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

Cerutti, M. Cristina; Escauriaza, Luis; Fabes, Eugene B.

Ann. Probab. 19, No. 2, 525-537 (1991).

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.

Reviewer: M.Capinski (Kraków)

MSC:

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

Keywords:

uniqueness of the martingale problem; Dirichlet problem; Poisson equation

Citations:

Zbl 0617.60075