Residence times and other functionals of reflected Brownian motion

We study the residence and local times for a Brownian particle confined by reflecting boundaries, and propose a general solution to the problem of finding the related probability distribution. Its Fourier transform (characteristic function) and Laplace transform (survival probability) are obtained in a compact matrix form involving the Laplace operator eigenbasis. Explicit combinatorial relations are derived for the moments, and the probability distribution is shown to be nearly Gaussian when the exploration time is long enough. When the eigenbasis (or a part of it) is known, the numerical computation of the residence time distributions is straightforward and accurate. The present approach can also be applied to investigate other functionals of reflected Brownian motion describing, in particular, restricted diffusion in an external field or potential (e.g., nuclei diffusing in an inhomogeneous magnetic field). Theoretical results for the local times are confronted with Monte Carlo simulations on the unit interval, disk, and sphere.

Figure 1

Three “strategies” for modeling the trajectory of the diffusing species with interactions in the $\epsilon$ vicinity of the interface (horizontal axis): (a) discrete random walks with step $\epsilon$, (b) Brownian flights with jumps at distance $\epsilon$, and (c) the reflected Brownian motion. In all three cases, the interaction is considered as a kind of reflection event.

Figure 2

Diffusive motion of species restricted inside a bounded confining domain $\Omega$ with reflecting boundary $\partial \Omega$. The residence time $\varphi$ indicates how long a diffusing species, starting somewhere in the domain (e.g., in the subset $A_0$), spends in a subset $A$ until time $t$. The local time shows how long the species spends in a close neighborhood ($\epsilon$ vicinity or $\epsilon$ sausage $\partial {\Omega }_{\epsilon }$) of the boundary $\partial \Omega$.

Figure 3

(Color online) Coefficient $b_{2,1}^{\epsilon }$ for different $\epsilon$ vicinities of the boundary of the unit interval.

Figure 4

(Color online) Probability distribution of the local time of reflected Brownian motion on the unit interval at times $t=1,5,10$ (stars, triangles, and circles, respectively), and its Gaussian approximation (26) with $b_{1,1}=2$ and $b_{2,1}=2∕3$ (dashed-dotted, dashed, and solid lines, respectively).

Figure 5

(Color online) Probability distribution of the local time of reflected Brownian motion on the unit interval at time $t=0.5$ (circles), and two Gaussian approximations: the basic approximation (26) with $b_{1,1}t=1$ and $b_{2,1}t=1∕3$ (dashed line); and a more accurate approximation with the same mean 1 but the variance $\prec {\varphi }^2\succ \simeq 0.31$, accounting for finite width $\epsilon =0.01$ (solid line).