Electron. J. Diff. Equ., Vol. 2014 (2014), No. 208, pp. 1-10.

Feynman-Kac theorem in Hilbert spaces

Irina V. Melnikova, Valentina S. Parfenenkova

Abstract:
In this article we study the relationship between solutions to Cauchy problems for the abstract stochastic differential equation $dX(t)=AX(t)dt + BdW(t)$ and solutions to Cauchy problems (backward and forward) for the infinite dimensional deterministic partial differential equation
$$ \pm\frac{\partial g}{\partial t}(t,x) + \frac{\partial g}{\partial x}(t,x)Ax + \frac{1}{2}\hbox{Tr}[(BQ^{1/2})^* \frac{\partial^2 g}{\partial x^2}(t,x) (BQ^{1/2})] = 0, $$
where g is the probability characteristic $g=\mathbb{E}^{t,x}[h(X(T))]$ in the backward case and $g=\mathbb{E}^{0,x}[h(X(t))]$ in the forward case. This relationship, that is the inifinite dimensional Feynman-Kac theorem, is proved in both directions: from stochastic to deterministic and from deterministic to stochastic. Special attention is given to the definition and interpretation of objects in the equations.

Submitted February 20, 2014. Published October 7, 2014.
Math Subject Classifications: 47D06, 60G15, 60H30.
Key Words: Semigroups of operators; infinite dimensional stochastic equations; diffusion processes; Kolmogorov equations.

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Irina V. Melnikova
Ural Federal University
Lenin av. 51, 620083 Ekaterinburg, Russia
email: Irina.Melnikova@usu.ru
Valentina S. Parfenenkova
Ural Federal University
Lenin av. 51, 620083 Ekaterinburg, Russia
email: vika8887@e1.ru

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