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Potential Analysis

, Volume 28, Issue 2, pp 139–162 | Cite as

Probabilistic Representations of Solutions of the Forward Equations

  • B. Rajeev
  • S. Thangavelu
Article

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.

Keywords

Stochastic differential equation Stochastic partial differential equation Evolution equation Stochastic flows Ito’s formula Stochastic representation Adjoints Diffusion processes Second order elliptic partial differential equation Monotonicity inequality 

Mathematics Subject Classifications (2000)

Primary 60H10 60H15 Secondary 60J60 35K15 

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Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Statistics and Mathematics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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