# 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 Copyright of this article belongs to Springer. 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. Division of Physical & Mathematical Sciences > Mathematics 31 Jul 2008 19 Sep 2010 04:48 http://eprints.iisc.ernet.in/id/eprint/15386