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Probabilistic Representations of Solutions of the Forward Equations

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Abstract

In this paper we prove a stochastic representation for solutions of the evolution equation

$$\partial _t \psi _t = \frac{1}{2}L^ * \psi _t $$

where L  ∗  is the formal adjoint of a second order elliptic differential operator L, with smooth coefficients, corresponding to the infinitesimal generator of a finite dimensional diffusion (X t ). Given ψ 0 = ψ, a distribution with compact support, this representation has the form ψ t  = E(Y t (ψ)) where the process (Y t (ψ)) is the solution of a stochastic partial differential equation connected with the stochastic differential equation for (X t ) via Ito’s formula.

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Authors and Affiliations

  1. Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, 560 059, India

    B. Rajeev

  2. Department of Mathematics, Indian Institute of Science, Bangalore, 560 012, India

    S. Thangavelu

Authors
  1. B. Rajeev
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  2. S. Thangavelu
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Correspondence to B. Rajeev.

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Rajeev, B., Thangavelu, S. Probabilistic Representations of Solutions of the Forward Equations. Potential Anal 28, 139–162 (2008). https://doi.org/10.1007/s11118-007-9074-0

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