J. Quastel and S.R.S. Varadhan
We study existence and uniqueness problems for semigroups and processes determined by time dependent second order linear partial differential operators of the form $$ L_t u = \nabla\cdot a\nabla u+\sigma c\cdot\nabla u\eqno(1.1) $$ when the coefficients $a$ and $c$ have minimal regularity, and $a$ is allowed to degenerate. The main assumption is that the square root of $a$ has a square integrable spacial gradient. Under this conditions we prove existence and uniqueness for the corresponding diffusion equation and existence of solutions of the martingale problem under standard energy conditions on $c$. Under the somewhat stronger assumption $\int_0^T dt\int dx\, Trace\left(\dot a a^{-1}\dot a\right) <\infty $ where $\dot a$ denotes ${\partial a\over\partial x_i}$ for any $i=1,\ldots,d$ and $Trace\left(\dot a a^{-1}\dot a\right)= \sum_{i,j,k,\ell=1}^d {\partial a_{jk}\over \partial x_i}(a^{-1})_{k\ell}{\partial a_{\ell j}\over\partial x_i}$ we also establish uniqueness for the martingale problem.