Abstract
We develop in this work a general version of paracontrolled calculus that allows to treat analytically within this paradigm a whole class of singular partial differential equations with the same efficiency as regularity structures. This work deals with the analytic side of the story and offers a toolkit for the study of such equations, under the form of a number of continuity results for some operators, while emphasizing the simple and systematic mechanics of computations within paracontrolled calculus, via the introduction of two model operations$\mathsf{E}$and $\mathsf{F}$. We illustrate the efficiency of this elementary approach on the example of the generalized parabolic Anderson model equation$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701},\end{eqnarray}$$on a 3-dimensional closed manifold, and the generalized KPZ equation$$\begin{eqnarray}(\unicode[STIX]{x2202}_{t}+L)u=f(u)\unicode[STIX]{x1D701}+g(u)(\unicode[STIX]{x2202}u)^{2},\end{eqnarray}$$driven by a$(1+1)$-dimensional space/time white noise.
Highlights
Thirty years after Lyons’ seminal work on controlled differential equations [22], it is well-understood that the construction of a robust approximation theory for continuous time stochastic systems, such as stochastic differential equations or stochastic partial differential equations, requires a twist in the notion of noise that allows to treat the resolution of such equations in a two-step process
We develop in this work a high order version of paracontrolled calculus that allows to treat analytically within this paradigm some parabolic singular partial differential equations that are beyond the scope of the original formulation of the theory
The high order paracontrolled expansion formula and the continuity results stated in Sections 2 and 3 respectively, and fully proved in Appendices B and C, make use of these operators and provide the spine of paracontrolled calculus
Summary
Thirty years after Lyons’ seminal work on controlled differential equations [22], it is well-understood that the construction of a robust approximation theory for continuous time stochastic systems, such as stochastic differential equations or stochastic partial differential equations, requires a twist in the notion of noise that allows to treat the resolution of such equations in a two-step process. Following our previous work [2], one can define from L only parabolic paraproduct and resonant operators that have good continuity properties in the scale of parabolic Holder spaces – see Appendix A.3. The high order paracontrolled expansion formula and the continuity results stated in Sections 2 and 3 respectively, and fully proved in Appendices B and C, make use of these operators and provide the spine of paracontrolled calculus. They are the main contributions of this work. Appendices B and C contain the proofs of a number of statements
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