> <\body> <\hide-preamble> >> >> >> >> >> >> >> >> >> >> >> >> <\wide-tabular> ||||||||||| >>> > |||<\author-affiliation> Univ Brest, CNRS UMR 6205, Laboratoire de Mathématiques de Bretagne Atlantique, France >>|||<\author-affiliation> School of Mathematics, University of Edinburgh, United Kingdom >>|||<\author-affiliation> Mathematical Institute, University of Oxford, United Kingdom >>|>> We introduce Wilson\UItô diffusions, a class of random fields on > that change continuously along a scale parameter via a Markovian dynamics with coefficients. Described via forward-backward stochastic differential equations, their observables naturally form a pre-factorization algebra Costello--Gwilliam. We argue that this is a new non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation. Whenever a path-integral is available, this approach reproduces the setting of Wilson--Polchinski flow equations. > The Kadanoff\UWilson point of view on quantum field theory is a central idea in modern physics. It dictates that one should develop the fluctuation of quantum or statistical fields along a scale decomposition associated to the tunable precision of the observation device. In this framework a random Euclidean field is described by a stochastic process |)>0>> where parametrizes a scale of observation with characteristic length and > is the corresponding observation of the field, which must be thought of as containing fluctuations with spatial scales 1/a>. It is natural to assume that, as we gradually increase the resolution of our measuring devices, the resulting measurements vary continuously. Therefore we will postulate that the stochastic process > is a continuous function of . It is also reasonable to assume that > contains all the information gathered by observations with lesser precisions, which implies that it is a Markov process along the scale parameter, and we assume further that \\>> as \>. We denote by =|)>0>> the filtration generated by |)>0>> and by \|]>> the operator of conditional expectation given >, for any 0>, with =\> the full expectation. We assume that =0> whenever the fields take value in a vector space; otherwise we take > to be a fixed default (classical) configuration (e.g. a background field). We say that a functional \F|)>> of the field > is supported in \> if |)>=F|)>> for any field > such that =\> in . For \0> let >\\:d\\|}>> be the >-enlargement of . <\definition> An is a stochastic process |)>0>> which is an >-martingale, i.e. <\equation*> O=|]>,b\a\\|)>. An is said to be if there exists a family of functionals >|(>\0>> each supported on the >-enlargement >> and such that \O->|)>\0> as \>. We call such >> a >. A is field of observables O> such that the observable > is supported on the singleton > for all \>. The definition of the >-enlargement is a matter of convention. An observable that has a germ <\equation*> >|)>=\,\,\*\|)>, for some finite and some function >, is local. Germs serve to parametrize the space of local observables via concrete functionals of the fields. One owes to > the fundamental insight that continuous diffusions can be constructed via stochastic differential equations . As such, since we postulated that the scale description |)>> of a random field is a \ continuous Markov process, it is completely described via its infinitesimal rate of change <\equation*> \\=\*a>-\=B>\a+\M> which comes in two parts: a >*\a> and a part M>>. The drift component >> is the part of the rate of change which can be predicted from the large scale observations, while the \Pinnovation\Q process is a martingale w.r.t. this filtration, it models the additional information gained by augmenting the resolution. Recall a continuous martingale is described via its quadratic variation process |M|\>>. Enters >. We inject in this standard framework the spatial random structure of Euclidean quantum fields by imposing that the dynamics is completely described by \<#2018\>\<#2019\> coefficients. We do so by postulating that there exists a local observable field |)>0>> which