A priori bounds for 2-d generalised Parabolic Anderson Model

Abstract

We show a priori bounds for solutions to $(\partial_t - Δ) u = σ(u) ξ$ in finite volume in the framework of Hairer's Regularity Structures [Invent Math 198:269--504, 2014]. We assume $σ\in C_b^2 (\mathbb{R})$ and that $ξ$ is of negative Hölder regularity of order $- 1 - κ$ where $κ< \barκ$ for an explicit $\barκ< 1/3$, 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 $β^2 \in (4 π, (1 + \barκ) 4 π)$. 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 $β$. 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.