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On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

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  • Tomasz Klimsiak

    (Nicolaus Copernicus University)

Abstract

We consider processes of the form [s,T]∋t↦u(t,X t ), where (X,P s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×ℝ d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.

Suggested Citation

  • Tomasz Klimsiak, 2013. "On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators," Journal of Theoretical Probability, Springer, vol. 26(2), pages 437-473, June.
    • Handle: RePEc:spr:jotpro:v:26:y:2013:i:2:d:10.1007_s10959-011-0381-4
    DOI: 10.1007/s10959-011-0381-4
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    References listed on IDEAS

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    1. Lejay, Antoine, 2004. "A probabilistic representation of the solution of some quasi-linear PDE with a divergence form operator. Application to existence of weak solutions of FBSDE," Stochastic Processes and their Applications, Elsevier, vol. 110(1), pages 145-176, March.
    2. Rozkosz, Andrzej, 1996. "Stochastic representation of diffusions corresponding to divergence form operators," Stochastic Processes and their Applications, Elsevier, vol. 63(1), pages 11-33, October.
    3. V. Bally & A. Matoussi, 2001. "Weak Solutions for SPDEs and Backward Doubly Stochastic Differential Equations," Journal of Theoretical Probability, Springer, vol. 14(1), pages 125-164, January.
    4. François Coquet & Jean Mémin & Leszek Słomiński, 2003. "On Non-Continuous Dirichlet Processes," Journal of Theoretical Probability, Springer, vol. 16(1), pages 197-216, January.
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