>\\*\+C**+w|)>,\,\*\|)>+G*+w|)>*\\*\|)>. So if we assume that \\*\|)>\C-1>>, then we have a well behaved representation of the resonant term +w|)>\\*\>, and |G*+w|)>\\*\|\<\|\|\>>-1>>||G|\<\|\|\>>>*|\|\<\|\|\>>>|)>*|w|\<\|\|\>>-1+2*\>|)>*|\*\|\<\|\|\>>-1>+|G*+w|)>|\<\|\|\>>>|\|\<\|\|\>>>>>||+|G*+w|)>|\<\|\|\>>>*|\\\*\|\<\|\|\>>-1>\C*C>*|w|\<\|\|\>>-1+2*\>|)>,>>>>> where we set <\equation*> C=|G|\<\|\|\>>>*andC>=sup>|\|\<\|\|\>>>|)>*|\\\*\|\<\|\|\>>-1>|)>. Let us now define <\equation*> \=G*+w|)>*\*\=G*+w|)>\\*\+G*+w|)>\\*\+G*+w|)>\\*\, so that and the paraproduct estimates yield <\equation> |\|\<\|\|\>>-1>\C*C>*|w|\<\|\|\>>-1+2*\>|)>, and satisfies +G*+w|)>*\*w>. So if we denote by |)>0>> the semigroup generated by )>>, then <\equation> w=P*u+P*\*\*s+P*+w|)>*\*w|)>*\*s, where we assumed that =0>. Applying the Schauder estimates for the fractional Laplacian (Lemma and Lemma) to, we obtain for all 0> that |w|\<\|\|\>>-1+2*\>>|*u+P*\*\*s+P*+w|)>*\*w|)>*\*s|\|>-1+2*\>>>||c*t-1|)>/2*\>*|u|\<\|\|\>>>+sup>-1|)>/|)>>|\|\<\|\|\>>-1>|)>|)>>>||+c*|G*+w|)>*\*w|\<\|\|\>>>>|-1+2*\|)>/|)>>>*\*s,>>>>> where 0> is a generic constant whose value may change in every step. But now recall from that |\|\<\|\|\>>-1>\C*C>*|w|\<\|\|\>>-1+2*\>|)>>. Moreover, if we choose \-1/2|)>> close enough to -1/2>, then +2*\-2\0> (recall that \5/6>), and therefore <\equation*> |G*+w|)>*\*w|\<\|\|\>>>>\|G|\<\|\|\>>>>*|\*w|\<\|\|\>>-2+2*\>\|G|\<\|\|\>>>>|w|\<\|\|\>>-1+2*\>. Thus, we get for all > that |)>>|w|\<\|\|\>>-1+2*\>|)>>|c|u|\<\|\|\>>>+c*C>*C*>-1|)>/|)>>|w|\<\|\|\>>-1+2*\>|)>|)>>>||+c*C*t|)>>*|)>>|w|\<\|\|\>>+1>|)>*s|-1+2*\|)>/|)>>*s|)>>>>>>>> Since -1+2*\|)>/|)>\1>, we have <\equation*> t|)>>**s|-1+2*\|)>/|)>>*s|)>>>\t-1+2*\|)>/|)>>\1 for >. Putting everything together, we conclude that <\equation*> |)>>|w|\<\|\|\>>-1+2*\>|)>\c|u|\<\|\|\>>>+c*C>*C*>-1|)>/|)>>|w|\<\|\|\>>-1+2*\>|)>|)>. In order to turn this into a bound on |w|\<\|\|\>>-1+2*\>>>, we use again a scaling argument. We extend the scaling transformation to the time variable in such a way that it leaves the operator invariant. More precisely, for \0> we set >*u=u*>*t,\*x|)>>, so that >=\>*\>*L>. Now let >=\>*u>, >=\>*w>, and >=\>*\>. Note that >:>|]>\>\>, where >=/*\|)>> is a rescaled torus, and that >> solves the equation <\equation*> L*w>=\>*\>*L*w=\>*\>*+G*|)>*\*w|)>=\>*\>*\+\-1>*G*>+\>|)>*\*w>. The same derivation as above shows that <\equation*> |\>*\|\<\|\|\>>-1>=|G*>+w>|)>*\>**\|)>|\<\|\|\>>-1>\C*C>>*|w>|\<\|\|\>>-1+2*\>|)>, where we get using Lemma and Lemma <\equation*> C>>=sup>|\>|\<\|\|\>>>|)>*|\>\\>**\|)>|\<\|\|\>>-1>|)>\\-1>*C>\\*C> as long as \>. Thus, we finally conclude that |)>>|w>|\<\|\|\>>-1+2*\>|)>>|c*|\>*u|\<\|\|\>>>+\-1>*c*C>*C*>-1|)>/|)>>|w>|\<\|\|\>>-1+2*\>|)>|)>>>||c|u|\<\|\|\>>>+\-1>*c*C>*C*>-1|)>/|)>>|w>|\<\|\|\>>-1+2*\>|)>|)>>>>>> for all \>. Since -1\0>, we get for small enough \0>, depending only on >> and > but not on >, that <\equation*> sup>|)>>|w>|\<\|\|\>>-1+2*\>|)>\2*|u|\<\|\|\>>>+1|)>. But >*>+\>|)>>, and therefore <\equation*> sup>*T|]>>|u|\<\|\|\>>>\>|u|\<\|\|\>>+C>. This provides the key ingredient for obtaining a uniform estimate on the full time interval >, and then the existence of global solutions to the Burgers equation. Uniqueness in the space of solutions with decomposition +w> with C-1+2*\>> can be handled easily along the lines above, and we obtain the following result: <\theorem> Let \5/6>, \-1/2|)>>, let 0>, and assume that >|)>\0>> is a family of smooth functions on \> with values in >, and C,,|)>|)>>. Suppose that there exist \C>> and \C-1>> such that >,>\\*\>|)>|)>> converges to ,\|)>> in -1>\C-1>>, where >> are solutions to >=\>> and >=0>, and where +(-\)>>. Let for \0> the function >> be the unique global solution to the Cauchy problem <\equation*> L*u>=G>|)>*\*u>+\>,u>=u, where \>>. Then there exists C>> such that >\u> in >>. The limit depends only on ,\,\|)>>, and not on the approximating family >,>\\*\>|)>|)>>. <\remark> As for s, the limit of the regularized solutions >> actually solves the equation <\equation*> L*u=G*\*u+\,u=u in the weak sense as long as we interpret the product *\*u> correctly. According to Remark, it is not important that =0>, and we could consider any other initial condition in \>> for some \> to obtain the same solution . However, of course the right choice of > may facilitate the proof of existence and uniqueness of paracontrolled solutions. <\remark> Of course, the solution to the fractional Burgers type equation also depends continuously on the initial condition >. It remains to show that if > is a space-time white noise, then the solution > to =\>, =0>, is in >> for all \\-1/2>, and that the area \\*\> is in -1>>. Some general results on the existence of the area for Gaussian processes indexed by a one-dimensional spatial variable are shown in. However, in the present setting it is relatively straightforward to construct the area ``by hand'', using Fourier analytic methods. In this section, we use > to denote the spatial Fourier transform, i.e. u|)>=>Te*k*x>u*\*x>. Recall that \> is a complex valued, centered Gaussian space-time distribution, whose covariance is formally given by <\equation*> [\(t,\\>,\|)>|)>|\>|]>=2*\*>*>*\*|)> for \,n|}>>, \>, \>, where > denotes the Dirac delta. If |)>0>=\|>>|)>|)>0>> denotes the semigroup generated by )>>, then =*\|)>*\*s>, >, from where a straightforward calculation yields the following result: <\lemma> The spatial Fourier transform \> of > is a complex-valued Gaussian process with zero mean and covariance <\equation*> [\(t,\\>,\|)>|)>|\>|]>=*>*>*-t|\|>>>-e|)>>>|)>/>|)>,>|0,>>|*>*>*t\t,>|>>>> for \,n|}>>, \>, and \>. As a consequence, \\|)>|\>|]>=2*\*>*=0>*>, and for 0> <\equation*> \\>|)>|\>|]>=\*>*>*>>-e>>-2*e>>+2*e>>|)>/>, where we write \=\|)>-\|)>> for all s\t\T>. In particular, <\equation> \\|\>|]>|\|>\>*|)>> for all \> and all 0>. Our first concern is to study the Hölder-Besov regularity of the process >. <\lemma> For any \\-1/2> and any 1>, the process > satisfies <\equation*> |\|\<\|\|\>>>|)>>|]>\\. <\proof> Let > and \0>. The case =0> can be treated using essentially the same arguments, except that then we need to distinguish