This paper is concerned with the mathematical analysis of terminal value problems (TVP) for a stochastic nonclassical diffusion equation, where the source is assumed to be driven by classical and fractional Brownian motions (fBms). Our two problems are to study in the sense of well-posedness and ill-posedness meanings. Here, a TVP is a problem of determining the statistical properties of the initial data from the final time data. In the case $0<\beta \le 1$, where $\beta$ 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-posedness results for the problems when $\beta >1$. The major analysis tools in this paper are based on properties of stochastic integrals with respect to the fBm.

本文关注的是随机非经典扩散方程的终值问题 (TVP) 的数学分析，其中假设源由经典和分数布朗运动 (fBms) 驱动。我们的两个问题是研究适定意义和不适定意义。在这里，TVP 是一个从最终时间数据中确定初始数据的统计特性的问题。在这种情况下$0<\beta \le 1$， 在哪里$\beta$是拉普拉斯算子的分数阶，我们证明这些在某些假设下是适定的。我们陈述了不适定性的定义，并在以下情况下获得问题的不适定性结果$\beta >1$. 本文中的主要分析工具是基于关于 fBm 的随机积分的性质。