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 H\"older norms of order $ \beta $ with $ 0 < \beta < H - \frac{1}{4} $. In the case $\frac{1}{2} < H < 1 $, the norms can be a H\"older norm of order $ \beta$ with $ H - \frac{1}{2} < \beta < H - \frac{1}{4} $. As an example, we compute the Onsager-Machlup functional for the stochastic pendulum equation