the cases and 0>, where is the argument in the Fourier transform. Using Gaussian hypercontractivity (, Theorem 3.50), we obtain for 1> that <\equation> |\>*\|\<\|\|\>>|)>>|]>\|>*\|\|>|]>|\<\|\|\>>|)>>. Using Fourier inversion and Lemma, we have >*\|\|>|]>>||)>*\>\>*\>|)>*e*|)>*x>\\|)>|\>|]>>>||>\>*>*-1|)>>\>*supp>|)>>*-1|)>>\>*2**|)>|)>>>>>>> for all \>. Hence, we obtain from that <\equation*> [\\*|)>-\*>>|)>>|]>\\-1>2*\*2*p>|\>*\|\>|)>>|]>\\-1>2*\*2*p>*>*2**|)>|)>>|)> for any \> and any 1>. For \\-1/2> there exists \> small enough so that the series converges. Since we can choose arbitrarily large, Kolmogorov's continuity criterion implies that > has a continuous version with |\|\<\|\|\>>*B>|)>>|]>\\> for all \\-1/2>. Now we use again that can be chosen arbitrarily large, so that the Besov embedding theorem, Lemma, shows that this continuous version takes its values in >|)>> for all \\-1/2>. Next, we construct the area \\*\>. <\lemma> Define <\equation*> \\\*\=\\*\>|)>k,\\n>=\1>\*\*\*\*\|)>k,\\n>. Then almost surely \\*\\C-1>;n>|)>> for all \\-1/2>. Moreover, if \> is such that \*\*x=1> and >=\>\\>, where >=\*\*\|)>>, then we have for all 1> that <\equation> lim\0>|\>\\*\>-\\\*\|\<\|\|\>>-1>>|]>=0. <\proof> Without loss of generality we can argue for \\*\>. The case \\*\> is easy, because Leibniz' rule yields \\*\=*\*\\|)>>. Let \>. Note that if is smaller than -N> for a suitable , and if \1>, then >**f*\*g|)>=0> for all >. Hence, the projection of \\*\> onto the >--th dyadic Fourier block is given by >*\\*\|)>=\1>\>**\*\*\*\|)>=\1>\i>*\>**\*\*\*\|)>.>>>>> To avoid case distinctions, we only argue for \N>, so that we can always assume 0>. The case \N> can be handled using essentially the same arguments. We can apply Gaussian hypercontractivity to obtain |>*\\*\-\>\\*\>|)>|)>|\<\|\|\>>|)>>|]>>>|\|\1>\i>*>**\*\*\*\-\*\>*\*\*\>|)>|)>|\|>|]>|\<\|\|\>>|)>>,>>>>> where we write >=\>\\> and similarly for >>. Let us start by estimating >|\1>\i>*\>(\*\(t,\*\*\-\*\>**\*\>|)>|\|>|]>|\>|\>>>|>|=\1>-j|\|>\1>\i>*\i>[\>(\*\|)>*\*\*\-\*\>(t,\**\*\>|)>>>||\>(\>*\|)>*\>*\*\-\>*\>(t,\*>*\*\>|)>|\>].>>>>> Taking the infinite sums outside of the expectation can be justified a posteriori, because for every finite partial sum we will obtain a bound on the >--norm below, which does not depend on the number of terms that we sum up. The Gaussian hypercontractivity then provides a uniform >--bound for all 2>, which implies that the squares of the partial sums are uniformly integrable, and thus allows us to exchange summation and expectation. Recall that =|)>*>u|)>v*|)>>, and *u|)>=\*k>, and therefore >(\*\|)>*\*\*\-\*\>(t,\**\*\>|)>>>||)>*>\>*e|k,x|\>>(\*\|)>*\*\*\-\*\>(t,\**\*\>|)>>>||)>*\>\>*e|k,x|\>>*\|)>*\*|)>*\*|)>\|)>|)>\*|)>>>|\**k|)>\**|)>|)>|)>.>>>>> From this expression it is clear that if we can show |\>\\*\>|\<\|\|\>>-1>>|]>\\>, then the convergence result in will follow by dominated convergence, because \> is bounded and \=1> by assumption. Using the covariance of \> that we calculated in Lemma, we obtain |\1>\i>*\>**\**\*\|)>|\|>|]>|\>|\>>>||\\1>-j|\|>\1>\i>*\i>*\>\>*|)>*\*\>*\|)>*\>|)>>>||\>>|2>>|\|>*>|\|>*|)>>>>||\\1>\i>*supp|)>,k\supp|)>>\>*|)>*2+\*\|)>>*>>>||\\>2>*2+\*\|)>>*>>>>>> for all \>. Since \5/6>, there exists \0> small enough so that the sum is finite, and we obtain <\equation*> \1>\i>*\>**\**\*\|)>|\|>|]>\2+\*\|)>>*>,|\>|\> and by the same arguments <\equation*> \1>\i>*\>**\*\*\*\*|\|>|]>\2+\*\|)>>*>.|\>|\> Since <\equation*> \*\|)>*\*\*\*|)>-\*\|)>*\*\*\*|)>=\*\|)>*\*\*\+\*\*\*\*\*|)>, we get for sufficiently small \0> and for arbitrarily large 1> that |\>*\\*\|)>|\<\|\|\>>|)>>|]>\2*-2-\*\|)>*p>**p>.>>>>> Now we use the same arguments as in the proof of Lemma to obtain the required >-bound for |\\\*\|\<\|\|\>>-1>>> with \\-1/2>. Now Lemma and Theorem give us the existence and uniqueness of solutions to the fractional Burgers type equation driven by space-time white noise: <\corollary> Let \5/6>, \-1/2|)>>, 0>, C>, \>|)>>, +(-\)>>, and let > be a space-time white noise on \> with values in >. Then there exists a unique solution to <\equation*> L*u=G*\*u+\,u=u, in the following sense: If \> with \*\*t=1> and if for \0> the function >> solves <\equation*> L*u>=G>|)>*\*u>+\>,u=u, where >=\*\*\|)>\\>, then >> converges in probability in >> to . <\remark> There is no problem in considering the equation on > rather than on >, and the analysis works exactly as in the one-dimensional case. The proof of Lemma shows that if > is a space-time white noise on \>, then the solution > to =\>, =0>, will be in >|)>> for every \\-d/2>. So as long as -d/2\1/3>, we can solve the Burgers equation on >. For the existence of the area \\*\> we need the additional condition -d/2-1\0>; see, Lemma5.4.3. But if -d/2\1/3>, then this is always satisfied. Consider now the following PDE on \> for some fixed 0>: <\equation> L*u=F*\,u=u, where -\>, the function is continuous from > to >, > is a spatial white noise, and \>> for suitable \>. Formally, this equation is very similar to the rough differential equation. The regularity of the spatial white noise > on > is \>> for all \0>. Since we are in dimension , we have \>>. The Laplacian increases the regularity by 2, so we expect that for fixed 0> we have \>>, and therefore the product *\> is ill-defined. However, let us assume that \-2>|)>> for some \\1>, and let \>> be such that =\>. Consider the paracontrolled ansatz <\equation*> u=F\\+u> with >\C>>, and where as in Section the paraproduct > is only acting on the spatial variables. If is of this form, then Lemma and Lemma imply that <\equation*> F*\=F\\+F\\+F*F*\\|)>+F*C,\,\|)>+F*>\\|)>+\|)> is well defined provided that \\|)>\C-2>>. Moreover, the algebraic rules for > and > acting on products imply that <\equation*> L*u=|)>\\+F\L*\-2*\*F\\*\+L*u>, and thus we find the following equation for >>: >=2*\*F\\*\-|)>\\+F\\+F*F*\\|)>>>|*C,\,\|)>+F*>\\|)>+\|)>.