On stochastic nonclassical diffusion equation with standard and fractional Brownian motion

Abstract

This paper is concerned with the mathematical analysis of terminal value problems for a stochastic non-classical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions. Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a terminal value problem is a problem of determining the statistical properties of the initial data from the final time data. In the case 0 < β ≤ 1, where β is the fractional order of a Laplace operator, we show that these are well-posed under certain assumptions. We state a definition of ill-posedness and obtain the ill-poseness results for the problems when β > 1. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fractional Brownian motion.