On weak convergence of stochastic heat equation with colored noise

Pavel Bezdek

Stochastic Processes and their Applications, 2016, vol. 126, issue 9, 2860-2875

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).

Keywords: The stochastic heat equation; Colored noise; Riesz kernel (search for similar items in EconPapers)
Date: 2016
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Citations: View citations in EconPapers (3)

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DOI: 10.1016/j.spa.2016.03.006

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