>>>>> We would like all the terms on the right hand side to be in -2>>. However, it is not easy to estimate the term |)>\\> in >> for any \>: the term *F> can be controlled in -2>>, but there are no straightforward estimates available for the time derivative *F> appearing in >. Indeed, it would be more convenient to treat the generalized parabolic Anderson model in a space-time parabolic Besov space adapted to the operator and to use the natural paraproduct associated to this space. An alternative strategy would be to stick with the simpler space -2>> and to observe that <\equation*> L*F=F*L*u-F**u|)>=F*F*\-F**u|)>, and that the terms on the right hand side can be analyzed using the paracontrolled ansatz. Since this strategy seems to require a lot of regularity from , we do not pursue it further. Instead, we keep working on -2>>, but we modify the paraproduct appearing in the paracontrolled ansatz. Let :\> be a positive smooth function with compact support and total mass , and for all -1> define the operator :C>\C>> by <\equation*> Q*f=>2*\**|)>*f*T|)>\0|)>*\*s. For > we have the following standard estimates, which we leave to the reader to prove: |Q*f|\<\|\|\>>>>\|f|\<\|\|\>>*L>>,|\*Q*f|\<\|\|\>>>>\2|)>*i>|f|\<\|\|\>>>*L>>,>>||*f-f|)>|\<\|\|\>>>>\2*i>|f|\<\|\|\>>>*L>>>>>>> for all > and all \>; for the second estimate we use that \*\*t=0>, and for the third estimate we use that > has total mass 1. With the help of >, let us define a modified paraproduct by setting <\equation*> f\g=*Q*f|)>*\*g for C>. It is easy to show that for this paraproduct we have essentially the same estimates as for the pointwise paraproduct g>, only that we have to bound uniformly in time; for example <\equation*> |g|)>|\<\|\|\>>>\|f|\<\|\|\>>*L>>|g|\<\|\|\>>>. for all >. For us, the following two commutator estimates are the most useful properties of >. <\lemma> Let 0>, \>, \>, and let C>\C/2>*L>> and C>>. Then <\equation> |L*v|)>-u\|\<\|\|\>>+\-2>>\|u|\<\|\|\>>/2>*L>>+|u|\<\|\|\>>>>|)>|v|\<\|\|\>>>>, as well as <\equation> |u\v-u\v|\<\|\|\>>+\>>\|u|\<\|\|\>>/2>*L>>|v|\<\|\|\>>>>. <\proof> For, observe that v|)>-u\=\v-2*\*u\\*v>. The second term on the right hand side is easy to estimate. The first term is given by <\equation*> \v=*Q*L*u|)>*\*v=*Q*u|)>*\*v. Observe that, as for the standard paraproduct, *Q*F|)>*\*v> has a spatial Fourier transform localized in an annulus >, so that according to Lemma it will be sufficient to control its *L>> norm. But |L*S*Q*u|\<\|\|\>>*L>>>||\*Q*S*u|\<\|\|\>>*L>>+|Q*\*S*u|\<\|\|\>>*L>>>>||2-2|)>*i>*|S*u|\<\|\|\>>/2>*L>>+|u|\<\|\|\>>>>|)>,>>>>> where we used the bounds. It is easy to see that |S*u|\<\|\|\>>/2>>\|u|\<\|\|\>>/2>>>, and therefore we obtain. As for, we have <\equation*> u\v-u\v=*S*u-S*u|)>*\*v, and again it will be sufficient to control the *L>> norm of each term of the series. But using once more, we obtain <\equation*> |*S*u-S*u|)>*\*v|\<\|\|\>>*L>>\2>*|S*u|\<\|\|\>>/2>*L>>*|\*v|\<\|\|\>>*L>>\2+\|)>>|u|\<\|\|\>>/2>*L>>|v|\<\|\|\>>>>, and the result is proved. Letting <\equation> u=F\\+u> and redoing the same computation as above gives <\equation> L*u>=\>=-\\|)>-F\\|]>+\\-F\\|]>+F\\+F\\. Lemma takes care of the first two terms on the right hand side. The term \\> can be controlled using the paraproduct estimates, so that it remains to control the resonant product \\>. In principle, this can be achieved by combining the decomposition described above with, which enables us to switch between the two paraproducts> and>. However, in that way we would pick up a superlinear estimate from Lemma, and thus would get a problem when trying to construct global in time solutions. We therefore have to be slightly more careful. <\lemma> Let \> and \|]>> be such that +\\2>. Let 0>, \C,|)>>, \C>>, C>>, and let C/\>>. Define >=u-F\\>. Then <\equation*> |\\|)>|\<\|\|\>>+\-2>\C*C>*|u|\<\|\|\>>>>+|u|\<\|\|\>>/2>*L>>+|u>|\<\|\|\>>+\>|)>, for all >, where <\equation> C>=|\|\<\|\|\>>-2>|)>*|\|\<\|\|\>>>>/\>|)>+|\\\|\<\|\|\>>-2>>*C=|F|\<\|\|\>>/\>>+|F|\<\|\|\>>/\>>/\>. We pay attention to indicate that, for fixed >, the estimate depends only on the +\>> norm of >>. This will come useful below when introducing the right norm to control the contribution of the initial condition. <\proof> We decompose F\\>|-F\\\|)>|)>\\+C*,F\\,\|)>>>|>|+F*\\-F\\|)>\\|]>+F*C,\,\|)>+F*F*\\|)>,>>>>> from where we can use Lemma and the commutator estimate Lemma to see that <\equation*> |F\\--F\\\|)>|)>\\|\<\|\|\>>+\-2>>\C*C>*|u|\<\|\|\>>>>+|u|\<\|\|\>>/2>*L>>|)>. It remains to treat the first term on the right hand side of, which we split into two parts: -F\\\|)>|)>\\>|n,>>|\1>>>>>>\*-F\\\|)>|]>*\*\>>|>|+n,>>|\1>>>>>>\*-F\\\|)>|]>*\*\>>>>> for >. For the first series, we use that +\-2\0> and simply estimate <\equation*> n,>>|\1>>>>>>\*-F\\\|)>|]>*\*\+\-2>>\n>2|)>>*C*C>\2|)>>*C*C>. For the second series in we get |\n,>>|\1>>>>>>\*-F\\\|)>|]>*\*\|)>>>>>||\n>*2+\-2|)>>+n>*2+\-2|)>>|)>|-F\\\|)>|)>|\<\|\|\>>+\>|\|\<\|\|\>>-2>>>||\2>*2+\-2|)>>|-F\\\|)>|)>|\<\|\|\>>+\>|\|\<\|\|\>>-2>>>>>> for all > and -1>. Now a slight modification of the proof of Lemma shows that for >> and +\>> we have <\equation*> |F*-F*\f|\<\|\|\>>+\>\|F|\<\|\|\>>/\>>*|f|\<\|\|\>>>/\>|)>*|g|\<\|\|\>>+\>+\>|)>, and applying this with \\|)>> and >> we deduce that <\equation*> -F\\\|)>|)>\\|)>+\-2>\C*C>*|)>>+2>|u>|\<\|\|\>>+\>+\>|)> for all >. It remains to optimize over the parameter . If |u>|\<\|\|\>>+\>\1>, we choose and obtain <\equation*> -F\\\|)>|)>\\|)>+\-2>\C*C>*|u>|\<\|\|\>>+\>|)>. If |u>|\<\|\|\>>+\>\1>, we choose with |)>>\|u>|\<\|\|\>>+\>>, so that <\equation*> 2|)>>+2>|u>|\<\|\|\>>+\>+\>\|u>|\<\|\|\>>+\>+|u>|\<\|\|\>>+\>|2-\>+\+\>. Since \\>, we can bound the exponent on the second term from above by -\/|)>>, and it is not hard to see that for \2> this expression is smaller or equal to 1. Therefore, we obtain also in that case <\equation*> -F\\\|)>|)>\\|)>+\-2>\C*C>*|u>|\<\|\|\>>+\>|)>, which completes the proof. Let us summarize our observations so far. <\lemma> Let \>, \,\|]>>, and 0>. Let \>>, \C,|)>>, let > be such that =\>, and let C/\>>. Then solves the PDE <\equation*> L*u=F*\,u=u\> on > if and only if \\+u>>, where >> solves <\equation*> L*u>=\>,u>=u-\\|)> on >, for >> as defined in . Moreover, for all > we have the estimate <\equation> |\>|\<\|\|\>>+\-2>\C*C>*|u|\<\|\|\>>>>+|u|\<\|\|\>>/2>*L>>+|u>|\<\|\|\>>+\>|)>, where > and >> are as defined in. Next, we would like to close the estimate, so that the right hand side only depends on >>. In order to estimate |u|\<\|\|\>>>>+|u|\<\|\|\>>/2>*L>>>, we observe that >+F\\> and that <\equation*> L*F\\=\\|)>-F\|)>|]>+F\\. Now it is easy to see that <\equation*> |L*\\|)>-F\|)>|\<\|\|\>>-2>>\|F|\<\|\|\>>*L>>|\|\<\|\|\>>>>\|F|\<\|\|\>>>>|\|\<\|\|\>>>> (compare also the proof of Lemma). Thus, we can apply the heat flow estimates Lemma, Lemma, and Lemma, to deduce |u|\<\|\|\>>>>+|u|\<\|\|\>>/2>*L>>>||u>|\<\|\|\>>>>+|u>|\<\|\|\>>/2>*L>>+|F\\|\<\|\|\>>>+|L*\\|)>|\<\|\|\>>-2>>>>|||u>|\<\|\|\>>>>+|u>|\<\|\|\>>/2>*L>>+|u|\<\|\|\>>>+|F|\<\|\|\>>>>*|\|\<\|\|\>>>>+|\|\<\|\|\>>-2>|)>.