Martingale Representation and Logarithmic-Sobolev Inequality for Fractional Ornstein-Uhlenbeck Measure

TL;DR

This paper establishes a martingale representation theorem and a Logarithmic-Sobolev inequality for the measure associated with a fractional Ornstein-Uhlenbeck process, using integration by parts and Bismut's method.

Contribution

It introduces a martingale representation and Logarithmic-Sobolev inequality for fractional Ornstein-Uhlenbeck measures, expanding the theoretical understanding of such stochastic processes.

Findings

Martingale representation theorem for fractional Ornstein-Uhlenbeck measure

Logarithmic-Sobolev inequality for this measure

Integration by parts formula derived via pull back and Bismut method

Abstract

In this paper, we consider the measure determined by a fractional Ornstein-Uhlenbeck process. For such measure, we establish a martingale representation theorem and consequently obtain the Logarithmic-Sobolev inequality. To this end, we also present the integration by parts formula for such measure, which is obtained via its pull back formula and the Bismut method.