Recurrence of Brownian Motion

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An interesting question to ask about a Brownian Motion process is whether or not it is recurrent. The answer to this question tells us whether or not a Brownian particle will re-enter some arbitrary spherical region infinitely many times.

The transition probabilities for a Brownian motion on \mathbb{R}^nare given by


p(x, t|x_0) = \left( \frac 1 {4 \pi D t} \right)^{n/2} \exp\left( - \frac {\|x - x_0\|^2}{4 D t}\right).

Let's exploit the spherical symmetry of this probability about x0 by defining r \equiv \|x-x_0\|, so we may write
p(r, t|x_0 = 0) = \left( \frac 1 {4 \pi D t} \right)^{n/2} \exp\left( - \frac {r^2}{4 D t}\right).

In order to determine recurrence properties, we must compute the probability that the particle lies within the sphere r < ε. We may approximate this arbitrarily closely by


p( r < \epsilon, t | r_0 = 0 ) = V_n(\epsilon) \left( \frac 1 {4 \pi D t} \right)^{n/2} \exp\left( - \frac {\epsilon^2}{4 D t} \right)

where Vn(ε) is the volume of the n-dimensional hypersphere of radius ε, by taking ε to be very small.

Now if we integrate p(r < ε,t | r0 = 0) over t, we get


\int_0^\infty dt\, p( r < \epsilon, t|r_0 = 0) = \left\{ \begin{matrix}\infty&:& n < 3\\\frac {V_n(\epsilon)}{4 \pi^{n/2} \epsilon^{n-2} D} \Gamma\left(\frac n 2 - 1\right)&:&n \geq 3\end{matrix}\right.

(Integration courtesy of Mathematica). This tells us that 1- and 2-dimensional Brownian motions are recurrent, but 3- and higher-dimensional Brownian motions are transient!

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