FEYNMAN-KAC TRANSFORM FOR ANOMALOUS PROCESSES

\bfA \bfb \bfs \bft \bfr \bfa \bfc \bft . We develop a new approach to the study of the Feynman-Kac transform for non-Markov anomalous process Yt = X_Et using methods from stochastic analysis, where X is a strong Markov process on a Lusin space \scrX and \{ Et, t \geq 0\} is the inverse of a driftless subordinator S that is independent of X and has infinite L\'evy measure. For a bounded function \kappa on \scrX and f in a suitable functional space over \scrX , we establish regularity of u(t, x) = \BbbE ^x\bigl[ exp ^\bigl( - ^\int₀^t \kappa (Ys)ds\bigr) f(Yt)\bigr] and show that it is the unique mild solution to a time fractional equation with initial value f. When X is a symmetric Markov process on \scrX , we further show that u is the unique weak solution to that time fractional equation. The main results are applied to compute the probability distribution of several random quantities of anomalous subdiffusion Y where X is a one-dimensional Brownian motion, including the first passage time, occupation time, and stochastic areas of Y .