A priori bounds for the generalised parabolic Anderson model

Chandra A.; de Lima Feltes G.; Weber H.

Abstract

We show a priori bounds for solutions to (Formula presented.) in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269–504, 2014]. We assume (Formula presented.) and that (Formula presented.) is of negative Hölder regularity of order (Formula presented.) where (Formula presented.) for an explicit (Formula presented.), 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 (Formula presented.). 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 fieldtheory (EQFT) on the torus in the regime (Formula presented.). 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 (Formula presented.). 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.