FEYNMAN-KAC TRANSFORM FOR ANOMALOUS PROCESSES

Zhen Qing Chen*, Weihua Deng, Pengbo Xu

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)

Abstract

\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 = XEt 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( - \int0t \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 .

Original languageEnglish
Pages (from-to)6017-6047
Number of pages31
JournalSIAM Journal on Mathematical Analysis
Volume53
Issue number5
DOIs
Publication statusPublished - 2021
Externally publishedYes

Keywords

  • Feynman-Kac transform
  • Markov process
  • anomalous process
  • fractional derivative
  • inverse subordinator
  • subordinator
  • time fractional equation

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