Hitting properties of generalized fractional kinetic equation with time-fractional noise

TL;DR

This paper investigates the hitting properties of solutions to generalized fractional kinetic equations driven by Gaussian noise, establishing bounds on hitting probabilities and demonstrating that points are polar in the critical dimension.

Contribution

It introduces new bounds for hitting probabilities of fractional kinetic equations and confirms that points are polar at the critical dimension, supporting a recent conjecture.

Findings

Derived mean square modulus of continuity for solutions

Established bounds for hitting probabilities using capacity and Hausdorff measure

Proved all points are polar in the critical dimension

Abstract

This paper studies hitting properties for the system of generalized fractional kinetic equations driven by Gaussian noise fractional in time and white or colored in space. We derive the mean square modulus of continuity and some second order properties of the solution. These are applied to deduce lower and upper bounds for probabilities that the path process hits bounded Borel sets in terms of the $\mathfrak{g}_q$-capacity and $g_q$-Hausdorff measure, respectively, which yield the critical dimension for hitting points. Further, based on some fine analysis of the harmonizable representation for the solution, we prove that all points are polar in the critical dimension. This provides strong evidence for the conjecture raised in Hinojosa-Calleja and Sanz-Sol\'e [Stoch PDE: Anal Comp (2022). https://doi.org/10.1007/s40072-021-00234-6].