Weak and Strong disorder for the stochastic heat equation and the continuous directed polymer in $d\ge 3$

Abstract

We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 Δu_{\eps,t} d t+ β\eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\ge 3$, where $B_{\eps,t}$ is a spatially smoothed (at scale $\eps$) space-time white noise, and $\beta >0$ is a parameter. We show the existence of a $\bar{\beta }\in (0,\infty )$ so that the solution exhibits weak disorder when $\beta <\bar{\beta }$ and strong disorder when $\beta >\bar{\beta }$. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.