How long does it take a Brownian particle to achieve a uniform probability distribution across a space? Imagine I have a point-like Brownian particle, with diffusion constant $D$, and I place it at some initial coordinate in a cage of known geometry.  Assuming the volume $V$ of the cage is "everywhere" accessible, how long does it take the particle to lose its "memory" of the initialization coordinate and achieve a uniform probability distribution across the cage?  How precisely can one measure this, and how important is the geometry of the cage?
 A: Let $\rho(t,x)$ be the probability that the particle is at location $x$ at time $t$.  $\rho$ satisfies the equation $\rho_t = \rho_{xx}$, with $\rho(0,x)=\delta(x-x_0)$ where $x_0$ is the starting position.  You have Neumann boundary conditions on the boundary of your domain.  Suppose the eigenfunctions of the negative Laplacian of your domain with Neumann boundary conditions are $\phi_0, \phi_1, \ldots$ with corresponding eigenvalues $\lambda_0 \leq \lambda_1 \leq \lambda_2 \cdots$.  $\phi_1$ is constant with $\lambda_0=0$.  The solution to the equation is
$$
\rho(t,x)= \sum_{j=0}^\infty c_j e^{- \lambda_j t} \phi_j (x)
$$
where $c_j$ is the inner product of $\delta(x-x_0)$ with $\phi_j$ which gives $c_j=\phi_j(x_0)$.  When $t$ goes to infinity you just get a constant for $\rho$.  Your question is (I think) equivalent to asking how long does it take for all the $j>0$ terms to die out.  Assuming $c_1 \neq 0$, this will be determined by $\lambda_1$; a bigger $\lambda_1$ means faster approach to uniform probability.  There is not going to be an explicit formula for $\lambda_1$ for most domains.  But you can get some intuition.  As Alice says, if you have two equal volumes with a narrow connection, then $\lambda_1$ is going to be very small and it will take a long time for $\rho$ to be flat.  (Unless you start exactly midway between the two volumes.  This corresponds to $c_1=0$.)
A: Seems like the timescale should depend on the number and sizes of spherical balls it takes to cover the entire space $V$.  For example if your space has a narrow neck connecting two balloons then it would take a while for your particle to diffuse from the balloon where it started through the neck to the other side.
I would have entered as a comment but can't yet (sorry).
A: For the initial distribution being delta function $\delta(y)$ the result will be the Green function of the  Neumann problem for the heat equation. If the domain is regular enough the function can be written as
$$
G(x,y,t)=\sum_{n=1}^\infty \varphi_n(x)\varphi_n(y)e^{-\lambda_n t}
$$
where $\lambda_n$ and $\varphi_n$ are eigenvalues and normed in $L_2$ eigenfunctions of the corresponding elliptic Neumann problem. The first eigenvalue $\lambda_1=0$. So the rate of convergence to constant is exponential and determined by the next value $\lambda_2>0$.
