Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains

Abstract

How long does a diffusing molecule spend in a close vicinity of a confining boundary or a catalytic surface? This quantity is determined by the boundary local time, which plays thus a crucial role in the description of various surface-mediated phenomena such as heterogeneous catalysis, permeation through semi-permeable membranes, or surface relaxation in nuclear magnetic resonance. In this paper, we obtain the probability distribution of the boundary local time in terms of the spectral properties of the Dirichlet-to-Neumann operator. We investigate the short-time and long-time asymptotic behaviors of this random variable for both bounded and unbounded domains. This analysis provides complementary insights onto the dynamics of diffusing molecules near partially reactive boundaries.

Statistics of boundary encounters by a particle diffusing outside a compact planar domain

“…n,L = −α K n (αL)I ′ n (αR) − I n (αL)K ′ n (αR) K n (αL)I n (αR) − I n (αL)K n (αR) , (D .2) with α = p/D and the prime denoting the derivative with respect to the argument (see similar computations in [23,24]). The decomposition (26) of the first-crossing time T ℓ implies that its probability density is obtained by convolution of two densities, which in the Laplace domain reads…”

“…In order words, while computing the statistics of the first-crossing times, we discard all trajectories that hit the circle of radius L up to time T ℓ . In Appendix D, we describe an extension of the spectral approach from [18,[23][24][25] to compute the density U L (ℓ, t|r 0 ) in the presence of the absorbing circle. Expectedly, the density U L (ℓ, t|r 0 ) is not normalized to 1, and 1 − ∞ 0 dt U L (ℓ, t|r 0 ) is the probability that the trajectory X t has crossed the circle of radius L before the associated boundary local time could reach the level ℓ.…”

“…In [18,23], alternative representations for the probability densities ρ(ℓ, t|x 0 ) and U(ℓ, t|x 0 ) were developed in terms of the eigenmodes of the Dirichlet-to-Neumann operator M p for an arbitrary Euclidean domain Ω with a smooth bounded boundary ∂Ω. For a given function f on ∂Ω, the operator M p associates another function g = (∂ n w)| ∂Ω on that boundary, where w satisfies the modified Helmholtz equation (p − D∆)w = 0 in Ω with Dirichlet boundary condition w| ∂Ω = f [59][60][61][62][63].…”

“…is the smallest eigenvalue of the Dirichlet-to-Neumann operator M p in the exterior of the disk, and K ν (z) is the modified Bessel function of the second kind [23]. The densities ρ(ℓ, t|R) and U(ℓ, t|R) follow immediately by taking the derivatives with respect to ℓ and t, respectively:…”

“…In turn, the integrals in Eqs. (21,23) exhibit logarithmically slow convergence at small z. A practical solution of this issue is discussed in Appendix B.…”