On weak convergence of stochastic heat equation with colored noise

STOCHASTIC PROCESSES AND THEIR APPLICATIONS

卷:126

Article

Bezdek, Pavel¹ 

[1] Univ Utah, Dept Math, Salt Lake City, UT 84112 USA

关键词: The stochastic heat equation;    Colored noise;    Riesz kernel;   

DOI  :  10.1016/j.spa.2016.03.006

来源: Elsevier

【 摘 要 】

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 partial derivative/partial derivative tu(t)(x) = k/2 partial derivative(2)/partial derivative x(2)u(t)(x) + sigma(u(t)(x))eta(alpha) with colored noise eta(alpha) will converge to the measure corresponding to the solution of the same equation but with white noise eta, as alpha up arrow 1. Function sigma is taken to be Lipschitz and the Gaussian noise eta(alpha), is assumed to be colored in space and its covariance is given by E [eta(alpha) (t, x)eta(alpha)(s, y)] = delta(t - s) f(alpha) (x - y) where f(alpha) is the Riesz kernel f(alpha) (x) proportional to 1/ vertical bar x vertical bar(alpha). 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 alpha, for alpha is an element of (0, 1). (C) 2016 Elsevier B.V. All rights reserved.

【 授权许可】

Free