On the Kolmogorov equation associated with Volterra equations and fractional Brownian motion

We consider a Volterra convolution equation in Rd perturbed with an additive fractional Brownian motion of Riemann-Liouville type with Hurst parameter H is an element of (0, 1). We show that its solution solves an infinite-dimensional stochastic differential equation (SDE) in the Hilbert space of square-integrable functions. Such an equation motivates our study of an unconventional class of SDEs requiring an original extension of the drift operator and its Fr & eacute;chet differentials. We prove that these infinite-dimensional SDEs generate a Markov stochastic flow which is twice Fr & eacute;chet 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 SDEs. 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 SDEs driven by the same infinite-dimensional noise. This problem, which is relevant to the theory of regularization by noise, remains open for future research. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).