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Stochastic billiards with Markovian reflections in generalized parabolic domains

Conrado da Costa, Mikhail V. Menshikov, Andrew R. Wade

Abstract

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sublinear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip $\mathbb{R}_+ \!\times S$ where $S$ is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the $\mathbb{R}_+$ component are of generalized Lamperti type, and the $S$ component is asymptotically Markov; this extends earlier work that dealt with finite $S$.

Stochastic billiards with Markovian reflections in generalized parabolic domains

Abstract

We study recurrence and transience for a particle that moves at constant velocity in the interior of an unbounded planar domain, with random reflections at the boundary governed by a Markov kernel producing outgoing angles from incoming angles. Our domains have a single unbounded direction and sublinear growth. We characterize recurrence in terms of the reflection kernel and growth rate of the domain. The results are obtained by transforming the stochastic billiards model to a Markov chain on a half-strip where is a compact set. We develop the recurrence classification for such processes in the near-critical regime in which drifts of the component are of generalized Lamperti type, and the component is asymptotically Markov; this extends earlier work that dealt with finite .

Paper Structure

This paper contains 19 sections, 31 theorems, 173 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Overview
  3. Motivation 1: Stochastic billiards
  4. Motivation 2: Random walks on half-strips
  5. Markov chains on a half-strip
  6. Asymptotic Markovianity
  7. Lamperti regimes and recurrence classification
  8. Stochastic billiards
  9. Model formulation and construction
  10. Assumptions and results
  11. Proofs for the half-strip model
  12. The strict Lamperti regime
  13. The Lamperti regime
  14. Proofs for the stochastic billiards model
  15. Displacement estimates
  16. ...and 4 more sections

Key Result

Proposition 2.1

Suppose that ass:non-confinement, ass:kernel, and ass:asymptotically-markov hold, and that ass:moments holds with $p>1$ and $q>0$. Suppose also that there exists $d \in {C}_{\mathrm{b}}(S)$ such that $\mu_1$ defined by eq:lamperti-mu-def satisfies $\lim_{x \to \infty} \sup_{u \in S} | \mu_1 (x,u) -

Figures (4)

  • Figure 1: Part of the region $\mathcal{D}_\gamma$ with $\gamma = 1/2$. A section of the particle's trajectory is indicated by the dotted line. It hits the boundary at the incoming angle indicated by the single-ruled angle, and exits at the angle indicated by the double-ruled angle, whose distribution is determined by the incoming angle according to a kernel ${\mathcal{K}}$.
  • Figure 2: Point $(z,h(1,z)) \in \partial \mathcal{D}_\gamma$ has inwards-pointing normal $n(z,1)$, making angle $\theta(z)$ with the vertical. The ray from $(z,h(1,z))$ at angle $\beta$ relative to the normal is parametrized by $\ell_t (z,1,\beta)$, $t >0$. If the particle hits $\mathcal{D}_\gamma$ at point $(z,h(1,z))$ at incoming angle $\alpha$, then it reflects at outgoing angle $\beta$ drawn from ${\mathcal{K}}(\alpha , \, \cdot \,)$. In the picture, both $\alpha$ and $\beta$ are positive.
  • Figure 3: Two examples of the computation of the new incoming angle $\Theta ( z, j , \beta )$ as given at \ref{['eq:inangle']}. In case ( a), the next collision point is on the opposite side of the domain, and $\Theta ( z, j , \beta ) = \beta + \theta(z) + \theta( \Lambda_1 ( z,j,\beta) )$. In case ( b), the next collision point is on the same side of the domain and $\beta>0$, so $\Theta ( z, j , \beta ) = \pi -\beta - \theta(z) + \theta( \Lambda_1 ( z,j,\beta) )$.
  • Figure 4: An illustration of the horizontal increment between successive collisions.

Theorems & Definitions (61)

  • Proposition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.7
  • Proposition 3.1
  • Theorem 3.2
  • Theorem 3.4
  • Remark 3.5
  • ...and 51 more