# What is the return probability for Brownian motion in three dimensions?

I would like to know the probability of return to the initial point in three dimensional Brownian motion. Does someone know an expression for the diffusion constant? (Suggestions of books on this would also be welcome)

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An expression for the diffusion constant in terms of what? – Ron Maimon May 8 '12 at 6:10
You may also be interested in reading this: Stochastic Differential Equations: An Introduction with Applications, Bernt Oksendal, Sixth Edition, Example 7.4.2, p.125. – user77066 Apr 6 '15 at 7:53

## 1 Answer

For Brownian motion, Langevin equation, Fokker-Planck equations, Stochastic process.. from the viewpoint of physicists, the following are standard references:

For a free particle that starts from rest at $t=0$ and position $\mathbf{x}_0$, the probability density to find the particle at position $\mathbf{x}$ at time $t$ is $$p(\mathbf{x},t | \mathbf{x}_0, 0) = \frac{1}{(4\pi D t)^{3/2}} e^{-\frac{(\mathbf{x} - \mathbf{x}_0)^2}{4Dt}}$$ where $$D = \frac{k_BT}{6 \pi \eta a}$$ is the diffusion constant in three dimensions for a particle of radius $a$ immersed in a fluid of viscosity $\eta$ at temperature $T$. The probability density to find the particle at the same point that it started from, i.e., $\mathbf{x} = \mathbf{x}_0$, is therefore $$p(\mathbf{x}_0,t | \mathbf{x}_0, 0) = \frac{1}{(4\pi D t)^{3/2}}$$

See also this question in math.stackexchange

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This is incorrect in the last part--- you are giving the probability density at zero, not the probability of return, which strictly vanishes in 3d, and is only nonzero in the continuum limit in 2d (but vanishes logarithmically there). – Ron Maimon May 8 '12 at 6:09
@RonMaimon, What I meant was the transition probability density. Perhaps I am wrong. Can you post some references which I can consult? I will remove that part of my answer if I can understand where I erred., – Vijay Murthy May 8 '12 at 14:48
It's not a big or difficult error--- it's normally not even an error--- but OP asked "what is the probability that the particle returned to the origin?" and this means "what is the probability that the particle was exactly at point 0 at any time" and this probability is exactly 0 because the origin is a mathematical point in 3d with no size and the RW has fractal dimension 2. Your expression is the probability per unit volume that the particle is found in a small box around the origin at any one time. – Ron Maimon May 8 '12 at 14:57
Agreed. Will change it. I thought you were referring to the continuum version of Polya's recurrence problems which is why your statement confused me. – Vijay Murthy May 8 '12 at 15:10