The backward problem of a stochastic PDE with bi-harmonic operator driven by fractional Brownian motion

Abstract

ABSTRACT This paper is concerned with a backward problem of a stochastic partial differential equation with bi-harmonic operator. The source term is driven by fractional Brownian motion. Based on the Gevrey-type space, the regularity of the mild solution is studied. However, this problem is ill-posed since it is unstable. The instability is discussed in the sense of expectation and variance. Moreover, a regularization method is proposed. The error estimation between the regularization solution and the mild solution is given using an a prior parameter choice rule.