>>>>> Plugging this into, we get <\equation*> |\>|\<\|\|\>>+\-2>\C*C>**C>+|u|\<\|\|\>>>+|u>|\<\|\|\>>>>+|u>|\<\|\|\>>/2>*L>>+|u>|\<\|\|\>>+\>|)>. Moreover, since >=u-\\|)>> and >=\>>, we can apply Lemma and Lemma to obtain <\equation*> t/2>|u>|\<\|\|\>>+\>\|u|\<\|\|\>>>+C*C>+sup>/2>|\>|\<\|\|\>>+\-2>|)>, so that our new estimate for >> reads <\equation*> t/2>|\>|\<\|\|\>>+\-2>\C*C>**C>+|u|\<\|\|\>>>+|u>|\<\|\|\>>>>+|u>|\<\|\|\>>/2>*L>>+sup>/2>|\>|\<\|\|\>>+\-2>|)>|)>, uniformly in >. It remains to control >> in /2>*L>\C>>. For s\t\T>, we have |u>-u>|\<\|\|\>>>>>||-|)>*P>|)>|\<\|\|\>>>>+P*\>*\*r>>>>||+-|)>*P*\>*\*r>>.>>>>> Applying Lemma to the first and third term and Lemma to the second term, we obtain |u>-u>|\<\|\|\>>>>>|/2>|u>|\<\|\|\>>>+/2+\/2>|\>|\<\|\|\>>+\-2>*\*r>>||+/2>*|P*\>|\<\|\|\>>>*\*r>>||/2>**C>+|u|\<\|\|\>>>|)>>>||+/2>*/2>*r/2>*\*r*sup>/2>|\>|\<\|\|\>>+\-2>|)>>>||+/2>*/2>*r/2>*\*r*sup>/2>|\>|\<\|\|\>>+\-2>|)>.>>>>> For the time integrals we have /2>*r/2>*\*r=/2>*r/2>*\*r\1>, so that <\equation*> |u>|\<\|\|\>>/2>*L>>\C*C>+|u|\<\|\|\>>>+sup>/2>|\>|\<\|\|\>>+\-2>|)>. Similar (but easier) arguments can be used to bound the >> norm of >>, and thus we obtain our final estimate for >>: <\equation> sup>/2>|\>|\<\|\|\>>+\-2>|)>\C*C>**C>|)>*|u|\<\|\|\>>>+sup>/2>|\>|\<\|\|\>>+\-2>|)>|)>. In order to use this estimate to bound >>, we will apply the usual scaling argument. More precisely, we set >*f=f**t,\*x|)>>, so that >=\*\>*L>. Now let >=\>*u>, >=\>*u>, >=\>*\>*\>, and >=\>*\>*\>. Note that >:|]>\>\>, where >=/*\|)>|)>> is a rescaled torus, and that >> solves the equation <\equation*> L*u>=\*F>|)>*\>*\=\>*F>|)>*\>,u>=u>. The scaling is chosen in such a way that |u>|\<\|\|\>>>\|u|\<\|\|\>>>>, |\>|\<\|\|\>>-2>>\|\|\<\|\|\>>-2>>>, |\>|\<\|\|\>>>>\|\|\<\|\|\>>>>>, and according to Lemma also |\>\\>|\<\|\|\>>-2>>\|\\\|\<\|\|\>>-2>>+|\|\<\|\|\>>>>|\|\<\|\|\>>-2>>, all uniformly in \>. In particular, >>\C>> and >*F>\\>*C> for all \>. Injecting these estimates into, we obtain <\equation*> sup>/2>|\,\>|\<\|\|\>>+\-2>|)>\1+|u>|\<\|\|\>>> for all sufficiently small \0> (depending only on >> and >, but not on >), where ,\>> is defined analogously to >>. From here we easily get the existence of paracontrolled solutions to. Similar arguments show that if C/\>>, then the map ,\,\,\\\|)>\u\C>> is locally Lipschitz continuous, and in particular there is a unique paracontrolled solution. So far, we argued under the assumption that there exist continuous functions >|)>> such that >,\>,\>\\>|)>> converges to ,\,\\\|)>> in -2>\C-2>\C-2>> as \0>. Here the superscript > refers to a smooth regularization of the noise, whereas in the previous section the superscript > referred to a scaling transform. From now on we will no longer consider scaling transforms, so that no confusion should arise. One further difficulty is that the resonant product >\\>|)>> does not converge in some relevant cases; in particular, if > is a spatial white noise. However, what we will show below is that for the white noise there exist constants >\> such that >\\>|)>-c>|)>> converges in probability in -2>>. In order to make the term >> appear in the equation, we can introduce a suitable correction term in the regularized problems and consider the renormalized PDE <\equation> L*u>=F>|)>*\>-c>*F>|)>*F>|)>. For this equation we use again the paracontrolled ansatz. The same derivation as for yields ,\>>|>,\>,\>|)>+F>|)>\\>-c>*F>|)>*F>|)>>>>>> for some bounded functional , and as in Lemma we decompose <\equation*> F>|)>\\>-c>*F>|)>*F>|)>=H>,u,\>,\>,\>|)>+F>|)>*F>|)>*>\\>-c>|)> for another bounded functional . We see that ,\>> only depends on >>, >>, and >\\>|)>-c>>. Thus, the convergence of >,\>,\>\\>-c>|)>> to ,\,\|)>> in -2>\C-2>\C-2>> implies that the solutions >|)>> to converge to a limit which only depends on >, >, and >, but not on the approximating family. <\theorem> Let \>, \,\|]>> and assume that >|)>\0>\C,|)>> and C/\>>. Suppose that there exist \-2>>, \C>>, and \C-2>> such that >,\>,>\\>|)>-c>|)>> converges to ,\,\|)>> in -2>\C>\C-2>>, where >> are solutions to >=\>>, and where >\> for all \0>. Let for \0> the function >> be the unique solution to the Cauchy problem <\equation*> L*u>=F>|)>*\>-c>*F>|)>*F>|)>,u>=u, where \>>. Then there exists C>> such that >\u> in >>. The limit depends only on ,\,\,\|)>>, and not on the approximating family >,\>,>\\>|)>-c>|)>>. As for the previous equations, is the unique paracontrolled weak solution to *\*\> with =u> if we interpret the renormalized product *\*\> in the right way, the initial condition > is of no importance, and depends continuously on >. It remains to study the regularity of the area \\>. As already indicated, we will have to renormalize the product by ``subtracting an infinite constant'' in order to obtain a well-defined object. To simplify the arguments below, we assume that > is given by |~>-|)>|~>>, where |~>> is a spatial white noise on >. Then \|)>>> is a complex valued, centered Gaussian process with covariance <\equation*> \\|)>|\>|]>=|)>*>*0>. Since > is a smooth additive perturbation of |~>>, this simplification will pose no problems, and we indicate below how to handle |~>> once we are able to handle >. Since > is a mean zero distribution, there exists a stationary solution > to =\>, given by <\equation*> \=>*\|)>*\*t. Then =\> by definition, and it is easily verified that \|)>> is a centered, complex valued Gaussian process with covariance <\equation*> \\|)>|]>=|)>>*>*0> and such that \|\>=\(-k)>. This yields, using Gaussian hypercontractivity and Besov embedding, that |\|\<\|\|\>>>|)>>|]>\\> for all \1> and 1>. Since *\> is a smooth function for 0>, the resonant term *\\\> is a smooth function, and therefore we could formally set \\=>*\\\|)>*\*t>. However, this expression is not well defined: <\lemma> For any > and 0> we have <\equation*> g=*\\\|)>|]>=**\\\|)>|]>=|)>*\>e>. In particular, > does not depend on the partition of unity used to define the > operator, and g*\*s=\> for all 0>. <\proof> Let >, 0>, and \-1>. Then >**\\\|)>|]>=\1>>***\|)>*\*\|)>|]>,>>>>> where exchanging summation and expectation is justified because it can be easily verified that the partial sums of >**\\\|)>> are uniformly >--bounded for any 1>. Now =e|\|\>>>, and therefore >***\|)>*\*\|)>|]>>||)>*\\>e*|k+k,x|\>>*\>*|)>*\*e>*\|)>\\|)>|]>>>|||)>*\>\>*\*e>*\>>|||)>*=-1>*\>\*\*e>.