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We study the macroscopic fractal properties of the deep valleys of thesolution 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 whitenoise 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 valleysof 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 thevalleys undergoes a phase transition at a point which does not depend on theinitial data. The key tool of our proof is a lower bound to the lower tailprobability of the parabolic Anderson equation. Such lower bound is obtainedfor the first time in this paper and will be derived by utilizing theconnection between the parabolic Anderson equation and the Kardar-Parisi-Zhangequation. Our techniques of proving this lower bound can be extended to othermodels in the KPZ universality class including the KPZ fixed point.

Fractal Geometry of the Valleys of the Parabolic Anderson Equation