Probabilistic representations of solutions of elliptic boundary value problem and non-symmetric semigroups (Q888188)

The authors consider a very general second order uniformly elliptic operator \[ L(u)=\frac 1 2 \sum_{i,j=1}^d\frac{\partial\;}{\partial x_j}\left(a_{ij}(x)\frac{\partial u}{\partial x_i}\right)+\sum_{i=1}^db_i(x)\frac{\partial u}{\partial x_i}+(c(x)-\operatorname{div}\hat{b}(x))u, \] with very low regularity assumptions on the coefficients. The aim of the paper is to prove existence and uniqueness of the (weak) solution for the Dirichlet problem using a probabilistic approach. In particular, the authors show that the solution have a probabilistic representation, and that such a weak solution is continuous up to the boundary. In the final section, they also give a probabilistic representation of the semigroup associated with the operator \(L\).