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Stochastic representation of solution to nonlocal-in-time diffusion

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  • Du, Qiang
  • Toniazzi, Lorenzo
  • Zhou, Zhi

Abstract

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.

Suggested Citation

  • Du, Qiang & Toniazzi, Lorenzo & Zhou, Zhi, 2020. "Stochastic representation of solution to nonlocal-in-time diffusion," Stochastic Processes and their Applications, Elsevier, vol. 130(4), pages 2058-2085.
    • Handle: RePEc:eee:spapps:v:130:y:2020:i:4:p:2058-2085
    DOI: 10.1016/j.spa.2019.06.011
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    Cited by:

    1. Zhang, Shuaiqi & Chen, Zhen-Qing, 2022. "Fokker–Planck equation for Feynman–Kac transform of anomalous processes," Stochastic Processes and their Applications, Elsevier, vol. 147(C), pages 300-326.

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