On the Kolmogorov equation associated with Volterra equations and Fractional Brownian Motion

TL;DR

This paper studies a class of SPDEs driven by fractional Brownian motion linked to Volterra equations, establishing their Markov properties, differentiability, and connection to Kolmogorov equations, with open problems in regularization-by-noise.

Contribution

It introduces an original extension of the drift operator for SPDEs driven by fractional Brownian motion and analyzes their Markov and differentiability properties.

Findings

SPDE solutions form a twice Fréchet differentiable Markov flow.

Associated Kolmogorov equations are solved in the infinite-dimensional setting.

Obstructions in mild formulation analysis highlight open problems in regularization-by-noise.

Abstract

We consider a Volterra convolution equation in $\mathbb{R}^d$ perturbed with an additive fractional Brownian motion of Riemann-Liouville type with Hurst parameter $H\in (0,1)$. We show that its solution solves a stochastic partial differential equation (SPDE) in the Hilbert space of square-integrable functions. Such an equation motivates our study of an unconventional class of SPDEs requiring an original extension of the drift operator and its Fr\'echet differentials. We prove that these SPDEs generate a Markov stochastic flow which is twice Fr\'echet differentiable with respect to the initial data. This stochastic flow is then employed to solve, in the classical sense of infinite dimensional calculus, the path-dependent Kolmogorov equation corresponding to the SPDEs. In particular, we associate a time-dependent infinitesimal generator with the fractional Brownian motion. In the final section, we show some obstructions in the analysis of the mild formulation of the Kolmogorov equation for SPDEs driven by the same infinite dimensional noise. This problem, which is relevant to the theory of regularization-by-noise, remains open for future research.