A priori bounds for the generalised parabolic Anderson model

We show a priori bounds for solutions to (𝜕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 ∈ 𝐶 2 𝑏 (ℝ) and that 𝜉 is of negative Hölder regularity of order −1 − 𝜅 where 𝜅 <̄ 𝜅 for an explicit̄ 𝜅 < 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 fieldtheory (EQFT) on the torus in the regime 𝛽 2 ∈ (4𝜋, (1 +̄ 𝜅)4𝜋). 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.