# The Onsager-Machlup functional associated with additive fractional noise

## TL;DR

This paper computes the Onsager-Machlup functional for solutions of stochastic differential equations driven by additive fractional noise, extending previous results to broader Hurst parameter ranges and norms.

## Contribution

It extends the computation of the Onsager-Machlup functional to cases with Hurst parameter H<1/2 and various norms, providing new conditions for its calculation.

## Key findings

- Computed Onsager-Machlup functional for H in (1/4, 1/2) for supremum and Hölder norms.
- Extended the functional computation to H<1/2 with a general condition on elements of the Cameron-Martin space.
- Generalized previous results by Moret and Nualart to broader Hurst parameter ranges.

## Abstract

We consider the solution of a stochastic differential equation with additive multidimensional fractional noise. In the case $\frac14<H<\frac12$, we compute the Onsager-Machlup functional (with respect to the driving fractional Brownian motion) for the supremum norm and the H\"older norms with exponent $\alpha \in \left(0,H-\frac14\right)$ for any element of the Cameron-Martin space $\mathcal H_H$, extending a previous result of Moret and Nualart. In the more general case $H<\frac12$ and $\alpha \in \left(0,H\right)$, we formulate a condition on $h\in\mathcal H_H$ under which the computation of the Onsager-Machlup functional $J\left(h\right)$ follows.