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Register a new userLocality for singular stochastic PDEs
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We develop in this note the tools of regularity structures to deal with singular stochastic PDEs that involve non-translation invariant differential operators. We describe in particular the renormalised equation for a very large class of spacetime dependent renormalization schemes.
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We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some operators are proved for that purpose. Unlike the regularity structures approach of the subject by Gerencser and Hairer, and Otto, Sauer, Smith and Weber, or Furlan and Gubinelli study of the two dimensional quasilinear parabolic Anderson model equation, we do not use parametrised families of models or paraproducts to set the scene. We use instead infinite dimensional paracontrolled structures that we introduce here.
We give a short essentially self-contained treatment of the fundamental analytic and algebraic features of regularity structures and its applications to the study of singular stochastic PDEs.
We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space of a space-time white noise, contains a bi-Laplacian in the drift. The lack of the maximum principle for the bi-Laplacian generates difficulties for the classical penalization method, which uses a crucial monotonicity property. Being inspired by the works of Debussche and Zambotti, we use a method based on infinite dimensional equations, approximation by regular equations and convergence of the approximated semi-group. We obtain existence and uniqueness of solution for nonnegative intial conditions, results on the invariant measures, and on the reflection measures.
We study the stochastic viscous nonlinear wave equations (SvNLW) on $mathbb T^2$, forced by a fractional derivative of the space-time white noise $xi$. In particular, we consider SvNLW with the singular additive forcing $D^frac{1}{2}xi$ such that solutions are expected to be merely distributions. By introducing an appropriate renormalization, we prove local well-posedness of SvNLW. By establishing an energy bound via a Yudovich-type argument, we also prove global well-posedness of the defocusing cubic SvNLW. Lastly, in the defocusing case, we prove almost sure global well-posedness of SvNLW with respect to certain Gaussian random initial data.
We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $Phi^4_3$ model. As a corollary, we prove that the Brownian bridge measure is the unique invariant measure for the KPZ equation with periodic boundary conditions.
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