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

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.

KEYWORDS:

• Stochastic PDE
• bi-harmonic equation
• fractional Brownian motion
• backward problem
• regularization

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35R30
• 35R60
• 65M32
• 60G22

Acknowledgments

The authors would like to offer their cordial thanks to the reviewers of this paper for their valuable comments and suggestions, without these suggestions there would be no present form of this paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The research is supported by the Fundamental Research Funds for the Central Universities (Nos. JB210706 and QTZX22052) and the National Natural Science Foundation of China (No. 61877046).