Thursday, June 6, 2024 - 09:30 in V2-210/216
A priori bounds for the nonlinear Parabolic Anderson Model
A talk in the Harmonic and Stochastic Analysis of Dispersive PDEs series by
Hendrik Weber from Universität Münster
| Abstract: |
We show a priori bounds that exclude finite time explosion for the solution of the nonlinear (or generalised) Parabolic Anderson model (gPAM) ($\partial_t - \Delta) u = \sigma (u) \xi$ in the framework of Hairer's regularity structures. gPAM is the archetype of an equation that can be treated using regularity structures and it was one of the first equations for which a local solution theory was developed using this theory.
While global existence in the linear case $\sigma (u) = u$ follows from Hairer's and Gubinelli-Imkeller-Perkowski’s original works, the case of non-linear $\sigma$ is significantly more challenging and has remained open so far. Our results apply to noise terms $\xi$ of (ir)regularity $> -1-\kappa$ for some $\kappa>0$, including in particular the case of 2-dimensional spacial white noise. As a corollary we obtain global existence and the existence of an invariant measure for the dynamic Sine-Gordon model on the two-dimensional torus in the regime $\beta^2 \in (4 \pi, (1 + \kappa ) 4 \pi)$.
This is joint work with Guilherme Feltes (Münster) and Ajay Chandra (Imperial College).
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