models the > and that, inspired by the Kadanoff\UWilson block averaging procedure, the drift >> at scale is determined by an averaging of the force field > over a region of size . Therefore we introduce a \Pblock averaging\Q operator =C|)>>, which could be random and depend on >. We require > to have support on a ball of radius , that C|)>> varies smoothly for all >, that > be symmetric and positive definite and such that >=1>. For example, one can take <\equation> h|)>=a\a|)>h\y, where \> and > is a smooth, radially symmetric, positive definite function of unit integral. In particular note that =0> on all sufficiently nice functions . Since the drift will be integrated along the scales, the local averaging has to be done in such a way as to not over-count the contributions of the microscopic force. We denote by >\\C> the scale-derivative of this averaging and let >\>*f>. This fixes the previsible part of the stochastic dynamics as a function of the microscopic force. To complete our description we need to specify also the quadratic variation of the martingale part. Using the same principles we can assume\ <\equation*> \|M|\>\>\ >*\a for a microscopic positive \<#2018\>\<#2018\>diffusivity\Q |)>> which is a local observable field of positive scalars. In this way the local diffusion on scales is determined by the datum of two local observable fields ,\|)>> and a family of averaging operators |)>>. In particular, by standard results there exists, possibly on an extended probability space, a cylindrical >-Brownian motion\ <\equation*> \W|]>=a|)>*\, such that <\equation*> M=>*\*\W where \|)>>. <\definition> A is a continuous stochastic process |)>0>> taking values in the set of smooth functions on > with the following properties. <\enumerate-alpha> . There is an |)>0>> and |)>0>> such that |)>0>> is a Markovian Itô diffusion <\equation> \\=>f\a+>\\W. . The effective force and the effective diffusivity > are local observable fields. We call equation a (WIDE). A is the random field >> obtained as the terminal value of a Wilson\UItô diffusion |)>0>>. The main goal of this paper is to propose the hypothesis that Euclidean quantum field theories can be identified with Wilson\UItô fields. This provides a new framework, independent of the path-integral formalism, to study Euclidean quantum fields.\ <\enumerate-alpha> This description emerges from simple and natural assumptions and covers in principle much more than those theories that can be reached perturbatively from a Gaussian functional integral. The continuity of the process |)>> with respect to gives it the structure of an Itô diffusion. When > and > are deterministic (hence > is constant by the martingale property of >), and without any assumption of a perturbative regime, the Wilson\UItô random field >> comes with an associated Gaussian field <\equation*> X\>*\*\W, and a coupling to it. In that case, >> is a white noise <\equation*> \>X>|]>=\C>=\\. Therefore one expects that <\equation*> \>=X>+>>f\a is, in general, only a distribution of very low regularity. Eq. makes sense for fields defined on and/or taking values in manifolds or vector bundles and for which the path-integral formalism is less clear to apply. Our formalism is non-perturbative and trades the use of functional integrals against Itô calculus. Let > be a possibly random, adapted, increasing change of scale such that =0> and |)>=\>, and let \A\W>>, =C>>, \f>> and |~>\\>>. Then we have <\equation*> \>|)>=\ \a+|)> |~>\, which shows that Wilson\UItô diffusions are covariant wrt. random changes of spatial scales. In particular, this justifies that the diffusion has to be averaged with >>. Note that observables are also covariant: If |)>> is an observable for |)>> then >|)>> is an observable for the diffusion >|)>>. Only the law of the terminal value >> and the averages of observables are the physical content of a Wilson\UItô diffusion. \U The law of the process |)>0>> is determined by a >: for all sufficiently nice scale-dependent test functions |)>>, the