Abstract
Upon its inception the theory of regularity structures allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by space-time white noise. When the driving noise is non-Gaussian the machinery of theory can still be used but must be combined with an infinite number of stochastic estimates in order to compensate for the loss of hypercontractivity. In this paper we obtain a more streamlined and automatic set of criteria implying these estimates which facilitates the treatment of some other problems including non-Gaussian noise such as some general phase coexistence models - as an example we prove here a generalization of the Wong-Zakai Theorem found by Hairer and Pardoux.
Highlights
Upon its inception the theory of regularity structures [7] allowed for the treatment for many semilinear perturbations of the stochastic heat equation driven by spacetime white noise
In [10, Remark 1.7] Hairer and Pardoux ask if an analogous statement can be proven if one replaces the mollified space-time white noise ξε(t, x) with ε−3/2ζ(ε−2t, ε−1x) where ζ is a non-Gaussian random field which is supported on smooth functions and satisfies a central limit theorem
The moment estimates we prove will be used as input for the theory of regularity structures developed in [7]
Summary
In the paper [10] the main focus was the convergence of smooth approximations u to the solution of the SPDE. In [10, Remark 1.7] Hairer and Pardoux ask if an analogous statement can be proven if one replaces the mollified space-time white noise ξε(t, x) with ε−3/2ζ(ε−2t, ε−1x) where ζ is a non-Gaussian random field which is supported on smooth functions and satisfies a central limit theorem. They conjectured that in addition to the renormalization seen in the Gaussian case one would see additional terms of order ε−. Since the proofs follow along the same lines, we refrain from redoing them here
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