Inverse problems for the fractional diffusion equation driven by fractional Brownian sheet

Abstract This paper explores the inverse problems of the space-time fractional diffusion equation driven by a space-time fractional Brownian sheet, including spatial stochastic source term, spatial deterministic source term, potential, and fractional order. In order to deal with the stochastic integral of fractional Brownian sheet with space-time Hurst indexes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi class="MJX-tex-calligraphic" mathvariant="script">H</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mo>∈</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:math> , we construct a unified Itô isometry framework for the first time, upon which we establish the regularity of the solution to the direct problem. Then, we ingeniously employ the generalized fractional Grönwall inequality, Sobolev embedding, and combine the statistical characteristics of the outward normal derivative of the weak solution on open subdomain at the boundary to achieve accurate recovery of the stochastic space-time source terms and prove the uniqueness of the space deterministic source term. Furthermore, by constructing a contraction mapping and exploiting the Banach fixed point theorem, we establish the uniqueness of the potential. Afterward, using the statistical characteristics of the solution at the terminal time, we establish the stability of the potential. Finally, leveraging the representation of the mild solution and the asymptotic properties of the Mittag-Leffler function, we examine the uniqueness and stability of fractional order at known terminal time. In scenario involving an unknown terminal time, we derive the uniqueness and stability of the terminal time.