Stochastic heat equation with rough dependence in space

2015/05/19 by Yaozhong Hu, Hu, Yaozhong, Jingyu Huang +7 · 4 citations
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · doi:10.48550/arxiv.1505.04924

openalex publication_date 2015/05/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper studies the nonlinear one-dimensional stochastic heat equation driven by a Gaussian noise which is white in time and which has the covariance of a fractional Brownian motion with Hurst parameter 1/4\textlessH\textless1/2 in the space variable. The existence and uniqueness of the solution u are proved assuming the nonlinear coefficient is differentiable with a Lipschitz derivative and vanishes at 0. In the case of a multiplicative noise, that is the linear equation, we derive the Wiener chaos expansion of the solution and a Feynman-Kac formula for the moments of the solution. These results allow us to establish sharp lower and upper asymptotic bounds for the moments of the solution.

Cited by

• Chung's LIL for the linear stochastic fractional heat equation at origin
• Temporal properties of the stochastic fractional heat equation with rough dependence in space
• Growth rates for the Hölder coefficients of the linear stochastic fractional heat equation with rough dependence in space
• General large deviations and functional iterated logarithm law for multivalued McKean-Vlasov stochastic differential equations

Related

• Stochastic Heat Equation Driven by Fractional Noise and Local Time
• Stochastic Heat Equation with general noise
• Spatial asymptotics for the parabolic Anderson model driven by a Gaussian rough noise
• Analytically weak solutions to stochastic heat equations with spatially rough noise
• Parabolic Anderson model with rough dependence in space