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

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 gq-capacity and gq-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-Solé [Stoch PDE: Anal Comp (2022).

Exact uniform modulus of continuity for $q$-isotropic Gaussian random fields

“…Proposition 3.1. Let B q the q-sheet defined in (17). Assume that q 2 is of class C 2 in (0, T ], and that dq 2 dτ is non-increasing.…”

“…The next proposition verifies that B q satisfies the local nondeterminism condition (LND): Proposition 3.2. Let B q the q-Brownian sheet defined in (17). Fix 0 < t < T , then for any…”

“…The proof of (21) finishes by a similar argument than (22), using ( 17), (24) and that Theorem 3.1. Let B q the q-Brownian sheet defined in (17). Fix 0 < t < T , assume that that q 2 is of class C 2 in (0, T ], and that dq 2 dτ is non-increasing.…”