Abstract
Abstract We obtain a generalisation of the Stroock–Varadhan support theorem for a large class of systems of subcritical singular stochastic partial differential equations 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 depends only 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 $\Phi ^4_3$ measure in finite volume has full support, and the associated Langevin dynamic is exponentially ergodic.
Highlights
The purpose of this article is to provide a far-reaching generalisation of the support theorem of Stroock and Varadhan [SV72]
T−, and we choose the scales such that λ δ τ λ δ τwhenever τ, τ ∈ T− with τ ≤ τ. To formalise this idea, we introduce in Definition 5.14 the notion of an attainable statement, and we show at the end of Section 5.4 that the necessary bound of Lemma 5.17 is attainable in this sense
The discrepancy stems from the fact that we use truncated kernels to define gε.) we have identified that the set H ⊆ G− for which we can construct a shift as before is equal to the coset f ξ ◦ H
Summary
The purpose of this article is to provide a far-reaching generalisation of the support theorem of Stroock and Varadhan [SV72]. It is possible to emulate a decrease in the renormalisation constant c (but not an increase!) by adding a small (in a distributional sense) highly oscillatory term to h This suggests that the support of the solution to equation (1.5) is given by the closure of the set of all solutions to (1.8). As a matter of fact, by considering perturbations of h of the type (1.7), but with an additional modulation of the highly oscillatory term, we will see in Theorem 1.15 that whatever the choice of renormalisation procedure, solutions to equation (1.5) have full support, so that this example exhibits some weak form of ‘hypoellipticity’
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
