The support of singular stochastic PDEs
File(s)
Accepted version
Author(s)
Hairer, Martin
Schönbauer, Philipp
Type
Working Paper
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 $\Phi^4_3$ measure in
finite volume has full support and that the associated Langevin dynamic is
exponentially ergodic.
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 $\Phi^4_3$ measure in
finite volume has full support and that the associated Langevin dynamic is
exponentially ergodic.
Date Issued
2021-12-06
Citation
2021
ISSN
2050-5086
Publisher
Cambridge University Press
Copyright Statement
©2021 The Author(s)
Sponsor
The Leverhulme Trust
Commission of the European Communities
The Royal Society
Identifier
http://arxiv.org/abs/1909.05526v2
Grant Number
RL-2012-020-Transfer In
615897
RP/R1/191065
Subjects
math.PR
math.PR
math.AP
Publication Status
Published