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

Abstract

We consider the smoothed multiplicative noise stochastic heat equation $$d u_{\eps,t}= \frac 12 \Delta u_{\eps,t} d t+ \beta \eps^{\frac{d-2}{2}}\, \, u_{\eps, t} \, d B_{\eps,t} , \;\;u_{\eps,0}=1,$$ in dimension $d\geq 3$, 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.