Singular perturbations to semilinear stochastic heat equations

2010/02/19 by Martin Hairer, Hairer, Martin
Economics, Econometrics and Finance · Mathematics · #35R60 #60H15 #Analysis of PDEs (math.AP) #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.AP #math.PR #msc:35R60 #msc:60H15

paper · pdf · doi:10.48550/arxiv.1002.3722

To appear in PTRF

openalex publication_date 2010/02/19 · arxiv created 2010/09/18 · arxiv updated 2010/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a class of singular perturbations to the stochastic heat equation or semilinear variations thereof. The interesting feature of these perturbations is that, as the small parameter epsilon tends to zero, their solutions converge to the 'wrong' limit, i.e. they do not converge to the solution obtained by simply setting epsilon = 0. A similar effect is also observed for some (formally) small stochastic perturbations of a deterministic semilinear parabolic PDE. Our proofs are based on a detailed analysis of the spatially rough component of the equations, combined with a judicious use of Gaussian concentration inequalities.

Citations

• An Introduction to Stochastic PDEs

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• Sampling conditioned hypoelliptic diffusions

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