We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential equations we prove the existence and uniqueness of solutions in the case where the right-hand side of the equation is monotone and satisfies mild integrability assumption, and the measure is smooth. We also study regularity of solutions under the assumption that the measure is smooth and has finite total variation. Some applications of our general results are given.Comment: Typos corrected. Two examples adde

We study the problem of existence and uniqueness of solutions of backward stochastic differential equations with two reflecting irregular barriers, L p data and generators satisfying weak integrability conditions. We deal with equations on general filtered probability spaces. In case the generator does not depend on the z variable, we first consider the case p = 1 and we only assume that the underlying filtration satisfies the usual conditions of right-continuity and completeness. Additional integrability properties of solutions are established if p ∈ (1, 2] and the filtration is quasi-continuous. In case the generator depends on z, we assume that p = 2, the filtration satisfies the usual conditions and additionally that it is separable. Our results apply for instance to Markov-type reflected backward equations driven by general Hunt processes.

We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of L p solutions for equations with generators monotone with respect to y and Lipschitz continuous with respect to z, and with data in L p spaces for p ≥ 1. We also prove that the solutions can be approximated via penalization method.Mathematics Subject Classifications (2010): Primary 60H20; Secondary 60F25.

We are mainly concerned with equations of the form −Lu = f (x, u) + µ, where L is an operator associated with a quasi-regular possibly nonsymmetric Dirichlet form, f satisfies the monotonicity condition and mild integrability conditions, and µ is a bounded smooth measure. We prove general results on existence, uniqueness and regularity of probabilistic solutions, which are expressed in terms of solutions to backward stochastic differential equations. Applications include equations with nonsymmetric divergence form operators, with gradient perturbations of some pseudodifferential operators and equations with Ornstein-Uhlenbeck type operators in Hilbert spaces. We also briefly discuss the existence and uniqueness of probabilistic solutions in the case where L corresponds to a lower bounded semi-Dirichlet form.

No abstract