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May 2018 Discretisations of rough stochastic PDEs
M. Hairer, K. Matetski
Ann. Probab. 46(3): 1651-1709 (May 2018). DOI: 10.1214/17-AOP1212


We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in time. As an application, we show that the dynamical $\Phi^{4}_{3}$ model on the dyadic grid converges after renormalisation to its continuous counterpart. 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.


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M. Hairer. K. Matetski. "Discretisations of rough stochastic PDEs." Ann. Probab. 46 (3) 1651 - 1709, May 2018.


Received: 1 December 2015; Revised: 1 June 2017; Published: May 2018
First available in Project Euclid: 12 April 2018

zbMATH: 06894783
MathSciNet: MR3785597
Digital Object Identifier: 10.1214/17-AOP1212

Primary: 60H15
Secondary: 65M12

Keywords: discretisations , invariant measure , Regularity structures , stochastic PDEs , Stochastic quantization equation

Rights: Copyright © 2018 Institute of Mathematical Statistics


Vol.46 • No. 3 • May 2018
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