Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Derui Sheng, Tau Zhou

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-Solé [Stoch PDE: Anal Comp (2022). https://doi.org/10.1007/s40072-021-00234-6].

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 -capacity and -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). https://doi.org/10.1007/s40072-021-00234-6].
Paper Structure (9 sections, 13 theorems, 152 equations)

This paper contains 9 sections, 13 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Well posedness of equation \ref{['FKE']}
  4. Time-fractional noise
  5. Background of potential theory
  6. Regularity of the solution
  7. Hitting properties for system of fractional kinetic equations
  8. Hitting probability
  9. Polarity of points in the critical dimension

Key Result

Theorem 2.2

For any $u_0\in L^1({\mathds{R}^d})$, the process $\{u(t, x)\}_{(t, x) \in[0, T] \times \mathds{R}^d}$ given by mildsol exists if and only if Assumption dalang-condition holds. In this case, for all $p> 0$ and $T>0$,

Theorems & Definitions (32)

  • Theorem 2.2: Well-posedness
  • proof
  • Lemma 2.3
  • Remark 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • Example 2.8: White noise
  • Example 2.9: Riesz kernel noise
  • Example 2.10: Fractional noise
  • ...and 22 more