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On weak convergence of stochastic heat equation with colored noise

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  • Bezdek, Pavel

Abstract

In this work we are going to show weak convergence of probability measures. The measure corresponding to the solution of the following one dimensional nonlinear stochastic heat equation ∂∂tut(x)=κ2∂2∂x2ut(x)+σ(ut(x))ηα with colored noise ηα will converge to the measure corresponding to the solution of the same equation but with white noise η, as α↑1. Function σ is taken to be Lipschitz and the Gaussian noise ηα is assumed to be colored in space and its covariance is given by E[ηα(t,x)ηα(s,y)]=δ(t−s)fα(x−y) where fα is the Riesz kernel fα(x)∝1/|x|α. We will work with the classical notion of weak convergence of measures, that is convergence of probability measures on a space of continuous function with compact domain and sup–norm topology. We will also state a result about continuity of measures in α, for α∈(0,1).

Suggested Citation

  • Bezdek, Pavel, 2016. "On weak convergence of stochastic heat equation with colored noise," Stochastic Processes and their Applications, Elsevier, vol. 126(9), pages 2860-2875.
    • Handle: RePEc:eee:spapps:v:126:y:2016:i:9:p:2860-2875
    DOI: 10.1016/j.spa.2016.03.006
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    Cited by:

    1. Giordano, Luca M. & Jolis, Maria & Quer-Sardanyons, Lluís, 2020. "SPDEs with linear multiplicative fractional noise: Continuity in law with respect to the Hurst index," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7396-7430.
    2. Li, Kexue, 2017. "Hölder continuity for stochastic fractional heat equation with colored noise," Statistics & Probability Letters, Elsevier, vol. 129(C), pages 34-41.

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