Nonlinear Young integrals via fractional calculus

TL;DR

This paper introduces a new approach to defining nonlinear Young integrals using fractional calculus, which are relevant in stochastic heat equations with random coefficients.

Contribution

It develops a fractional calculus-based framework for nonlinear Young integrals and their iterated versions, expanding the mathematical tools for stochastic analysis.

Findings

Defined nonlinear Young integrals via fractional calculus.

Established a method for iterated nonlinear integrals.

Connected the integrals to stochastic heat equations with random coefficients.

Abstract

For H\"older continuous functions $W(t,x)$ and $\varphi_t$, we define nonlinear integral $\int_a^b W(dt, \varphi_t)$ via fractional calculus. This nonlinear integral arises naturally in the Feynman-Kac formula for stochastic heat equations with random coefficients. We also define iterated nonlinear integrals.