A Stochastic Fractional Calculus with Applications to Variational Principles

TL;DR

This paper introduces a new stochastic fractional calculus framework and extends the calculus of variations to stochastic processes, providing theoretical foundations and practical examples including quantum mechanics applications.

Contribution

It develops the first stochastic fractional calculus of variations and derives a stochastic fractional Euler-Lagrange equation, broadening the scope of variational calculus.

Findings

Derived a stochastic fractional Euler-Lagrange equation

Presented examples from quantum mechanics and numerical simulations

Extended classical and fractional calculus of variations to stochastic processes

Abstract

We introduce a stochastic fractional calculus. As an application, we present a stochastic fractional calculus of variations, which generalizes the fractional calculus of variations to stochastic processes. A stochastic fractional Euler-Lagrange equation is obtained, extending those available in the literature for the classical, fractional, and stochastic calculus of variations. To illustrate our main theoretical result, we discuss two examples: one derived from quantum mechanics, the second validated by an adequate numerical simulation.