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

Question

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)

Answer 1

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

• Brownian Motion: Fluctuations, Dynamics, and Applications
• The Fokker-Planck Equation: Methods of Solutions and Applications
• Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences
• The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering
• Stochastic Processes in Physics and Chemistry

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

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