27/04/2022, 17:00 — 18:00 — Online
Hubert Lacoin, Instituto de Matemática Pura e Aplicada
Existence of solution and localization for the stochastic heat equation with multiplicative Lévy white noise
We consider the following stochastic partial differential equation in $\mathbb R^d$ $ \partial_t u = \Delta u + \xi \cdot u $ where the unknown $u$ is a function of space and time. The operator $\Delta$ denotes the usual Laplacian in $\mathbb R^d$ and $\xi$ is a space-time Lévy white noise. This equation has been extensively studied in the case where $\xi$ is a Gaussian White noise. In that case, it is known that the equation is well posed only when the space dimension $d$ is equal to one.
In our talk, we consider the case where $\xi$ is a Lévy white noise with no diffusive part and only positive jumps. We identify necessary and sufficient conditions on the Lévy jump measure for the existence of solutions to the equation. We further discuss the connection between the SHE and continuum directed polymer models.
Joint work with Quentin Berger (Sorbonne Université, Paris) and Carsten Chong (Columbia University, New York)
