Existence of solution and localization for the SHE with multiplicative Lévy white noise.

Resumen: 

We consider the following stochastic PDE in $$ \partial_t u = \Delta u + \xi \cdot u $$ where $u$ 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, 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 a solution to the equation. We further discuss the connection between the SHE and continuum directed polymer models.Joint work with Q. Berger (Université de Paris) and C. Chong (Columbia).

Posted on Sep 2, 2021 in Seminario de Probabilidades de Chile, Seminars