>>>>> For \1> we have *\=0>. This implies, independently of >, that <\equation*> g=*\\\|)>|]>=\>\*\*e>=|)>*\>e>, while *\\\|)>-\**\\\|)>|)>]=0>. <\remark> The same calculation shows that if \>, and if >=\*\*\|)>\\>, then <\equation*> *\>\\>|)>|]>=**\>\\>|)>|]>=|)>*\>e>*\**k|)>|\|>. The diverging time integral motivates us to study the renormalized product \\->g*\*t>, where >g*\*t> is an infinite constant: <\lemma> Set <\equation*> *\*\|)>=>*\\\-g|)>*\*t. Then |\*\*\|\<\|\|\>>-2>|]>\\> for all \1>, 1>. Moreover, if \> satisfies \*\*x=1>, and if >=\*\*\|)>\\> with >=\*\*\|)>> for \0>, and >=>P*\>*\*t>, then <\equation*> lim\0>|\*\*\->\\>-c>|)>|\<\|\|\>>-2>|]>=0 for all 1>, where for > <\equation*> c>=>*\>|]>=>\\>|]>=>*\>\\>|]>*\*t=|)>*\>\**k|)>|\|>|>. <\proof> We split the time integral into two components, \*\*t> and >\*\*t>. The second integral can be treated without relying on probabilistic estimates: Given >, we have >*\\\-g|)>*\*t->*\>\\>-*\>\\>|]>|)>*\*t-2>>>|>|P*\\\-P*\>\\>|\<\|\|\>>>*\*t+>\>e>\|1-\|\**k|)>|\|>*\*t>>|>|P*-\>|)>*|+2>|\|\<\|\|\>>-2>+|P*\>|\<\|\|\>>+2>|\<\|\|\>>*\-\>|\<\|\|\>>-2>|)>*\*t+\>>|>\|1-\|\**k|)>|\|>,>>>>> Since \=0>, the estimate |P*\>|\<\|\|\>>+2>\t|\>|\<\|\|\>>-2>> of Lemma holds uniformly over 0>, and thus the time integral is finite. The convergence in |)>> now easily follows from the dominated convergence theorem. We will treat the integral from 0 to 1 using similar arguments as in the proof of Lemma. To lighten the notation, we will only show that |*\\\-g|)>*\*t|\<\|\|\>>-2>|]>\\>. The difference <\equation*> *\\\-g|)>*\*t-*\>\\>-*\>\\>|]>|)>*\*t-2>|]> can be treated with the same arguments, we only have to include some additional factors of the form \**k|)>|\|>> in the sums below. The convergence of the expectation can then be shown using dominated convergence. Let > and define =P*\\\-g>. By Gaussian hypercontractivity we obtain for 1> and -1> that <\equation> |\*\|\<\|\|\>>|)>>|]>\|*\|\|>|]>|\>|)>>. By Lemma we have <\equation> *\|\|>|]>=Var**\\\|)>|)>, for all -1>, where |)>> denotes the variance. Now **\\\|)>=|)>*,k\>\1>e*|k+k,x|\>>*\*+k|)>*\|)>*e|\|>>\|)>*\|)>\|)>,>>>>> and therefore |**\\\|)>|)>>>||=|)>*,k\>,k\>\1>-j|\|>\1>e*|k+k,x|\>>*\*+k|)>*\|)>*e|\|>>*\|)>>>||\e*|k+k,x|\>>*\*+k|)>*\|)>*e|\|>>*\|)>*cov\|)>\|)>,\|)>\|)>|)>,>>>>> where exchanging summation and expectation can again be justified a posteriori by the uniform >--boundedness of the partial sums, and where denotes the covariance. Since |^>|)>>> is a centered Gaussian process, we can apply Wick's theorem (, Theorem1.28) to deduce ||^>|)>*|^>|)>,|^>|)>*|^>|)>|)>=|)>*=-k>*=-k>+=-k>*=-k>|)>*\0>*\0>,>>>>> and therefore |**\\\|)>|)>>>||=|)>*,k\\>\1>-j|\|>\1>i>*i>*\*+k|)>*\|)>*\|)>*\>|)>*\>|)>*e|\|>>|\>>>|||+i>*i>*\*+k|)>*\|)>*\|)>*\>|)>*\>|)>*e|\|>-t|\|>>|]>.>>>>> There exists 0> such that >\e>> for all supp|)>> and for all -1>. In the remainder of the proof the value of this strictly positive may change from line to line. If \1>, then we also have >\e>> for all supp|)>>. Thus >|**\\\|)>|)>)>>|>|\,j>i>*j\i\j>*,k\\>|)>>*+k|)>*|)>>|)>*|)>>|)>*e>>>||\m>2*2*e>\|t>*m>e>\|t>*e>,>>>>> where we used that \e|)>*2>> for any \c>. Now let \1>. We apply Jensen's inequality and combine , , and to obtain |\|\<\|\|\>>-2>>|]>\-1>2-2|)>*m*2*p>|\*\|\<\|\|\>>|)>>|]>|)>>>>|t-1>2-2|)>*m*2*p>*2*e>|)>>\t>|)>-1|)>>*e|)>>*\*x|)>>.>>>>> The change of variables *2> then yields |\|\<\|\|\>>-2>>|]>\t*-1|)>>*>y-1|)>-1>*e>*\*y|)>>.>>>>> If \1/2>, the integral is finite for all sufficiently large , and therefore |\|\<\|\|\>>-2>>|]>\t>>, so that |\|\<\|\|\>>-2>>|]>*\*t\\> for all \1>. Gaussian hypercontractivity allows us to conclude that also <\equation*> \*\*t-2>>|]>\\ for all 1>. The result now follows from the Besov embedding theorem, Lemma. <\corollary> Let |~>> be a spatial white noise, and let >, >, and *\*\> be as defined above. Set |~>=\+t*|)>|~>> (so that |~>=|~>>) and <\equation*> |~>*\*|~>|)>=*\*\|)>+|)>*\|~>|)>+t*|)>*|~>\\|)>+t*|)>|~>|)>. If \> satisfies \*\*x=1>, and if |~>>=\>\|~>> and |~>>=\>\|~>>, where >=\*\*\|)>> for \0>, then \0>||~>*\*|~>-|~>>\|~>>-c>|)>|\<\|\|\>>-2>>|]>=0> for all 0> and 1>, where >> is as defined in Lemma. Combining the existence of the renormalized product *\*\> with Theorem, we obtain the existence and uniqueness of solutions to the generalized parabolic Anderson model: <\corollary> Let \>, \,\|]>>, C/\>>, \>>, -\>, and let > be a spatial white noise on >. Then there exists a unique solution to <\equation*> L*u=F*\*\,u=u, in the following sense: For \> with \*\*t=1> and for \0> consider the solution >> to <\equation*> L*u>=F>|)>*\>-c>*F>|)>*F>|)>,u>=u, on |)>\>, where >=\*\*\|)>\\>, and where >> is as defined in Lemma. Then for all 0>, >|)>> converges in probability in >> to . <\remark> Concerning the convergence of >\\>|)>>, let us make the following remark: Since >=\>>, we have <\equation*> >\\>|)>=>\L*\>|)>=*L*>\\>|)>-*\>\\*\>|)>, from which we see that the only problem in passing to the limit is given by the second term on the right hand side. This integration by parts formula is the crucial difference with what happens in the case, which otherwise shares many structural properties with the model. The fact that is a second order operator generates the term *\>\\*\>|)>> in the above computation, which is absent in case of the operator >. This term, whose