The third moment for the parabolic Anderson model

TL;DR

This paper derives explicit formulas for the third moment of the parabolic Anderson model with delta initial data and uses these to analyze phase transitions in intermittency fronts.

Contribution

It provides explicit formulas for the third moment of the model and applies them to identify phase transitions in intermittency fronts.

Findings

Explicit formulas for the third moment of the solution.

Identification of phase transition for the intermittency front of order three.

Application of contour integral evaluation to stochastic PDEs.

Abstract

In this paper, we study the {\it parabolic Anderson model} starting from the Dirac delta initial data: \[ \left(\frac{\partial}{\partial t} -\frac{\nu}{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = \lambda u(t,x) \dot{W}(t,x), \qquad u(0,x)=\delta_0(x), \quad x\in\mathbb{R}, \] where $\dot{W}$ denotes the space-time white noise. By evaluating the threefold contour integral in the third moment formula by Borodin and Corwin [2], we obtain some explicit formulas for $\mathbb{E}[u(t,x)^3]$. One application of these formulas is given to show the exact phase transition for the intermittency front of order three.