Electronic Journal of Probability

Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise

Xavier Bardina, Maria Jolis, and Lluís Quer-Sardanyons

Full-text: Open access

Abstract

In this paper, we consider a quasi-linear stochastic heat equation with spatial dimension one, with Dirichlet boundary conditions and controlled by the space-time white noise. We formally replace the random perturbation by a family of noisy inputs depending on a parameter that approximate the white noise in some sense. Then, we provide sufficient conditions ensuring that the real-valued mild solution of the SPDE perturbed by this family of noises converges in law, in the space of continuous functions, to the solution of the white noise driven SPDE. Making use of a suitable continuous functional of the stochastic convolution term, we show that it suffices to tackle the linear problem. For this, we prove that the corresponding family of laws is tight and we identify the limit law by showing the convergence of the finite dimensional distributions. We have also considered two particular families of noises to that our result applies. The first one involves a Poisson process in the plane and has been motivated by a one-dimensional result of Stroock. The second one is constructed in terms of the kernels associated to the extension of Donsker's theorem to the plane.

Article information

Source
Electron. J. Probab., Volume 15 (2010), paper no. 39, 1267-1295.

Dates
Accepted: 9 August 2010
First available in Project Euclid: 1 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1464819824

Digital Object Identifier
doi:10.1214/EJP.v15-792

Mathematical Reviews number (MathSciNet)
MR2678391

Zentralblatt MATH identifier
1225.60100

Subjects
Primary: 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60G60: Random fields 60H15: Stochastic partial differential equations [See also 35R60]

Keywords
stochastic heat equation white noise weak convergence two-parameter Poisson process Donsker kernels

Rights
This work is licensed under aCreative Commons Attribution 3.0 License.

Citation

Bardina, Xavier; Jolis, Maria; Quer-Sardanyons, Lluís. Weak Convergence for the Stochastic Heat Equation Driven by Gaussian White Noise. Electron. J. Probab. 15 (2010), paper no. 39, 1267--1295. doi:10.1214/EJP.v15-792. https://projecteuclid.org/euclid.ejp/1464819824


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