Probabilistic Representations of Solutions of the Forward Equations

Rajeev, B and Thangavelu, S (2008) Probabilistic Representations of Solutions of the Forward Equations. In: Potential Analysis, 28 (2). pp. 139-162.

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Abstract

In this paper we prove a stochastic representation for solutions of the evolution equation $\partial_t \psi_t= \frac{1}{2}L^\ast \psi_t$ where $L^\ast$ 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 $\psi_0 = \psi$, a distribution with compact support, this representation has the form $\psi_t = E(Y_t(\psi))$ where the process $(Y_t(\psi))$ is the solution of a stochastic partial differential equation connected with the stochastic differential equation for $(X_t)$ via Ito's formula.

Item Type:	Journal Article
Publication:	Potential Analysis
Publisher:	Springer
Additional Information:	Copyright of this article belongs to Springer.
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.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	31 Jul 2008
Last Modified:	19 Sep 2010 04:48
URI:	http://eprints.iisc.ac.in/id/eprint/15386

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