convergence is equivalent to the convergence of the positive term *\>|\|>>, cannot have simple cancellation properties and it is the origin for the need of introducing an additive renomalization when considering the model. Our previous analysis easily implies that the solutions to the modified problem <\equation*> L*u>=F>|)>*\>+F>|)>*F>|)>**\>|\|> will converge as soon as >\\> in -2>>, without any requirements on the bilinear term >\\>> . In the following, we describe the concepts from Littlewood--Paley theory which are necessary for our analysis, and we recall the definition and some properties of Besov spaces. For a general introduction to Littlewood--Paley theory, Besov spaces, and paraproducts, we refer to the nice book of Bahouri, Chemin, and Danchin . Littlewood--Paley theory allows for an efficient way of characterizing the regularity of functions and distributions. It relies on the decomposition of an arbitrary distribution into a series of smooth functions whose Fourier transforms have localized support. Let \,\\> be nonnegative radial functions on >, such that <\enumerate-roman> the support of > is contained in a ball and the support of > is contained in an annulus; +0>\**z|)>=1> for all >; |)>\supp**\|)>|)>=\> for 1> and **\|)>|)>\supp**\|)>|)>=\> for \1>. We call such ,\|)>> , and we frequently employ the notation <\equation*> \=\*\=\**\|)>*j\0. For the existence of dyadic partitions of unity see , Proposition 2.10. The Littlewood--Paley blocks are now defined as <\equation*> \*u=u|)>=u|)>*\*u=u|)>*j\0. Then *u=K\u>, where =\> for all -1>. In particular, *u> is a smooth function for every -1>. We also use the notation <\equation*> S*u=j-1>\*u. It is easy to see that -1>\*u=lim\> S*u> for every >. For \>, the Hölder-Besov space >> is given by >=B>,\>,|)>>, where for |]>> we define <\equation*> B>,|)>=,|)>:|u|\<\|\|\>>>>=-1>>*|\*u|\<\|\|\>>>|)>|)>\\|}>, with the usual interpretation as >> norm in case >. The |\|\>>> norm is taken with respect to Lebesgue measure on >. While the norm |\|\>>>> depends on the dyadic partition of unity ,\|)>>, the space >> does not, and any other dyadic partition of unity corresponds to an equivalent norm. We write |\|\>>> instead of |\|\>>,\>>>. If \|)>\>, then >> is the space of |\|\>> times differentiable functions, whose partial derivatives up to order |\|\>> are bounded, and whose partial derivatives of order |\|\>> are (-|\|\>>)-Hölder continuous (see p. 99 of ). Note however, that for > the Hölder-Besov space > is strictly larger than >. We will use without comment that |\|\>>\|\|\>>> for \\>, that |\|\>>>\|\|\>>> for \0>, and that |\|\>>\|\|\>>>> for \0>. We will also use that |S*u|\<\|\|\>>>>\2>|u|\<\|\|\>>>> for \0> and >>. We denote by >> the set of all distributions such that *u\>> for all \>. If the difference *-u|)>> converges to in >> for all \>, then we say that |)>> converges to in >>. The following Bernstein inequalities are tremendously useful when dealing with functions with compactly supported Fourier transform. <\lemma> >Let > be an annulus and let > be a ball. For any >, \0>, and p\q\\> we have that <\enumerate-numeric> if L|)>> is such that u|)>\\>, then <\equation*> max\:|\|>=k>|\>*u|\<\|\|\>>>\\-|)>>|u|\<\|\|\>>>; if L|)>> is such that u|)>\\>, then <\equation*> \|u|\<\|\|\>>>\max\:|\|>=k>|\>*u|\<\|\|\>>>. For example, it is a simple consequence of the Bernstein inequalities that |\*u|\<\|\|\>>-k>\|u|\<\|\|\>>>> for all \> and >. We point out that everything above and everything that follows can (and will) be applied to distributions on the torus. More precisely, let |)>> be the space of distributions on >. Any |)>> can be interpreted as a periodic tempered distribution on >, with frequency spectrum contained in > -- and vice versa. For details see, Chapter 3.2. In particular, *u> is a periodic smooth function, and therefore |\*u|\<\|\|\>>>>=|\*u|\<\|\|\>>>|)>>>. In other words, we can define <\equation*> >|)>=>:u*|)>-|}> for \>. For \> however, this definition is not very useful, because no nontrivial periodic function is in > for \>. Therefore, general Besov spaces on the torus are defined as <\equation*> B>|)>=|)>:|u|\>>|)>>=-1>>*|\*u|\>|)>>|)>|)>\\|}>, where we set <\equation*> \*u=|)>*>e|k,x|\>>*\>u|)>=>>u|)>, and where >> and >> denote Fourier transform and inverse Fourier transform on the torus. The two definitions are compatible: we have >|)>=B>,\>|)>>. Strictly speaking we will not work with >|)>> for \,\|)>>. But we will need the Besov embedding theorem on the torus. <\lemma> Let p\p\\> and q\q\\>, and let \>. Then >,q>|)>> is continuously embedded in -d*-1/p|)>>,q>|)>>, and >,q>|)>> is continuously embedded in -d*-1/p|)>>,q>|)>>. For the embedding theorem on > see, Proposition2.71. The result on the torus can be shown using the same arguments, see for example. In both cases, the proof is based on the Bernstein inequalities, Lemma. The following characterization of Besov regularity for functions which can be decomposed into pieces that are well localized in Fourier space will be useful below. <\lemma> (Lemmas2.69 and2.84 of) <\enumerate-numeric> Let > be an annulus, let \>, and let |)>> be a sequence of smooth functions such that u> has its support in >, and such that |u|\<\|\|\>>>>\2>> for all . Then <\equation*> u=-1>u\>and|u|\<\|\|\>>>\sup-1>>|u|\<\|\|\>>>>|}>. Let > be a ball, let \0>, and let |)>> be a sequence of smooth functions such that u> has its support in >, and such that |u|\<\|\|\>>>>\2>> for all . Then <\equation*> u=-1>u\>and|u|\<\|\|\>>>\sup-1>>|u|\<\|\|\>>>>|}>. <\proof> It u> is supported in >, then *u\0> only for j>. Hence, we obtain <\equation*> |\*u|\<\|\|\>>>>\i>|\*u|\<\|\|\>>>>\sup-1>>|u|\<\|\|\>>>>|}>*i>2>\sup-1>>|u|\<\|\|\>>>>|}>*2>. If u> is supported in >, then *u\0> only for j>. Therefore, <\equation*> |\*u|\<\|\|\>>>>\i>|\*u|\<\|\|\>>>>\sup-1>>|u|\<\|\|\>>>>|}>*i>2>\sup-1>>|u|\<\|\|\>>>>|}>*2>, using \0> in the last step. Here we discuss the action of some important linear operators on Besov spaces. We start with the rescaling of the spatial variable: <\lemma> For \0> and > we define the scaling transformation >*u|)>=u*\|)>>. Then <\equation*> |\>*u|\<\|\|\>>>\max >|}>|u|\<\|\|\>>> for