# 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

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 $$p(\mathbf{x}_0,t | \mathbf{x}_0, 0) = \frac{1}{(4\pi D t)^{3/2}}$$