Initial measures for the stochastic heat equation

2011/10/18 by Daniel Conus, Conus, Daniel, Mathew Joseph +5
Computer Science · Economics, Econometrics and Finance · Mathematics · #35R60 #60H15 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.1110.4079

openalex publication_date 2011/10/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a family of nonlinear stochastic heat equations of the form ∂_t u=Lu + σ(u)W, where W denotes space-time white noise, L the generator of a symmetric Lévy process on \R, and σ is Lipschitz continuous and zero at 0. We show that this stochastic PDE has a random-field solution for every finite initial measure u₀. Tight a priori bounds on the moments of the solution are also obtained. In the particular case that Lf=cf" for some c>0, we prove that if u₀ is a finite measure of compact support, then the solution is with probability one a bounded function for all times t>0.

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