all \\> and all >>. <\proof> Let >> and let >*u=u**x|)>> for some \0>. Note that >*\=\*\*\>>, and therefore >*\*u=\>*\**\|)>*u=\**\*\|)>*\>*u>, which implies that the Fourier transform of >*\*u> is supported in the annulus *2> (where > is the annulus in which > is supported). In particular, if 0>, we have *\>*\*u\0> only if \\*2>. Thus, there exist 0> such that |\*\>*u|\<\|\|\>>>>>|\\*2\b*2>|\*\>*\*u|\<\|\|\>>>>\\\*2\b*2>|\*u|\<\|\|\>>>>>>|||u|\<\|\|\>>>*\\*2\b*2>2*j>\|u|\<\|\|\>>>*\>*2*k>>>>>> for all 0>. For we can simply bound <\equation*> |\*\>*u|\<\|\|\>>>>\*2\1>|\*\>*\*u|\<\|\|\>>>>\|u|\<\|\|\>>>**2\1>2*j>\|u|\<\|\|\>>>*max >|}>. Next, we are concerned with the action of Fourier multipliers on Besov spaces. <\lemma> Let \>. Let > be a continuous function, such that > is infinitely differentiable everywhere except possibly at 0, and such that > and all its partial derivatives decay faster than any rational function at infinity. Assume also that \\L>. Then <\equation*> |\**\|)>*u|\<\|\|\>>+\>\,\>\>|u|\<\|\|\>>>*and|\**\|)>*u|\<\|\|\>>>\,\>\>|u|\<\|\|\>>>>. for all \>, \0>, and >. <\proof> Let \> with support in an annulus be such that *\=\>, where ,\|)>> is our dyadic partition of unity. Then we have for 0> that <\equation*> \**\|)>*\*u=**\|)>*\**\|)>|)>|]>\\*u, and therefore Young's inequality implies <\equation*> |\**\|)>*\*u|\<\|\|\>>>>\**\|)>*\**\|)>|)>|\|>>*2>|u|\<\|\|\>>>=**\*\|)>*\|)>|\|>>*2>|u|\<\|\|\>>>. Hence, it suffices to prove **\*\|)>*\|)>|\|>>\\>*2>>. But **\*\|)>*\|)>|\|>>\\|)>**\*\|)>*\|)>|\|>>>\|)>***\*\|)>*\|)>|)>|\|>>>>>|||)>***\*\|)>*\|)>|\<\|\|\>>>\*\|)>*max\:|\|>\2*d>|\>*\**\*\|)>|\<\|\|\>>>|)>|)>>.>>>>> By assumption, > is smooth away from 0, and > and all its partial derivatives decay faster than any rational function at infinity. Thus, there exists ,\|)>\0> such that <\equation*> sup1>|)>+2*d>*>*\|\|>\C for all \> with |\|>\2*d>. Since |)>> is bounded away from , there exists a minimal \>, such that >*\\1> for all supp|)>>. Thus, we obtain <\equation*> **\*\|)>*\|)>|\|>>\C**\|)>**\|)>-2*d>\,\>*\|)>>\2>*\> for all j>. On the other side, we get for j> |\**\|)>*\*u|\<\|\|\>>>>>||**\|)>|)>|\<\|\|\>>>*|\*u|\<\|\|\>>>>\>2>|u|\<\|\|\>>>=*2|)>>*\>*2+\|)>>|u|\<\|\|\>>>>>||*2>|)>>*\>*2+\|)>>|u|\<\|\|\>>>\>\>*2+\|)>>|u|\<\|\|\>>>,>>>>> where we used that \0>. The estimate for L>> follows from the same arguments. <\remark> If the support of u> has a ``hole'' at 0, that is if there exists a ball > centered at 0 such that u> is supported outside of >, then the estimates of Lemma hold uniformly in \0> and not just for \>. This is an immediate consequence of the proof above. As an application, we derive the smoothing properties of the heat kernel generated by the fractional Laplacian. <\lemma> Let \>, let )>> be the fractional Laplacian with periodic boundary conditions on *\>, and let |)>0>> be the semigroup generated by )>>. Then for all 0>, >, \>, \0>, and > we have <\equation*> |P*u|\<\|\|\>>+\>\t/|)>>|u|\<\|\|\>>>*and|P*v|\<\|\|\>>>\t/|)>>|v|\<\|\|\>>>>. If u> is supported outside of a ball centered at 0, then these estimates are uniform in 0> and not just in >. <\proof> The semigroup is given by =\*|)>>*\|)>> with =e>>>. Now > and its derivatives decay faster than any rational function at >. For \1>, \> is the density of a symmetric >-stable random variable, and therefore in >. For \1> it is easily shown that \|)>\> is bounded, and therefore in >. Thus, the estimates follow from Lemma. <\lemma> Let > and |)>0>> be as in Lemma. Let \>, \>, and let >>. Then we have for all 0> <\equation*> |-Id|)>*u|\<\|\|\>>>>\t/|)>>|u|\<\|\|\>>>. <\proof> For the uniform estimate of -Id|)>*u>, we write -Id> as convolution operator: if =e>>> and =*\>, then -Id|)>*u|\|>>||)>>*K|)>>>|)>*-u|)>*\*y|\|>>>||t|)>>*K|)>>>|)>*>|u|\<\|\|\>>>*\*y\t/|)>>|u|\<\|\|\>>>,>>>>> where we identified >> with the space of Hölder continuous functions. Based on Lemma and Lemma, we derive the following Schauder estimates: <\lemma> Let > and |)>0>> be as in Lemma. Assume that C>> for some \> and 0>. Letting =P*v*\*s>, we have <\equation> t>|V|\<\|\|\>>+2*\>\sup>>|v|\<\|\|\>>>|)> for all \> and all >. If \+2*\\1>, then we also have <\equation> |V|\<\|\|\>>+2|)>/|)>>*L>>\sup>|v|\<\|\|\>>>. <\proof> Consider *V> for some 0> and let \>. We decompose the integral into two parts: <\equation*> \*V=P**v|)>*\*s=>P**v|)>**\*s+>P**v|)>**\*s. Letting >>|v|\<\|\|\>>>|)>>, we estimate the first term by >P**v|)>**\*s|\|>>>>|>2>*|v*|\<\|\|\>>>*\*s\2>*M*>>*\*s>>||>*t>*/t>*s|>>\M*2>*t>*\,>>>>> using /t|)>>|\|>\\/t> in the last step. On the other side, we can use Lemma to estimate the second term for \0> by |>P**v|)>**\*s|\|>>>\>s>*2+2*\*|)>|)>>*|v*|\<\|\|\>>>*\*s>>||\M*2+2*\*|)>|)>>*>*s|s>*>>=M*2+2*\*|)>|)>>*t-\>*/t>*s|s>*>>>>||\M*2+2*\*|)>|)>>*t>*\>=M*2+2*\|)>>*>*\|)>>*t>.>>>>> If >\t/2>, we can take =2>> to obtain |\*V|\<\|\|\>>>>\M*t>*2+2*\|)>>>. If >\t/2>, we have |\*V|\<\|\|\>>>>\M*2>*t>\M*t>*2+2*\|)>>>, and the first claim follows. As for the second claim, note that for s\t\T> we have <\equation*> V-V=-Id|)>*V+P*v*\*r, and therefore we can apply Lemma to obtain V>|\<\|\|\>>>\|-Id|)>*V|\<\|\|\>>>>+|P*v|\<\|\|\>>>>*\*r>>||+2|)>/|)>>|V|\<\|\|\>>+2>+|v*|>\*r\+2|)>/|)>>*sup>|\<\|\|\>>*v|\<\|\|\>>>,>>>>> where we used that +2|)>\> and that \T>. This yields the second claim. When dealing with s, the convolution with the (fractional) heat kernel has a natural correspondence in the integral map. <\lemma> Let -1>|)>> for some \>. Then there exists a unique >|)>> such that *U=u> and =0>. This antiderivative satisfies <\equation> -U|\|>\>|u|\<\|\|\>>-1> for all > with \1>.We will use the notation =u*\*s> to denote this map, which is an extension of the usual definite integral. If the support of is contained in for some 0>, then >> and <\equation*> |U|\<\|\|\>>>\T|u|\<\|\|\>>-1>. <\proof> The second statement about compactly supported follows from the first statement by identifying >> with the space of bounded Hölder continuous functions. As for the first statement, we define <\equation*> U=-1>\*u*\*s. If we can show, then is indeed in >> and therefore in particular in >. Since the derivative > is a continuous operator on >, we then conclude that *U=\*u=u>. Let therefore > with \1>. We have <\equation*> \*u*\*r|\|>\2|)>>|u|\<\|\|\>>-1>*. If 0>, then *u=\*\**u|)>>, where > is the Fourier multiplier with symbol *z|)>>, and therefore <\equation*> \*u*\*r|\|>=*\*u-\*\*u|\|>\2*|\*u|\<\|\|\>>>>\2>|u|\<\|\|\>>-1>, where we used the Bernstein inequality, Lemma. If > is such that >\\2+1>>, then we use the first estimate for j> and the second estimate for j>, and obtain -U|\|>>|-1>\*u*\*r|\|>\j>2|)>>|u|\<\|\|\>>-1>*+j>2>|u|\<\|\|\>>-1>>>||*|)>>*+2*\>|)>|u|\<\|\|\>>-1>\>|u|\<\|\|\>>-1>.