A discretization of Caputo derivatives with application to time fractional SDEs and gradient flows

TL;DR

This paper introduces a new discretization method for Caputo derivatives, linking it to Volterra integrals, and demonstrates its effectiveness in analyzing time fractional stochastic differential equations and gradient flows.

Contribution

The paper proposes a novel discretization scheme for Caputo derivatives based on deconvolving schemes for Volterra integrals, with proven stability and applicability to fractional SDEs and gradient flows.

Findings

The discretization scheme preserves key properties like positivity and stability.

It ensures unique limiting measures for fractional Langevin equations with convex potentials.

Numerical solutions converge to strong solutions of fractional gradient flows.

Abstract

We consider a discretization of Caputo derivatives resulted from deconvolving a scheme for the corresponding Volterra integral. Properties of this discretization, including signs of the coefficients, comparison principles, and stability of the corresponding implicit schemes, are proved by its linkage to Volterra integrals with completely monotone kernels. We then apply the backward scheme corresponding to this discretization to two time fractional dissipative problems, and these implicit schemes are helpful for the analysis of the corresponding problems. In particular, we show that the overdamped generalized Langevin equation with fractional noise has a unique limiting measure for strongly convex potentials and establish the convergence of numerical solutions to the strong solutions of time fractional gradient flows. The proposed scheme and schemes derived using the same philosophy can be useful for many other applications as well.