The Onsager-Machlup functional for distribution dependent SDEs driven by fractional Brownian motion

Abstract

In this paper, we compute the Onsager-Machlup functional for distribution dependent SDEs driven by fractional Brownian motions with Hurst parameter $H\in (\frac{1}{4},1)$. In the case $\frac{1}{4}<H<\frac{1}{2}$, the norm can be either the supremum norm or Hölder norms of order ${\hat{I}}^2$ with $0<{\hat{I}}^2<H-\frac{1}{4}$. In the case $\frac{1}{2}<H<1$, the norms can be a Hölder norm of order ${\hat{I}}^2$ with $H-\frac{1}{2}<{\hat{I}}^2<H-\frac{1}{4}$. As an example, we compute the Onsager-Machlup functional for the stochastic pendulum equation

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