Integration by parts and time reversal for diffusion processes (Q1823551)

Let \(\{X_ t\), \(0\leq t\leq 1\}\) be a d-dimensional diffusion process which is a solution to \(dX_ t=b(t,X_ t)dt+\sigma (t,X_ t)dW_ t,\) with the generator \[ L_ t=2^{-1}a^{ij}(t,x)\nabla_{ij}+b^ i(t,x)\nabla_ i,\quad a=\sigma \sigma^*. \] The main objective of the article is to answer the following question: when is the time-reversed process \(\bar X_ t=X_{1-t}\) a diffusion process? Assuming that \(X_ t\) has a density \(p_ t(x)\), the authors show that a necessary and sufficient condition for \(\bar X_ t\) to be a diffusion process is the local integrability of the distributional derivatives \[ \sum \nabla_ j(a^{ij}(t,x)p_ t(x)),\quad i=1,2,...,d. \] Moreover, they also identify the generator of \(\bar X_ t\) to be \[ \bar L_ t=2^{-1}\bar a^{ij}(t,x)\nabla_{ij}+\bar b^ i(t,x)\nabla_ i,\quad where\quad \bar a^{ij}(1-t,x)=a^{ij}(t,x) \] \[ and\quad \bar b^{ij}(1-t,x)=- b^ i(t,x)+(1/p_ t(x))\nabla_ j(a^{ij}(t,x)p_ t(x)). \]