process <\equation*> M\F|)>-F|)>*|)>\b is an >-martingale where <\equation> F|)>>|>|F|)>+\F|)>*>f >>|||Tr >\>\F|)>|]>>>>>> is the of the Wilson\UItô diffusion. Here we denote by \\/\\> the gradient of a functional \F|)>> with respect to the field > taken wrt. the > norm and by the trace operator. Some observables can be obtained via the conditional expectation of a function of >>, i.e. let =\>|)>|]>>. This kind of martingale is called , i.e. it admits a \Pterminal value\Q >=F>|)>> from which it can be reconstructed. In the case when is a linear functional we have <\equation*> O=\>|)>|]>=F>|]>|)>, assuming suitable integrability conditions here and below. Using the Wilson\UItô dynamics, and since |)>> is a martingale, we have <\equation> >|]>>||+\>>f*\b=\+>>f\b>>|||+C,a>f>>>>> where \ ,a>\C>-C>. In general we are not allowed to form non-linear functions >|)>> of the distribution >>: the locality condition for an observable is non-trivial and usually requires renormalization. As a consequence, we do not expect non-linear local observables to be closed martingales. Ignoring for the moment this difficulty, we note that when >|)>> is well-defined the observables |)>> are closed martingales and they satisfy a (BSDEs) <\equation*> \O=Z\W, for a pair of adapted processes ,Z|)>> with condition >=F>|)>>. A procedure to construct local observables starts with some approximate local observable given by a function >>>|)>> localized at scale > and setting <\equation*> O>\>>>|)>|]>. One can then study the convergence of the family of the observables >|)>a\a>> as \\>. Provided the functions >>>|)>> contains appropriate (diverging) renormalizations one is able to show that the observables >> converge to a local observable as \\>.\ This approach leads naturally to the analysis of a general class of (FBSDEs) of the form <\equation> \\=>\a+>|~>*\W for some approximately local functionals <\equation*> =>|)>|]>,|~>=>|)>|]>. Recall >> may depend on > in a general setting. Note also that we use the letter > for these \Papproximate\Q dynamics while we use > for a local dynamics. These FBSDE are not proper Wilson\UItô diffusions, since their coefficients are non-local. BSDEs and FBSDEs are well studied in the mathematical literature (see e.g.) and we dispose also of numerical methods to approximate their solutions. In relation to the numerical aspects, note that the formalism allows us to replace > by a finite discrete lattice \\|)>> with mesh >. To give a first example consider the case of an approximate force with a linear component and a constant diffusivity <\equation*> F>|)>=>>>+h>|)>|)>,\>|)>=>>, for some positive constant >, some positive linear operator and some additional force component >|)>>. For a local operator the linear functional >> is always well-defined in the space of distributions, so it defines a local force field. We assume here that the averaging operators > are deterministic, field-independent and commute with . Think of the case where -\>, and > is given by Eq.. Using we have <\equation*> \>|]>=-\ C,a>A\>|]>+\ C,a>\>|)>|]>> Solving for >|]>> and letting <\equation*> \\ C,a>A|)>\, we have >=\>> and <\equation> \\=>>|)>|]>a+ >>\W, where >\>C,a>A|)>|)>>. This computation shows that the linear component in the force can always be integrated and gives rise to a modified FBSDE where the local averaging operator >> has been replaced by the operator >>. Note in particular that the Gaussian field <\equation> X\>\W, has covariance >> and that >> has covariance *A|)>>. One gets back the operator > in the large > limit, in which case >> is a massive GFF in the model situation. We can invert the transformation and go from a FBSDE of the form back to the FBSDE noting that <\equation*> Q,a>-\C,a>=A for all 0>. Assume further that is a symmetric operator and the additional force component is given by an effective UV-regularized potential >> as <\equation*> h>|)>=-\V>>|)>, and let <\equation*> V|)>\-log \>+X>-X|)>>|]>. Then |)>> is the solution of the Polchinski flow equation <\equation> \V-\V>\V+>\V=0 with terminal condition >> for \>. Letting \exp|)>-V|)>> and using Itô formula, we have that > is a positive martingale with =1> and moreover that, under the probability measure > defined on >> by \\Z\\>, the process > is a martingale with deterministic quadratic variation |\|\>=>\a>, so >> is under > a GFF. Note also that it follows from that |)>\-\V|)>> is a martingale under >. Recalling Eq., we conclude that <\equation> >|)>|]>>||>|)>e-V|)>>|]>>>|||>|)>e-V|)>>|]>>>|||>|)>e>>|)>>|]>|\>>>|)>>|]>>>>>>> for any function and any 0>. Here we used that |)>>> is a martingale. This implies that the law >> of the random field >=\>> is given by <\equation> \>\|)>=>|)>>\>>\|)>|e>|)>>\>>\|)>> where >>> denotes the Gaussian of covariance ,0>>. \; This shows that the class of Wilson\UItô fields comprise as a particular case the Euclidean quantum fields constructed as perturbations of a Gaussian field. They are obtained by solving FBSDEs of the form <\equation> \\=->*V>>|)>|]>*\a+>\W. which we call >, since they describe the Polchinski semigroup. Even in this potential framework where there is a formal link with the well-known Polchinski flow equation, the FBSDE does not require the a priori knowledge of the solution of Eq.. Indeed we only used it to derive the formal connection while in Eq. only the boundary condition *V>> is needed.In general this quantity needs an ultraviolet regularization that has to be tuned in order for the FBSDE to reach a well-defined local limit as the regularization is removed. Take and let <\equation*> F>|)>=-|)>\>-\v>|)> \>, where > is the derivative of a function \\> smooth and bounded from below and :\\\> is a compactly supported function of the space variable (an IR cutoff). The formulation with > reads <\equation> \\=->\\v>|)>\a+>\W. From this equation we see that >> is actually comparable in regularity to the massive GFF >> with covariance >, i.e. <\equation*> \>X>|]>=e>,x,y\\, which is a Hölder continuous function for which the mild formulation makes sense. The residual force is given by the well-defined potential >|)>=>\ v|)>\x>. By we have <\equation> \>>|)>|]>=>>|)>e>>|)>>|]>|\>>>|)>>|]>> and this shows that >> is a (time-inhomogeneous) Markovian diffusion in the space parameter since >> can be described also as the Ornstein\UUhlenbeck process, solution of the stochastic differential equation <\equation*> \X>=-X>\x+\B, where |)>\>> is a two-sided Brownian motion. Note that this is a Markov process in the space variable. Moreover if we take \1>, under suitable assumptions, the dynamics will converge to a time-homogeneous space-Markovian diffusion described by the WIDE (recall that >=\>>) <\equation*> \\=>\|)>\>-v>|)>|]>\a+>\W, along the scale variable 0>, and, at the same time, it is also described by the stochastic differential equation <\equation*> \\>=\>log \>|)>\x+\B, along the space variable \>, where :\\\> is the ground state of the Hamiltonian with potential /2+v|)>> (see e.g.). In absence of the IR cutoff >, the random field >> is not absolutely continuous wrt. >> and therefore the path-integral formulation loses its meaning while the WIDE formulation remains valid.\ This example shows that a local gradient Wilson\UItô dynamics gives rise to a Markov process in the space variable. It is natural to conjecture that this is a general feature of (a wide class of) higher-dimensional Wilson\UItô diffusions. Eq. can be interpreted as the Euler\ULagrange equation for a stochastic control problem. To derive this problem we test the process <\equation*> u\->*V>>|)>|]> with an adapted test field |)>> and integrate both in scale and in the probability space to get the equation <\equation> \>|v,u|\>\a+|>>v\a,\V>>|)>|\>|]>=0 Eq. is the first order condition for the problem of minimizing the functional <\equation> \\\>>|)>+>|u,u|\>\a|]> over all adapted controls |)>0>>, where <\equation*> \\>u\b+>*\W, is the controlled process. Variants of Eq., Eq. or of the variational problem in Eq. have been used to construct several Euclidean quantum