The third moment for the parabolic Anderson model

In this paper, we study the {\it parabolic Anderson model} starting from the Dirac delta initial data: $\left(\frac{\partial}{\partial t} -\fracν{2}\frac{\partial^2}{\partial x^2} \right) u(t,x) = λu(t,x) \dot{W}(t,x), \qquad u(0,x)=δ_0(x), \quad x\in\mathbb{R},$ 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.

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