| Summary: | 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.
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