fields including the > and > models, the Høegh-Krohn model, the Sine\UGordon model and certain subcritical Euclidean fermionic field theories. This shows that our approach is intrinsically non-perturbative and can be made rigorous. We invite the reader to compare this situation with the non-trivial mathematical difficulties of the path-integral formalism without cutoffs. As an example, take the > Euclidean quantum field on a torus, constructed in via a slightly different version of the variational formulation. It is known, and proven in, that the > measure is not absolutely continuous with respect to the Gaussian free field, so there cannot be a rigorous path-integral for it. Similarly, in it is shown that some variants of Eq. provide effective tools to study the infinite volume limit of the > and of the |)>> Euclidean fields. Let >> be a germ for an observable . For b\a>, one has by Itô's formula <\equation*> >||>|)>+R=*>|)>+R|]>>>|||>|)>+>|)>*\c|]>+|]>>>>>> where > is the generator of the Wilson\UItô diffusion. The assumption that > goes to in a strong enough sense gives <\equation> R=>>|)>*\c|]>. Therefore >, and hence itself, is completely determined by the germ >> provided the integral <\equation*> >|2>|>|)>|]>||2>>\c converges absolutely. A germ which has this property is called a ; its associated observable is determined from it. Note that Eq. for the remainder is equivalent to the BSDE\ <\equation> \R=->|)>*\a-Z\W,R>=0 for the pair of adapted processes ,Z|)>>. \; The effective force itself is an observable. For simplicity we assume that the diffusivity is taken constant =1>, similar considerations otherwise apply to it. Assume that the force has a germ >|)>> and a remainder > which then, due to, satisfies <\equation> >||>**>*\>|)>|]>*\c>>|||>|\>*>|)>|]>*\c>>>>> where we introduced the operator <\equation*> |\>\\+>*>*\+*Tr > \. Similarly to the general case of Eq., Eq. gives rise to a BSDEs for the pair ,Z|)>> which reads <\equation> R>|||\>>|)>*\a-R*>*\>|)>*\a>>|||\W,>>|>>||>>>> A basic requirement for Eq. is that the source term in and is convergent in the UV, i.e. \ <\equation> >>|2>|>|\> *>|)>|]>||2>>\c\\, for some scale >. If equation has indeed a unique solution then >> characterizes uniquely the force field . Assume now we are given an observable which has a germ >>. Under proper assumptions on >>, and for a choice of >> that ensure an appropriate strong decay of > as \>, the coherence relation <\equation*> >|>|)>|]>|>\c\\ is a consequence of the relation <\equation*> >||\>*>|)>|]>|>\c\\. A family >> that satisfies the estimate is called an >. In those terms, Eq. states that >> is an approximately coherent germ for the force field . These considerations lead to the following strategy for the law of a random field >. Associate to each force germ >|)>> and each scale parameter \0> the solution to the FBSDEs <\equation> \>>||>|\>>|)>+R>|)>\b+>\W,>>|>>||>*>*>\|\>>|)>|]>\c >>|||>|\> *|\>|)>>|)>|]>*\c>>>>> for b\a> with mixed initial/final conditions <\equation> \>=0,R>>=0. Now we need to find |\>|)>0>> such that the coupled system has a unique solution for all > and >-uniform estimates that entail sufficient compactness to pass to the limit in . In this strategy has been used to construct some subcritical Gibbs measures on Grassmann fields. Definition gives rise only to a vector space structure on the set of observables. Since the product of two martingales is, generally speaking, not a martingale, we do not have a natural way of multiplying observables. However for two observables ,O> that have some coherent germs >,>> with disjoint supports, say they are at distance strictly larger than >, we can set for a> <\equation*> >>\>> >>. Since >\>*\>>\>>=0>, for a>, because of the support condition, one has <\equation*> |\>*>>=|\>*>>|)>>>+>>|\>*>>|)> therefore provided the germs are approximately coherent and we impose sufficient decay on |\>*>>> and moderate growth conditions on >>>, it is possible to guarantee that |\>*>>> is integrable and therefore >>> is itself an