A priori bounds for 2-d generalised Parabolic Anderson Model

Abstract

We show a priori bounds for solutions to $({\partial }_t-\Delta )u=\sigma (u)\xi$ in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269--504, 2014]. We assume $\sigma \in C_b^2(\mathbb{R})$ and that $\xi$ is of negative Hölder regularity of order $-1-\kappa$ where $\kappa <\bar{\kappa }$ for an explicit $\bar{\kappa }<1/3$, and that it can be lifted to a model in the sense of Regularity Structures. Our main results guarantee non-explosion of the solution in finite time and a growth which is at most polynomial in $t>0$. Our estimates imply global well-posedness for the 2-d generalised parabolic Anderson model on the torus, as well as for the parabolic quantisation of the Sine-Gordon Euclidean Quantum Field Theory (EQFT) on the torus in the regime ${\beta }^2\in (4\pi ,(1+\bar{\kappa })4\pi )$. We also consider the parabolic quantisation of a massive Sine-Gordon EQFT and derive estimates that imply the existence of the measure for the same range of $\beta$. Finally, our estimates apply to Itô SPDEs in the sense of Da Prato-Zabczyk [Stochastic Equations in Infinite Dimensions, Enc. Math. App., Cambridge Univ. Press, 1992] and imply existence of a stochastic flow beyond the trace-class regime.

some changes to exposition, accepted version

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