Stochastic Convolution-Type Heat Equations with Nonlinear Drift

Abstract

In this article, we study the solution of a class of stochastic convolution-type heat equations with nonlinear drift. For general initial condition and coefficients, we prove existence and uniqueness by using the characterization theorem and Banach's fixed-point theorem. We also give an implicit solution, which is a well-defined generalized stochastic process in a suitable distribution space. Finally, we investigate the continuous dependence of the solution on the initial data as well as the dependence on the coefficient.

Keywords:

• Convolution product
• Generalized functions
• Generalized valued stochastic processes
• Stochastic heat equation

Mathematics Subject Classification:

• Primary 60H40
• Secondary 60H40, 46F25, 46G20

Acknowledgments

We thank our colleagues and friends Ludwig Streit and Margarida Faria for the hospitality during a very pleasant stay at CCM of Madeira University in December 2002 and August 2003 where the main body of this work was produced. The third author thanks Habib Ouerdiane and Mohamed Erraoui for the warm hospitality during a very pleasant and fruitful stay at the Faculté des Sciences de Tunis in September 2002 and in Marrakech, respectively. Financial support of the project convénio ICCTI/Proc. 4.1.5/Marrocos “Análise Estocástica em Dimensáo Infinita,” project Luso/Tunisino, Conénio ICCTI/Tunisia, proc. 4.1.5 Tunisia, “Analyse en Dimension Infinie et Stochastique: Theorie et Applications” and FCTPOCTI is gratefully acknowledged.