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Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, I: The half line

Mirko D'Ovidio

Abstract

We extend the results obtained in \cite{Dov22} by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain $Ω$ with non-local dynamic conditions on the boundary $\partial Ω$. Due to the pioneering nature of the present research, we propose here the apparently simple case of $Ω=(0, \infty)$ with boundary $\{0\}$ of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.

Fractional Boundary Value Problems and Elastic Sticky Brownian Motions, I: The half line

Abstract

We extend the results obtained in \cite{Dov22} by introducing a new class of boundary value problems involving non-local dynamic boundary conditions. We focus on the problem to find a solution to a local problem on a domain with non-local dynamic conditions on the boundary . Due to the pioneering nature of the present research, we propose here the apparently simple case of with boundary of zero Lebesgue measure. Our results turn out to be instructive for the general case of boundary with positive (finite) Borel measures. Moreover, in our view, we bring new light to dynamic boundary value problems and the probabilistic description of the associated models.
Paper Structure (15 sections, 10 theorems, 222 equations, 1 figure)

This paper contains 15 sections, 10 theorems, 222 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Presentation of the results
  3. Plan of the work
  4. Motivations
  5. Preliminaries
  6. The Caputo-Dzherbashian fractional derivative
  7. The process $X$
  8. The random times $L$ and $H$
  9. The fractional initial value problem
  10. Time changes for FIVPs
  11. The fractional boundary value problem
  12. The process $\bar{X}$ and the FBVP
  13. On the occupation times for $\bar{X}$
  14. On the holding times of $\bar{X}$
  15. On the lifetime of $\bar{X}$

Key Result

Theorem 3.1

Let us consider the solution $v$ to the problem P1-FIVP and the solution $w$ to the problem P2-FP: Moreover, where $R_\lambda$ has been defined in potXdef and the following probabilistic representation holds true

Figures (1)

  • Figure 1: A representation of $\bar{X}$ on $[0, \infty)$ as a motion on the path of $X^+$ (a Brownian motion reflected at zero). The plateau is given by the inverse of $\bar{V}_t = t + H \circ (\eta/\sigma) \gamma^+_t$. As the local time at zero $\gamma^+$ increases the jump of $H$ produces a plateau for $\bar{V}^{-1}_t$. According with this plateau, the process $X^+ \circ \bar{V}^{-1}_t$ spends more time on the boundary point $\{0\}$. The path exhibits a delayed reflection. The delay is given by $H$ which is independent from $X^+$. This delay is the holding time with Mittag-Leffler distribution.

Theorems & Definitions (27)

  • Remark 1.1
  • Theorem 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 1
  • proof
  • Remark 4.1
  • Remark 4.2
  • Lemma 2
  • ...and 17 more