A priori bounds for the generalized parabolic Anderson model in the full subcritical regime

Abstract

We show a priori bounds for the generalized Parabolic Anderson Model $({\partial }_t-\Delta )u=\sigma (u)\diamond \xi$ in finite volume in the full subcritical regime. The approach is based on the observation that, for constant initial data, the time interval over which the solution remains close to its initial condition can be chosen uniformly in the value of the initial condition. A comparison principle is then used to extend this estimate to general bounded initial data.

7 pages; simplified proof using a comparison principle, eliminating the auxiliary field satisfying the transport equation