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The aim of this paper is to give a stochastic representation for the solutionto a natural extension of the Caputo-type evolution equation. Thenonlocal-in-time operator is defined by a hypersingular integral with a(possibly time-dependent) kernel function, and it results in a model whichserves a bridge between normal diffusion and anomalous diffusion. We derive thestochastic representation for the weak solution of the nonlocal-in-time problemin case of nonsmooth data. We do so by starting from an auxiliary Caputo-typeevolution equation with a specific forcing term. Numerical simulations are alsoprovided to support our theoretical results.

Stochastic representation of solution to nonlocal-in-time diffusion