分数阶布朗运动驱动的时间分数阶扩散方程的随机源反问题
An inverse random source problem for the time fractional diffusion equation driven by a fractional Brownian motion
摘要：
研究了时间分数阶扩散方程的随机源反问题，其中源被假定为分数阶布朗运动驱动.在给定随机源的情况下，直接问题是研究随机时间分数阶扩散方程。反问题是从最终时间数据的期望和方差确定源的统计特性。对于正问题，我们证明了它是适定性的，并且在一定条件下有唯一的温和解。对于反问题，证明了其唯一性，并刻画了其不稳定性。分析的主要成分是基于Mittag-Leffler函数的性质和与分数布朗运动相关的随机积分。
英文摘要：
This paper is concerned with the mathematical analysis of the inverse random source problem for the time fractional diffusion equation, where the source is assumed to be driven by a fractional Brownian motion. Given the random source, the direct problem is to study the stochastic time fractional diffusion equation. The inverse problem is to determine the statistical properties of the source from the expectation and variance of the final time data. For the direct problem, we show that it is well-posed and has a unique mild solution under a certain condition. For the inverse problem, the uniqueness is proved and the instability is characterized. The major ingredients of the analysis are based on the properties of the Mittag--Leffler function and the stochastic integrals associated with the fractional Brownian motion.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
