[Paper Review] Expected exit times of Brownian motion from planar domains: Complements to a paper of Markowsky

Mark W. Coffey|arXiv (Cornell University)|Mar 22, 2012

Mathematical Dynamics and Fractals18 references3 citations

TL;DR

This paper extends Markowsky's work on expected exit times of Brownian motion from planar domains using conformal mapping and potential theory. It provides exact analytical solutions for expected exit times in wedge-shaped regions, half-discs, n-gons, and other symmetric domains, deriving closed-form expressions involving generalized hypergeometric functions and special functions like the Appell F1 function and dilogarithms.

ABSTRACT

We supplement a very recent paper of G. Markowsky concerned with the expected exit times of Brownian motion from planar domains. Besides the use of conformal mapping, we apply results from potential theory. We treat the case of a wedge-shaped region exactly, subsuming an example of Markowsky. A number of other results are presented, including for a half disc, $n$-grams, and a variety of other regions.

Motivation & Objective

• To extend Markowsky's recent results on expected exit times of Brownian motion from simply connected planar domains.
• To provide exact analytical solutions for expected exit times in symmetric and polygonal domains using conformal mapping and potential theory.
• To derive closed-form expressions for expected exit times in wedge-shaped regions, half-discs, and regular n-gons.
• To apply potential theory and Green's function methods to solve the Poisson equation with Dirichlet boundary conditions.
• To explore connections between probabilistic exit times and special functions such as generalized hypergeometric functions and the Appell F1 function.

Proposed method

• Utilizes conformal mapping techniques to transform the domain of interest into the unit disc, enabling use of known results for Brownian motion exit times.
• Applies the formula $ E_{f(0)}[ au(f(D))] = \frac{1}{2} \sum_{n=1}^{\infty} |a_n|^2 $, derived from the optional stopping theorem for martingales, to compute expected exit times.
• Solves the Poisson equation $ \nabla^2 u = -2 $ in the domain with Dirichlet boundary conditions using the Green's function $ G(x,y) $, where $ E_x[\tau] = \int_R G(x,y) dy $.
• Employs special functions such as the generalized hypergeometric function $ {}_4F_3 $, the Appell F1 function, and the dilogarithm $ \text{Li}_2 $ to express solutions in closed form.
• Derives integral representations and series expansions for logarithmic and Green's function integrals using orthogonality and Fourier series techniques.
• Uses Lemma 3 and Proposition 6 to evaluate angular integrals involving logarithmic terms, leading to expressions in terms of dilogarithms and trigonometric series.

Experimental results

Research questions

• RQ1 What is the exact expected exit time for Brownian motion starting at the origin in a wedge-shaped region with angle $ \pi/m $?
• RQ2 How can the expected exit time for a half-disc be computed exactly using potential theory and conformal mapping?
• RQ3 What closed-form expression exists for the expected exit time in a regular n-gon using generalized hypergeometric functions?
• RQ4 How do the solutions for symmetric polygonal domains (e.g., equilateral triangle, rectangle) relate to eigenfunctions of the Laplacian with Dirichlet boundary conditions?
• RQ5 Can the Green's function method be systematically applied to derive expected exit times for non-symmetric or irregular planar domains?

Key findings

• The expected exit time for Brownian motion in a wedge of angle $ \pi/m $ is derived exactly, subsuming Markowsky’s example and generalizing it via conformal mapping.
• For a half-disc of radius $ r_0 $, the expected exit time from the center is $ \frac{1}{2}(r_0^2 - r^2) $, recovering the known result via Green's function integration.
• The expected exit time for a regular m-gon $ U_m $ is given by $ E_0[\tau(U_m)] = \frac{m^2}{2B^2(1/m, 1-2/m)} \cdot {}_4F_3\left(\frac{1}{m}, \frac{1}{m}, \frac{2}{m}, \frac{2}{m}; 1+\frac{1}{m}, 1+\frac{1}{m}, 1; 1\right) $, involving the Beta and generalized hypergeometric functions.
• The paper derives a new integral identity involving the dilogarithm: $ \int_0^\pi \ln(r^2 + \rho^2 - 2r\rho\cos(\theta \mp \phi)) d\phi = 2\pi \ln r \pm i[\cdots] $, expressed in terms of $ \text{Li}_2 $ functions.
• An exact solution for the Poisson equation $ \nabla^2 u = -2 $ in a disc is recovered via Green's function integration, yielding $ u(r) = \frac{1}{2}(r_0^2 - r^2) $, confirming consistency with known results.
• The paper provides a systematic method to evaluate angular integrals of logarithmic terms using Fourier expansions and Pochhammer symbol identities, enabling the derivation of special function representations.