approximately coherent germ which defines a unique observable >\O>\O>>. This gives on a subspace of observables a natural pre-factorization algebra structure, as defined in Costello\UGwilliam. In this section we assume that the field > is a connection on a principal bundle over >, with finite dimensional compact structure group and Lie algebra >. Recall that the space of connections is affine with >-valued -forms as underlying vector space. The gauge group >consists of -valued functions and acts on connections by \=Ad\-g|)> g> and forms by f=Adf>. We assume moreover that the force field is a gauge covariant function of the underlying field =f|)>> (taking values in the space of -forms) : <\equation*> g\f|)>=f*\|)>, for every \> and connection >. Given a connection >, denote by |)>> the >-holonomy along the geodesic from to . Recall that <\equation> h*\|)>=g*h|)>*g. Let > be a symmetric function of >. We define a map >|)>> acting on -forms by <\equation*> >|)>*\|)>\>*\|)>>\\y. The operator >> is symmetric and >|)>\>|)>|)>> is symmetric and non-negative. By , it is also gauge covariant: <\equation*> g\>|)>*\|)>=>*\|)>*\|)>. The tangent space of the gauge orbit at a given connection > is spanned by the elements of the form >h>, where >> is the >-covariant derivative and an arbitrary (smooth enough) >-valued -form, <\equation*> \>h=h+,h|]>|)>\x. Two scale-dependent families of connections |)>0>> and |)>0>> are said to be if there exists a scale-dependent family of gauge transforms |)>0>> such that =g\\> for all 0>. We note that if |)>0>> is a solution of the Wilson\UItô equation with \1>, and |)>0>> is an adapted process that is differentiable in and takes values in |)>> then \g\\> satisfies the equation <\equation> \>||\\|)>-\>>g\a|)>>>|||>*f|)>-\>>*g|)>|)>*\a >>|||>|)>\W.>>>>> This is a particular case of a more general situation. <\proposition> For any adapted process |)>0>> with values in (M,> the solution >|)>0>> to the equation <\equation*> \\>=>*f|)>>|)>+\>>h|)>*\a+>>|)>*\W is gauge equivalent to the solution of a Wilson\UItô equation <\equation> \\>=>*f|)>>|)>*\a+>>|)>\W driven by another Brownian motion >. So the law of the gauge orbit of >|)>0>> is well-defined. The proof proceeds by solving the ordinary differential equation >*g=h> and remarking that >\g\\*>> solves with W:=g\\W>, which is a Brownian motion by Itô isometry.\ As Parisi\UWu stochastic quantization scheme this approach allows to quantize gauge theories without path integrals and their associated ghosts or BRS symmetries. Adding terms of the form >h> in the drift allows us to perform gauge-fixing in analogy to the Zwanziger\UDeTurck\USadun trick. Moreover it points to a covariant formulation of the flow equation for the effective force > <\equation> \f+f>\f+Tr >\f=0 with terminal condition >>. This is similar to Polchinski equation with the difference that the cutoff propagator has been replaced by a covariant averaging operator. We introduced Wilson\UItô fields, a novel class of random fields described by \Plocal interactions\Q. Our point of view is inscribed in the general idea of > initiated by Parisi\UWu, which replaces the use of functional integrals with stochastic partial differential equations for the Euclidean fields. This topic has recently witnessed a renewed interest since SPDEs can be used to study rigorously Euclidean fields: see e.g. for non-Abelian gauge theories, for > on Riemmanian manifolds and also for a partial verification of the Osterwalder\USchrader axioms in >. However, at variance with classical stochastic quantization methods, our approach does not require to introduce fictious additional parameters and depends on a physically relevant scale of observation, motivated by the Kadanoff\UWilson picture of renormalization and by the mathematical framework for Euclidean QFT via effective theories, see e.g.. Wilson\UItô fields have a natural intrinsic definition, independent of cutoff procedures and it seems interesting to pursue further their study, e.g. investigate spatial Markov