Antoine Lejay, Massimiliano Gubinelli and Samy Tindel
In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t$, where $X$ is a $\gamma$-Hölder continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $\gamma$ in $(0,1)$), and then when $B$ satisfies some Lipschitz type conditions (for $\gamma>1/2$). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.
Antoine Lejay, Massimiliano Gubinelli and Samy Tindel. Young integrals and SPDEs. Papers of Massimiliano Gubinelli, 2004, version 1. https://mgubi.org/papers/paper/young-integrals-spdes/
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