Time reversal of degenerate diffusions

J. Quastel

We consider the problem of when the time reversal of a diffusion is a diffusion. Haussman and Pardoux [Ann. Prob. 14, 1986] showed that this is the case, and computed the generator of the reversed process, under the assumption that either the diffusivity is uniformly elliptic, or, in the degenerate case (no uniform ellipticity) if the diffusion matrix $a^{ij}$ satisfies $a^{ij}_{x_i,x_j}\in L^\infty$. Using results of the author and Varadhan [Comm. Pure Appl. Math. 50, 1997] we extend this to cases where $a^{ij}$ have only one weak derivative in a suitable sense, the drift satisfies a finite entropy condition, and the density is apriori bounded.