Uniqueness for some diffusions with discontinuous coefficients. (English) Zbl 0737.60038

The authors show the uniqueness of the martingale problem for a second order operator \(L\) where \(A\) is the matrix of its coefficients under the assumption that \(A\) is bounded, symmetric, uniformly positive definite and continuous on \(\mathbb{R}^ n\) except a countable set with at most one cluster point. In the case that \(A\) is continuous the result was established by Stroock and Varadhan and for some piecewise constant \(A\) by R. F. Bass and E. Pardoux [Probab. Theory Relat. Fields 76, 557-572 (1987; Zbl 0617.60075)]. The proof is based on the ideas of Bass and Pardoux, the key result being the uniqueness of a Dirichlet problem for the Poisson equation \(Lu=-f\), considered in the first part of the paper.


60G46 Martingales and classical analysis
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B10 Periodic solutions to PDEs
35B45 A priori estimates in context of PDEs
35J99 Elliptic equations and elliptic systems
60G12 General second-order stochastic processes
60G15 Gaussian processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J35 Transition functions, generators and resolvents


Zbl 0617.60075
Full Text: DOI