Stochastic representation of solution to nonlocal-in-time diffusion

Qiang Du, Lorenzo Toniazzi, Zhi Zhou

Research output: Journal article publicationJournal articleAcademic researchpeer-review

11 Citations (Scopus)


The aim of this paper is to derive a stochastic representation of the solution to a nonlocal-in-time evolution equation (with a historical initial condition), which serves a bridge between normal diffusion and anomalous diffusion. We first derive the Feynman–Kac formula by reformulating the original model into an auxiliary Caputo-type evolution equation with a specific forcing term subject to certain smoothness and compatibility conditions. After that, we confirm that the stochastic formula also provides the solution in the weak sense even though the problem data is nonsmooth. Finally, numerical experiments are presented to illustrate the theoretical results and the application of the stochastic formula.

Original languageEnglish
Pages (from-to)2058-2085
Number of pages28
JournalStochastic Processes and their Applications
Issue number4
Publication statusPublished - Apr 2020


  • Feynman–Kac formula
  • Historical initial condition
  • Nonlocal evolution

ASJC Scopus subject areas

  • Statistics and Probability
  • Modelling and Simulation
  • Applied Mathematics


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