FEYNMAN-KAC TRANSFORM FOR ANOMALOUS PROCESSES

We develop a new approach to the study of the Feynman-Kac transform for nonMarkov anomalous process Y-t = X-Et using methods from stochastic analysis, where X is a strong Markov process on a Lusin space X and {E-t, t >= 0} is the inverse of a driftless subordinator S that is independent of X and has infinite Levy measure. For a bounded function kappa on X and f in a suitable functional space over X, we establish regularity of u(t, x) = E-x[exp(- integral(t)(0) kappa(Y-s)ds) f(Y-t)] 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 X, 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.