The support of singular stochastic PDEs

Abstract

We obtain a generalisation of the Stroock-Varadhan support theorem for a large class of systems of subcritical singular stochastic PDEs driven by a noise that is either white or approximately self-similar. The main problem that we face is the presence of renormalisation. In particular, it may happen in general that different renormalisation procedures yield solutions with different supports. One of the main steps in our construction is the identification of a subgroup $\mathcal{H}$ of the renormalisation group such that any renormalisation procedure determines a unique coset $g\circ \mathcal{H}$. The support of the solution then only depends on this coset and is obtained by taking the closure of all solutions obtained by replacing the driving noises by smooth functions in the equation that is renormalised by some element of $g\circ \mathcal{H}$. One immediate corollary of our results is that the ${\Phi }_3^4$ measure in finite volume has full support and that the associated Langevin dynamic is exponentially ergodic.

Accepted version

References in corpus (2)

• Geometric stochastic heat equations
• Malliavin Calculus and Density for Singular Stochastic Partial Differential Equations

Cited by in corpus (6)

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• The ${\Phi }_3^4$ measure has sub-Gaussian tails
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• Characterization of the support for Wick powers of the additive stochastic heat equation
• Synchronisation by noise for the stochastic quantisation equation in dimensions $2$ and $3$