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Martin boundary of a degenerate Reflected Brownian Motion in a wedge

Maxence Petit

Abstract

We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green's functions associated with the process. These Laplace transforms are expressed as an infinite sum of products using the compensation method. We also derive the asymptotics of the Green's functions along all possible paths and determine the (minimal) Martin boundary. Finally, we provide explicit formulae for all the corresponding harmonic functions.

Martin boundary of a degenerate Reflected Brownian Motion in a wedge

Abstract

We consider an outward degenerate drifted Brownian motion in the quarter plane with oblique reflections on the boundaries. In this article, we explicitly compute the Laplace transforms of the Green's functions associated with the process. These Laplace transforms are expressed as an infinite sum of products using the compensation method. We also derive the asymptotics of the Green's functions along all possible paths and determine the (minimal) Martin boundary. Finally, we provide explicit formulae for all the corresponding harmonic functions.

Paper Structure

This paper contains 24 sections, 37 theorems, 168 equations, 8 figures.

Table of Contents

  1. Introduction and main results
  2. Definition of the process
  3. Functional equation, kernel and analytic continuation
  4. Compensation method
  5. Heuristic of the compensation method
  6. Parabola and automorphisms
  7. Explicit expression of Laplace transforms via the compensation approach
  8. Laplace inverse and saddle-point method
  9. Laplace inverse theorem and saddle-point.
  10. Contour of steepest descent
  11. Shift of the integration contours and contribution of the poles
  12. Negligibility of some integrals
  13. Proof of Theorem \ref{['thm1']}
  14. Asymptotics in Theorem \ref{['thm1']}
  15. Non nullity of $h_\alpha(z_0)$.
  16. ...and 9 more sections

Key Result

Theorem 1

Assume (drift) to (simple). Then, the Green's density function $g^{z_0}$ of this process has the following asymptotics when $\alpha \to \alpha_0$ and $r\to\infty$. where ${c_{\alpha_0} = \frac{1}{\sqrt{2\pi(\cos(\alpha_0) + \sin(\alpha_0))}}}$, $c^*$ and $c^{**}$ are positive explicit constants only depending on the parameters of the degenerate reflected Brownian motion (see eq:c*) and where $h_{

Figures (8)

  • Figure 1: Reflections $R_1, R_2$ on the edges, the drift $\mu$ and the direction $v$ of the degenerate Brownian motion. The process starting from $z_0$ does never reach the hatched zone.
  • Figure 2: Example of a typical path (restricted to a finite time) of the drifted degenerate Brownian motion. The initial point is marked in orange.
  • Figure 3: Geometrical interpretation of $(x(\alpha), y(\alpha))$, $\alpha^*$ and $\alpha^{**}$.
  • Figure 4: Parabola $\mathcal{P}$ and points $(a_n, b_n)$ on the parabola.
  • Figure 5: Martin boundary $\Gamma$ when $0 < \alpha^*$ and $\alpha^{**} < \pi/2$.
  • ...and 3 more figures

Theorems & Definitions (80)

  • Theorem 1: Asymptotics in the quadrant, general case
  • Theorem 2: Explicit expressions of Martin harmonic functions
  • Theorem 3: Martin Boundary
  • Definition 2.1: Degenerate reflecting Brownian motion
  • Theorem 4: Existence, uniqueness and Strong Markov property
  • proof
  • Proposition 2.2: Transience
  • proof
  • Proposition 2.3: Densities and Laplace transforms
  • proof
  • ...and 70 more