>>>>> Uniqueness is easy since every distribution with zero derivative is a constant function. When applying the scaling argument to solve equations, we need to control the resonant product of the rescaled data. This can be done by relying on the following commutator estimate. <\lemma> Let ,\\> and >. Then we have uniformly in \> <\equation*> |\>*g|)>->*f|)>\>*g|)>|\<\|\|\>>+\>\max +\>,1|}>|f|\<\|\|\>>>|g|\<\|\|\>>>, and thus >*\\|)>->\|)>\>\|)>> extends to a bounded bilinear operator from >\>> to +\>>. <\proof> We have >*\=\>*\|)>=\**\|)>*\>> for all -1>. Let > and \> be such that =\*2>. Then <\equation> \>*g|)>=\1>>|k>>>>>>\>**f*\*g|)>+\1>>|k>>>>>>\**\*\|)>*\>*f*\**\*\|)>*\>*g. The first sum is spectrally supported in a ball centered at zero (which is independent of ), and therefore <\equation*> \1>>|k>>>>>>\>**f*\*g|)>+\>\\1>>|k>>>>>>2-j*\>|f|\<\|\|\>>>|g|\<\|\|\>>>\max +\>,1|}>|f|\<\|\|\>>>|g|\<\|\|\>>>. The second sum is the resonant paraproduct >*f*|~>*\>*g|)>> with respect to the dyadic partition of unity *\|)>,\(\\>, except that the sum only starts in . By Lemma we can therefore bound <\equation*> \1>>|k>>>>>>\**\*\|)>*\>*f*\**\*\|)>*\>*g->*f|)>\>*g|)>+\>\|f|\<\|\|\>>>|g|\<\|\|\>>>. Next, we prove that it is possible to ``pull the time integral inside the paraproduct'': <\lemma> Let ,\\> with +\\1>. Let >,n>|)>> and >,|)>>. Then <\equation*> \*v|)>*\*r-u*-v|)>|\|>\+\>|u|\<\|\|\>>>|v|\<\|\|\>>>, for all > with \1>, where we write f*\*r=f*\*r-f*\*r>. <\proof> Fix > with \1>. We can rewrite <\equation*> \*v|)>*\*r-u*-v|)>=*u-u|]>*\*\*v*\*r. We will use two different estimates, one for large and one for small . First note that *u-u|]>*\*\*v*\*r|\|>>|*u-S*u|]>*\*\*v*\*r|\|>>>||+*u-u|]>*\*\*v*\*r|\|>.>>>>> Now *u-S*u|\|>\>|u|\<\|\|\>>>>, and therefore *u-u|]>*\*\*v*\*r|\|>>|>*2|)>>*\*r+2>*2|)>>*\*r|)>|u|\<\|\|\>>>|v|\<\|\|\>>>>>|||)>>*>+2-\|)>>*|)>|u|\<\|\|\>>>|v|\<\|\|\>>>.>>>>> On the other side, it follows from integration by parts that *u-u|]>*\*\*v*\*r|\|>>|*u-S*u|]>*\*\*v*\*r|\|>>>|>|+*u-u|]>*\*\*v*\*r|\|>>>|>|*u-S*u|)>*\*v|\|>+\*S*u*\*v*\*r|\|>>>|>|+*u-u|)>**v-\*v|)>|\|>>>|>|>*2>++\-\>*2>+2+\|)>>|)>|u|\<\|\|\>>>|v|\<\|\|\>>>,>>>>> for all \+\|)>>, where for the middle term we applied Lemma, which gives us \*S*u*\*v*\*r|\|>>|+\-\>*|\*S*u*\*v|\<\|\|\>>+\-\-1>>>||+\-\>*2+\-\-1|)>>*|\*S*u*\*v|\<\|\|\>>>>>>||+\-\>*2>|u|\<\|\|\>>>|v|\<\|\|\>>>.>>>>> Let now \> be such that >\\2+1>>. We use estimate for j> and for j> to obtain |\*v|)>*\*r-u*-v|)>|\|>\j>|)>>*>+2-\|)>>*|)>|u|\<\|\|\>>>|v|\<\|\|\>>>>>||+j>>*2>++\-\>*2>+2+\|)>>|)>|u|\<\|\|\>>>|v|\<\|\|\>>>>>||\|u|\<\|\|\>>>|v|\<\|\|\>>>*+\>,>>>>> where we used that +\\1>. <\bibliography|bib|alpha|bib/all> <\bib-list|FGGR12> Hajer Bahouri, Jean-Yves Chemin, and Raphael Danchin. . Springer, 2011. Zdzisªaw Brze¹niak, Massimiliano Gubinelli, and Misha Neklyudov. Global evolution of random vortex filament equation. , 26(9):2499, 2013. Hakima Bessaih, Massimiliano Gubinelli, and Francesco Russo. The evolution of a random vortex filament. , 33(5):1825--1855, 2005. Jean-Michel Bony. Calcul symbolique et propagation des singularites pour les équations aux dérivées partielles non linéaires. , 14:209--246, 1981. Rémi Catellier and Khalil Chouk. Paracontrolled distributions and the 3-dimensional stochastic quantization equation. , 2013. Michael Caruana and Peter Friz. 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Hölder-Continuous Rough Paths by Fourier Normal Ordering. , 298(1):1--36, 2010. > <\references> <\collection> > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > <\auxiliary> <\collection> <\associate|bib> Gubinelli2004 Bony1981 Lyons1998 Catellier2013 Gubinelli-KPZ-2013 Unterberger2010a Unterberger2010 Nualart2011 Perkowski2014Thesis Gubinelli2013 Lyons1998 Lyons2002 Lyons2007 Friz2010 Gubinelli2006 Gubinelli2012a Deya2012 Caruana2009 Caruana2011 Friz2011b Diehl2012 Teichmann2011 Bessaih2005 Brzezniak2010 Hairer2011 Hairer2013 Hairer2013b Hairer2012 Hairer2013 Hairer2012 Hairer2013b Kardar1986 Chouk2013 Hairer2013a Bony1981 Bony1981 Bahouri2011 Bahouri2011 Bony1981 Bony1981 Triebel2006 Friz2010 Janson1997 Hairer2011 Friz2012 Janson1997 Perkowski2014Thesis Janson1997 Bahouri2011 Bahouri2011 Bahouri2011 Bahouri2011 Schmeisser1987 Bahouri2011 Chemin2006 Bahouri2011 <\associate|toc> |math-font-series||1Introduction> |.>>>>|> |Relevant literature. |.>>>>|> > |Alternative approaches. |.>>>>|> > |Plan of the paper. |.>>>>|> > |Acknowledgments. |.>>>>|> > |Notation and conventions. |.>>>>|> > |math-font-series||2Paracontrolled calculus> |.>>>>|> |2.1Bony's paraproduct |.>>>>|> > |2.2Paracontrolled distributions and RDEs |.>>>>|> > |2.3Commutator estimates and paralinearization |.>>>>|> > |math-font-series||3Rough differential equations> |.>>>>|> |3.1Interpreting our RDE solutions |.>>>>|> > |3.2Alternative approach |.>>>>|> > |3.3Structure of solutions to RDEs |.>>>>|> > |The stable form of an RDE. |.>>>>|> > |Rough paths as a transformation group. |.>>>>|> > |Geometric conditions. |.>>>>|> > |3.4Connections to rough paths and existence of the area |.>>>>|> > |math-font-series||4Rough Burgers equation> |.>>>>|> |4.1Construction of the area |.>>>>|> > |math-font-series||5A generalized parabolic Anderson model> |.>>>>|> |5.1Renormalization |.>>>>|> > |5.2Regularity of the area and renormalized products |.>>>>|> > |math-font-series||6Besov spaces and paraproducts> |.>>>>|> |6.1Littlewood-Paley theory and Besov spaces |.>>>>|> > |6.2Linear operators acting on Besov spaces |.>>>>|> > |math-font-series||7Some more commutator estimates> |.>>>>|> |math-font-series||Bibliography> |.>>>>|>