Dirichlet forms and semilinear elliptic equations with measure data

Abstract

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

Lp solutions of backward stochastic differential equations with jumps

“…For results on BSDEJs in other interesting directions, see [44,43] for second-order BSDEs with jumps and the related fully-nonlinear PIDEs; see Cohen and Elliott [20,21,24] for BSDEs driven by Markov chains; see Kharroubi et al [47] for (minimal) solutions to BSDEs with constrained jumps and related quasi-variational inequalities; see Aazizi and Ouknine [1] for a class of constrained BSDEJs and its application in pricing and hedging American options; see Klimsiak and Rozkosz [50,51] for a general (non-Markovian) BSDE and a related semilinear elliptic equation with measure data whose operator is associated with a regular semi-Dirichlet form; see [54,37] for BSDEJs with singular terminal data and their applications to optimal position targeting and a non-Markovian liquidation problem respectively; see also [35] for numerical simulation of BSDEJs by Wiener chaos expansion among other.…”

Existence and large-time asymptotics for solutions of semilinear parabolic systems with measure data

“…By what has already been proved, the pair (Y 0 , Z 0 ) has integrability properties under which the solution to BSDE x (ζ, f + dμ) is unique (see [18]). Therefore, from [15], it follows that (Y 0 , Z 0 ) has the representation Letting m → ∞ in (6.9) and using (6.19), we obtain From this and (6.10), (6.11), (6.20)-(6.23), we get (6.7).…”

Renormalized solutions of semilinear elliptic equations with general measure data

“…It is worth noting here that in case µ is a smooth measure it is possible to define duality solutions for general operators corresponding to transient regular Dirichlet forms, i.e. without the additional assumption that there exists the Green function for A (see [7]).…”

“…His approach was adapted to fractional Laplacian in [5,15]. The general formulation for operators A generated by Markov semigroups was introduced in [6] (see also [7] for the case of smooth measure data).…”