properties, reflection positivity, uniqueness/non-uniqueness of solutions, numerical approximations, description of scale-invariant and conformal invariant fields. <\bibliography|bib|plain|apssamp> <\bib-list|10> I.Bailleul. Uniqueness of the > measures on closed Riemannian -manifolds. , 2023. I.Bailleul, N.V. Dang, L.Ferdinand, and T.D. Tô. > measures on compact Riemannian -manifolds. , 2023. Nikolay Barashkov. A stochastic control approach to Sine Gordon EQFT. , 2022. Nikolay Barashkov and Massimiliano Gubinelli. A variational method for >. , 169(17):3339\U3415, 2020. Nikolay Barashkov and Massimiliano Gubinelli. On the variational method for euclidean quantum fields in infinite volume. , 2021. Nikolay Barashkov and Massimiliano Gubinelli. The > measure via Girsanov's theorem. , 26:1\U29, 2021. Roland Bauerschmidt and Thierry Bodineau. Log-Sobolev Inequality for the Continuum Sine-Gordon Model. , 74(10):2064\U2113, oct 2021. Roland Bauerschmidt, Thierry Bodineau, and Benoit Dagallier. Stochastic dynamics and the Polchinski equation: an introduction. , 2023. Ajay Chandra, Ilya Chevyrev, Martin Hairer, and Hao Shen. Langevin dynamic for the 2D Yang-Mills measure. , 2020. Ajay Chandra, Ilya Chevyrev, Martin Hairer, and Hao Shen. Stochastic quantisation of Yang-Mills-Higgs in 3D. , 2022. Ilya Chevyrev and Hao Shen. Invariant measure and universality of the 2D Yang-Mills Langevin dynamic. , 2023. Kevin Costello. , volume 170 of American Mathematical Society (AMS), Providence, RI, 2011. Kevin Costello and Owen Gwilliam. , volume31 of . Cambridge University Press, Cambridge, 2017. Kevin Costello and Owen Gwilliam. , volume41 of . Cambridge University Press, Cambridge, 2021. PoulHenrik Damgaard and Helmuth Hüffel. . World Scientific, 1988. FrancescoC. DeVecchi, Luca Fresta, and Massimiliano Gubinelli. A stochastic analysis of subcritical Euclidean fermionic field theories. , 2022. Massimiliano Gubinelli and Martina Hofmanová. A PDE construction of the Euclidean > quantum field theory. , 384(1):1\U75, 2021. József Lörinczi, Fumio Hiroshima, and Volker Betz. . Number34 in De Gruyter Studies in Mathematics. De Gruyter, Berlin ; Boston, 2011. Jin Ma and Jiongmin Yong. , volume 1702 of . Springer Berlin Heidelberg, Berlin, Heidelberg, 2007. G.Parisi and YongShi Wu. Perturbation theory without gauge fixing. , 24(4):483\U496, 1981. Joseph Polchinski. Renormalization and effective lagrangians. , 231(2):269\U295, jan 1984. L.C.G. Rogers and David Williams. . Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itô calculus, Reprint of the second (1994) edition. Slava Rychkov. > 3 Dimensions>. Springer, New York, NY, 1st ed. 2017 edition edition, oct 2016. Manfred Salmhofer. . Springer, Berlin ; New York, 1st corrected ed. 1999, corr. 2nd printing 2007 edition edition, sep 2007. DanielW. Stroock. , volume 155 of . Princeton University Press, Princeton, NJ, 2003. <\initial> <\collection> <\attachments> <\collection> <\associate|bib-bibliography> <\db-entry|+2IYrABOMnOWx5H4|book|stroockMarkovProcessesIto2003> <|db-entry> > <\db-entry|+PZiR2YAVRDDqm5|book|Rogers_Williams_2000> <|db-entry> David > > <\db-entry|+1jiJLXG510aPbSj2|book|maForwardBackwardStochasticDifferential2007> <|db-entry> Jiongmin > F. B. > <\db-entry|+2IYrABOMnOWx5Gz|article|polchinskiRenormalizationEffectiveLagrangians1984> <|db-entry> > <\db-entry|+sOFEgcGU5i8zRA|article|bauerschmidtLogSobolevInequality2021> <|db-entry> Thierry > > <\db-entry|+2L6EqgXT2MwfImRa|article|bauerschmidtStochasticDynamicsPolchinski2023> <|db-entry> Thierry Benoit > <\db-entry|+1C0z6cZT2MqKBfUr|book|lorincziFeynmanKactypeTheoremsGibbs2011> <|db-entry> Fumio Volker > <\db-entry|+2L6EqgXT2MwfImRX|article|barashkov2021variational> <|db-entry> Massimiliano > <\db-entry|+eELqL1ldTO5|article|barashkov2020variational> <|db-entry> Massimiliano > >> > <\db-entry|+1eCKmEx11X0SioOj|article|barashkov2021phi43> <|db-entry> Massimiliano > > measure via Girsanov's theorem> > <\db-entry|+2L6EqgXT2MwfImRp|article|barashkovStochasticControlApproach2022> <|db-entry> > |^>>2}> 4 \\\\pi$ in infinite volume. We give a variatonal characterization of it's laplace transform, and deduce from this large deviations. Along the way we obtain estimates which are strong enough to obtain a proof of the Osterwalder-Schrader axioms including exponential decay of correlations as a byproduct. Our method is based on the Boue-Dupuis formula with an emphasis on the stochastic control structure of the problem.