Stochastic Shell Models driven by a multiplicative fractional Brownian--motion

TL;DR

This paper establishes the existence and uniqueness of solutions for a stochastic shell-model driven by infinite-dimensional fractional Brownian motion with Hurst parameter greater than 1/2, using variational and compactness methods.

Contribution

It introduces a novel approach to define and analyze solutions of shell-models driven by fractional noise with non-trivial coefficients, extending previous stochastic analysis techniques.

Findings

Proved existence and uniqueness of variational solutions.

Established convergence of approximations driven by piecewise linear noise.

Demonstrated the solution as the unique pathwise mild solution.

Abstract

We prove existence and uniqueness of the solution of a stochastic shell--model. The equation is driven by an infinite dimensional fractional Brownian--motion with Hurst--parameter $H\in (1/2,1)$, and contains a non--trivial coefficient in front of the noise which satisfies special regularity conditions. The appearing stochastic integrals are defined in a fractional sense. First, we prove the existence and uniqueness of variational solutions to approximating equations driven by piecewise linear continuous noise, for which we are able to derive important uniform estimates in some functional spaces. Then, thanks to a compactness argument and these estimates, we prove that these variational solutions converge to a limit solution, which turns out to be the unique pathwise mild solution associated to the shell--model with fractional noise as driving process.