## View abstract

### Session S04 - Random Walks and Related Topics

Friday, July 16, 15:00 ~ 15:30 UTC-3

## The Stochastic Heat Equation with Lévy white noise and its continuum polymer counterpart

### Hubert Lacoin

#### IMPA, Brazil   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak73cf78f53177a006ff76d542ed0be23e').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy73cf78f53177a006ff76d542ed0be23e = 'l&#97;c&#111;&#105;n' + '&#64;'; addy73cf78f53177a006ff76d542ed0be23e = addy73cf78f53177a006ff76d542ed0be23e + '&#105;mp&#97;' + '&#46;' + 'br'; var addy_text73cf78f53177a006ff76d542ed0be23e = 'l&#97;c&#111;&#105;n' + '&#64;' + '&#105;mp&#97;' + '&#46;' + 'br';document.getElementById('cloak73cf78f53177a006ff76d542ed0be23e').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy73cf78f53177a006ff76d542ed0be23e + '\'>'+addy_text73cf78f53177a006ff76d542ed0be23e+'<\/a>';

The object of the talk is the following stochastic PDE $\partial_t u = \Delta u + \xi \cdot u$ in $[0,\infty)×\mathbb R^d$. $\Delta$ denotes the usual Laplacian in $\mathbb R^d$ and $\xi$ is a space-time white noise. The equation above has been extensively studied in the case where $\xi$ is a Gaussian White noise. In that case, the equation is well defined only when $d=1$. In that case, a probability measure on space time trajectory associated to the solutions of the equation has been introduced as the continuum directed polymer model.

In our talk, we consider the case where $\xi$ is a Lévy white noise with no diffusive part. We identify necessary and sufficient conditions on $\xi$ for the existence of solution to the equation, and discuss its intermittency properties, which are intimately related to the localization properties of the associated continuum polymer.

Joint work with Quentin Berger (Université de Paris) and Carsten Chong (Columbia University).