Determine the prefactor for arbitrary starting points and wedge angles

Determine the stationary harmonic Robin solution and the associated survival-probability prefactor B_alpha(qx_0) for a partially reactive infinite wedge of arbitrary opening angle and a generic starting point.

Background

The paper derives the long-time survival asymptotics for a Brownian particle in an infinite wedge with partially reactive sides by matching an outer effectively absorbing solution to an inner stationary harmonic Robin problem. For wedge angles alpha = pi/n, the inner problem is solved exactly through a terminating harmonic polynomial, which yields an explicit prefactor for arbitrary starting points.

For noninteger pi/alpha, the authors propose a conjectural formula for the connection coefficient b_nu and obtain an explicit apex prefactor if that formula holds. However, the harmonic function u_nu, and consequently the prefactor for a generic starting point away from the apex, is not explicitly determined.

References

For a generic starting point, however, an explicit determination of the harmonic function $u_\nu(\rho,\phi)$, and hence of $B_\alpha(qx_0)$, remains an open problem.

— Survival in a partially reactive wedge  (2609.02665 - Grebenkov, 2 Sep 2026) in Section 'Arbitrary angle'