Moment bounds for SPDEs with non-Gaussian fields and application to the Wong-Zakai problem

Abstract

Upon its inception the theory of regularity structures [Hai14] 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 the 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, as was done in [HS15]. 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 [HX16], [SX16] -as an example we prove here a generalization of the Wong-Zakai Theorem found in [HP15].

Another look into the Wong–Zakai theorem for stochastic heat equation

“…Several theories have been developed to make sense of singular equations with multiplication of distributions, see [Hai13,Hai14,GIP15,Kup16,OW16] (and the references therein). One example is the Wong-Zakai theorem for stochastic PDEs [HL18, HP15,CS17], which is an infinite dimensional analogue of [WZ65a,WZ65b,SV72]. In this article, we revisit this problem for a special case: the Stochastic Heat Equation (SHE) in one space dimension:…”

Paracontrolled distributions on Bravais lattices and weak universality of the 2d parabolic Anderson model

“…Gq :" C 8 ω p p Gq with periodic boundary conditions. The dual space S 1 ω p p Gq can be equipped with a Fourier transform F´1 G as in (9) such that F G , F´1 G become isomorphisms between S 1 ω pGq and S 1 ω p p Gq that are inverse to each other. For a proof of these statements we refer to Lemma A.1.…”

“…Moreover, in the Gaussian case all moments of polynomials of the noise are equivalent, and therefore it suffices to control variances. In the non-Gaussian case we can still regroup in terms of Wick polynomials [37,30,9,43], but a priori the moments are no longer comparable and new methods are necessary. In [37] the authors used martingale inequalities to bound higher order moments in terms of variances.…”

Weak universality of dynamical $$\Phi ^4_3$$ Φ 3 4 : non-Gaussian noise

“…The p-th moment of a random object is represented by a sum of certain labelled graphs, each obtained by "gluing together" the nodes of p identical "trees" in a certain way, and each of the identical trees represents that random object. When p is arbitrarily large which makes the labelled graph to be arbitrarily large, it is in general very hard to keep track of all the conditions that need to be checked; so in [18,Section 4,5] and [9,Section 3], the verification procedure was further reduced to checking some conditions for every "sub-tree" of a single tree (instead of the conditions for the large graph consisting of p trees with p arbitrarily large) that represents the random object (see Assumption 4.4 and Theorem 4.6 below). This significantly simplifies the verification; see for example [18] for application to the KPZ equation.…”

“…As for the values of these constants, it is easy to see that C ( ) 2,2,π diverges logarithmically when π = { (1, 1), (1, 1)} (see [15, Remark 1.17 and Section 10.5]), while all other C ( ) k, ,π 's converge to a finite limit. In fact, we have that if π = {(1, 1), (1, 1)} then 9) for some universal constant C log 2 , while for π being the single contraction of all four points together, we have…”