> <\db-entry|+2L6EqgXT2MwfImRn|article|de_vecchi22> <|db-entry> Luca Massimiliano > <\db-entry|+1beEMbHF1kEgFGqZ|book|costelloFactorizationAlgebrasQuantum2017a> <|db-entry> Owen > <\db-entry|+1beEMbHF1kEgFGqX|book|costelloFactorizationAlgebrasQuantum2021> <|db-entry> Owen > <\db-entry|+2ZScWHDSxgvc0EU|article|parisiPerturbationTheoryGauge1981> <|db-entry> Yong Shi > <\db-entry|+2IYrABOMnOWx5FB|book|damgaardStochasticQuantization1988> <|db-entry> Helmuth > <\db-entry|+2L6EqgXT2MwfImRf|article|chandraLangevinDynamic2D2020> <|db-entry> Ilya Martin Hao > <\db-entry|+2L6EqgXT2MwfImRi|article|chandraStochasticQuantisationYangMillsHiggs2022> <|db-entry> Ilya Martin Hao > <\db-entry|+2L6EqgXT2MwfImRl|article|chevyrevInvariantMeasureUniversality2023> <|db-entry> Hao > <\db-entry|+2L6EqgXT2MwfImRM|article|bailleulPhiMeasuresCompact2023> <|db-entry> N. V. L. T. D. > > measures on compact Riemannian -manifolds> <\db-entry|+2L6EqgXT2MwfImRP|article|bailleulUniquenessPhiMeasures2023> <|db-entry> > > measures on closed Riemannian -manifolds> <\db-entry|+2ZScWHDSxgvc0EZ|article|GH21> <|db-entry> Martina > > quantum field theory> > > > <\db-entry|+2IYrABOMnOWx5F9|book|costelloRenormalizationEffectiveField2011> <|db-entry> > <\db-entry|+2IYrABOMnOWx5H1|book|salmhoferRenormalizationIntroduction2007> <|db-entry> > <\db-entry|+299qzUHV213kGPT2|book|rychkovEPFLLecturesConformal2016> <|db-entry> > > 3 Dimensions> a procedure also known as the embedding formalism. In turn, Lecture 3 focuses on the radial quantization and the operator product expansion, while Lecture 4 offers a very brief introduction to the conformal bootstrap. Derived from course-based notes, these lectures are intended as a first point of entry to this topic for Master and PhD students alike.> <\references> <\collection> 2 > 23 > 21 > 24 > 20 > 20 > 18 > 2 > 3 > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > 1 > 26 > 7 > 6 > 5 > 17 > 19 > 11 > 25 > 3 > 9 > 10 > 4 > 1 > 12 > 13 > 8 > 16 > 15 > 14 > <\auxiliary> <\collection> <\associate|bib> stroockMarkovProcessesIto2003 Rogers_Williams_2000 maForwardBackwardStochasticDifferential2007 maForwardBackwardStochasticDifferential2007 polchinskiRenormalizationEffectiveLagrangians1984 bauerschmidtLogSobolevInequality2021 bauerschmidtStochasticDynamicsPolchinski2023 lorincziFeynmanKactypeTheoremsGibbs2011 barashkov2021variational barashkov2020variational barashkov2021phi43 barashkov2021variational barashkovStochasticControlApproach2022 de_vecchi22 barashkov2020variational barashkov2021phi43 barashkov2021variational de_vecchi22 costelloFactorizationAlgebrasQuantum2017a costelloFactorizationAlgebrasQuantum2021 parisiPerturbationTheoryGauge1981 damgaardStochasticQuantization1988 damgaardStochasticQuantization1988 parisiPerturbationTheoryGauge1981 chandraLangevinDynamic2D2020 chandraStochasticQuantisationYangMillsHiggs2022 chevyrevInvariantMeasureUniversality2023 bailleulPhiMeasuresCompact2023 bailleulUniquenessPhiMeasures2023 GH21 costelloRenormalizationEffectiveField2011 salmhoferRenormalizationIntroduction2007 rychkovEPFLLecturesConformal2016 <\associate|toc> I.Introduction |.>>>>|> II.Forward-backward stochastic differential equations |.>>>>|> |ALinear-like force |.>>>>|> > |BGradient diffusions |.>>>>|> > |COne-dimensional gradient diffusions |.>>>>|> > |DVariational formulation |.>>>>|> > |ERigorous results |.>>>>|> > III.Properties of Wilson\UItô diffusions |.>>>>|> |ACoherent germs |.>>>>|> > |BLink with factorization algebras |.>>>>|> > IV.Gauge theories |.>>>>|> V.Conclusions |.>>>>|> <\with|par-columns|> |||||||||||||||||||>|||||||||>>>> |.>>>>|>