Fractal Geometry of the Valleys of the Parabolic Anderson Equation

Abstract

We study the macroscopic fractal properties of the deep valleys of the solution of the $(1+1)$-dimensional parabolic Anderson equation $${\partial \over \partial t}u(t,x) =\frac{1}{2} {\partial^2 \over \partial x^2} u(t,x) + u(t,x)\dot{W}(t,x),t>0, x\in {\bf R},\quad u(0,x) \equiv u_0(x),x\in {\bf R}, $$ where $\dot{W}$ is the time-space white noise and $0<\inf_{x\in {\bf R}} u_0(x)\leq \sup_{x\in {\bf R}} u_0(x)<\infty.$ Unlike the macroscopic multifractality of the tall peaks, we show that valleys of the parabolic Anderson equation are macroscopically monofractal. In fact, the macroscopic Hausdorff dimension (introduced by Barlow and Taylor [J. Phys. A 22 (1989) 2621--2628; Proc. Lond. Math. Soc. (3) 64 (1992) 125--152]) of the valleys undergoes a phase transition at a point which does not depend on the initial data. The key tool of our proof is a lower bound to the lower tail probability of the parabolic Anderson equation. Such lower bound is obtained for the first time in this paper and will be derived by utilizing the connection between the parabolic Anderson equation and the Kardar-Parisi-Zhang equation. Our techniques of proving this lower bound can be extended to other models in the KPZ universality class including the KPZ fixed point.