Discretisations of rough stochastic PDEs
type: Item
instance of: scholarly article
title: Discretisations of rough stochastic PDEs
zbMATH Open document ID: 1406.60094
DOI: 10.1214/17-AOP1212
author: Y. Aharonov
published in: The Annals of Probability
publication date: 2018-06-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.06937
full work available at URL: https://projecteuclid.org/euclid.aop/1523520026
review text: The aim of this paper is to develop a general framework for spatial discretization of the class of parabolic stochastic partial differential equations (PDEs) of the form \[ \partial_tu=Au+F(u,\xi), \] where \(A\) is an elliptic differential operator, \(\xi\) is a rough noise, and \(F\) is a nonlinear function in \(u\) which is affine in \(\xi\). In the discretization, the spatial variable takes values in the dyadic grid with mesh tending to zero. The solutions of the initial PDE are provided in the framework of the theory of regularity structures, and they are functions in time. A particular example, prototypical for the class of the equations under consideration, is the dynamics model which is called \(\Phi^4\) model in dimension 3, or \(\Phi^4_3\), that is the same. This model is considered on the torus, and operator \(A\) contains Laplace operator. It is proved that the dynamical \(\Phi^4_3\) model, discretized to the dyadic grid, converges after renormalization to its continuous counterpart. The convergence is in probability and in the norm of the functional spaces introduced in the main theorem. This result in particular implies that, as expected, the \(\Phi^4_3\) measure with a sufficiently small coupling constant is invariant for this equation and that the lifetime of its solutions is almost surely infinite for almost every initial condition.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H40
Mathematics Subject Classification ID: 65M12
zbMATH DE Number: 6894783
zbMATH Keywords: stochastic PDEs
zbMATH Keywords: discretisations
zbMATH Keywords: regularity structures
zbMATH Keywords: stochastic quantization equation
zbMATH Keywords: invariant measure
zbMATH Keywords: dynamics on torus
zbMATH Keywords: convergence in probability
reviewed by: Yuliya S. Mishura
MaRDI profile type: Publication
arXiv ID: 1511.06937
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OpenAlex ID: W2964183012
Wikidata QID: Q129986911
description: scientific article