Global solutions to the stochastic heat equation with superlinear accretive reaction term and polynomially growing multiplicative noise

2024/09/23 by Michael Salins, Salins, Michael
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2409.15527

openalex publication_date 2024/09/23 · openalex created_date 2024/10/26 · openalex updated_date 2026/07/28

Abstract

We prove that mild solutions to the stochastic heat equation with superlinear accretive forcing and polynomially growing multiplicative noise cannot explode under two sets of assumptions. The first set of assumptions allows both the deterministic forcing and multiplicative noise terms to grow polynomially, as long as the multiplicative noise is sufficiently larger. The second set of assumptions imposes an Osgood condition on the deterministic forcing and allows the multiplicative noise to grow polynomially. In both cases, the multiplicative noise cannot grow faster than u^(3)/(2), as this would cause explosion.

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