Diffusion Semigroups and Diffusion Processes Corresponding to Degenerate Divergence Form Operators

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.