A priori bounds for 2-d 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 ∈ Cb² (R) and that ξ is of negative H\"older regularity of order - 1 - κ where κ < κ̄ for an explicit κ̄< 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 β² ∈ (4 π, (1 + κ̄) 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 It\^o 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.