Let {X(w),s ≦ w ≦ t} be a
Gaussian Markov stochastic process with continuous sample functions. Examples of
such processes are the Wiener, Orns+ˇein-Uhlenbeck, and Doob-Kac processes. An
operator valued function space integral is defined for each process. This was
done for the Wier.er process by R. H. Cameron and D. A. Storvick. For
functionals of the form F(x) =exp{∫st𝜃(t − w,x(w))dw} where 𝜃(t,u) is
bounded and almost everywhere continuous, the special inlegrals satisfy
integral equations related to the generalized Schroedinger equations studied
by the first author. For the Wiener process, a “backwards time” equation
is coupled with the Cameron-Storvick equation to give a pair of integral
equations.
