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Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations

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Abstract

We prove that under natural assumptions on the data strong solutions in Sobolev spaces of semilinear parabolic equations in divergence form involving measure on the right-hand side may be represented by solutions of some generalized backward stochastic differential equations. As an application we provide stochastic representation of strong solutions of the obstacle problem by means of solutions of some reflected backward stochastic differential equations. To prove the latter result we use a stochastic homographic approximation for solutions of the reflected backward equation. The approximation may be viewed as a stochastic analogue of the homographic approximation for solutions to the obstacle problem.

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Authors and Affiliations

  1. Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland

    Tomasz Klimsiak

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  1. Tomasz Klimsiak
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Correspondence to Tomasz Klimsiak.

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Research supported by the Polish Minister of Science and Higher Education under Grant No. N N201 372 436.

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Klimsiak, T. Strong Solutions of Semilinear Parabolic Equations with Measure Data and Generalized Backward Stochastic Differential Equations. Potential Anal 36, 373–404 (2012). https://doi.org/10.1007/s11118-011-9235-z

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  • DOI: https://doi.org/10.1007/s11118-011-9235-z

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