Question

Let's say I have a Brownian particle of some radius $r_b$ and coefficient of diffusion $D$, freely moving about in a toroidal/doughnut-shaped chamber with inner and outer radius $R_{inner}$ and $R_{outer}$, respectively. As such, the circle that can be rotated to generate the torus has radius $R_{chamber} = \frac{1}{2}(R_{outer}-R_{inner})$, and $r_b < R_{chamber}$, though the particle is not necessarily point-like. Now, let's also say that someone places an invisible, circular reflecting boundary (of radius $R_{chamber}$) somewhere inside the toroid s.t. the Brownian particle can no longer return to a previous position by travelling in a strictly clockwise or counterclockwise fashion around the torus.

Here one should presumably be able to watch the trajectory of the Brownian particle to assign a probability for the location of the reflecting boundary. In practice though, how should one proceed? Provided the particle's diffusion coefficient $D$, how does the one's confidence in the position of the reflecting boundary change over time?

Clarification - The heart of this question is: while it's obviously possible to do so, how does one actually go about mapping the position of a reflecting barrier (of known geometry or not) using the trajectory of a Brownian particle?

Answer 1

If a Brownian motion hits a reflecting barrier like that, it is almost surely going to hit multiple times. In fact, it will almost surely hit in a Cantor set of times and locations, just as an unreflected one-dimensional Brownian motion almost surely returns to the value at $t=0$. This type of behavior almost surely doesn't happen if there is no barrier. The probability that the maximums of an unreflected one-dimensional Brownian motion on disjoint intervals are equal is $0$.

Let $cw(t)$ be the angle of the plane through the axis which is tangent to the particle on the clockwise side at time $t$, and let $ccw(t)$ be the analogue for the counterclockwise side. If the minimum of $cw(t)$ is achieved multiple times, then almost surely the particle has reflected off a boundary at $\min (cw(t))$. If the maximum of $ccw(t)$ is achieved multiple times, then almost surely the particle has reflected off a boundary at $\max (ccw(t))$. If neither occurs then you have restricted the possible locations of the boundary, excluding an interval $(\min(cw(t)),\max(ccw(t)))$.