Fractional Diffusion Bridges
Yuzuru Inahama
TL;DR
This work addresses conditioning stochastic systems driven by fractional Brownian motion to a prescribed terminal point by forming fractional diffusion bridges via quasi-sure Malliavin analysis. It develops a quasi-sure refinement of canonical rough-path lifts (Besov and Hölder frameworks) and constructs bridge measures, including the Brownian special case that recovers classical diffusion bridges under nondegeneracy. A Freidlin–Wentzell type small-noise Large Deviation Principle is established for scaled bridges under an ellipticity condition, with the rate function given by a Cameron–Martin energy minimization over skeleton paths. The results extend to the Young regime H>1/2 and align with classical diffusion-bridge theory in that setting, offering a rigorous conditioning framework for non-Markovian fractional diffusions.
Abstract
Consider ``stochastic differential equations" driven by fractional Brownian motion with Hurst parameter H (1/4 <H< 1). Their solutions are sometimes called fractional diffusion processes. The main purpose of this paper is conditioning these processes to reach a given terminal point. We call the conditioned processes fractional diffusion bridges. Our main tool for mathematically rigorous conditioning is quasi-sure analysis, which is a potential theoretic part of Malliavin calculus. We also prove a small-noise large deviation principle of Freidlin-Wentzell type for scaled fractional diffusion bridges under a mild ellipticity assumption on the coefficient vector fields.