# What is the characteristic functional for Brownian motion on a sphere?

I'm a physicist, somewhat familiar with stochastic processes, but I'm a little unsure of what follows. What I basically have is a complicated quantity involving a vector that is equivalent to diffusion on a unit sphere. I need to average this quantity, and I am able to do so if I have the characteristic functional of the process. Here are the details:

Suppose we have standard Brownian motion in $d$ dimensions, represented by the usual Langevin equation

$\frac{d}{dt} \mathbf{x}(t) = \boldsymbol{\xi}(t)$,

where $\boldsymbol{\xi}(t)$ is Gaussian noise. Then the characteristic functional (not function) is a Gaussian functional,

$P[\mathbf{k}] = \langle e^{i \int_{0}^{\infty} \mathbf{k}(t') \cdot \mathbf{x}(t') \, dt'} \rangle = e^{i \int_{0}^{\infty} k_{i}(t') \langle x_{i}(t') \rangle \, dt' - \frac{1}{2} \int_{0}^{\infty} \int_{0}^{\infty} k_{i}(t') \langle x_{i}(t') x_{j}(t'') \rangle k_{j}(t'') \, dt' dt''}$.

In the above equation, repeated Cartesian indices are implicitly summed.

I'd like the equivalent of $P[\mathbf{k}]$ for Brownian motion on a unit sphere in $d$ dimensions (or even just $3$ dimensions). From what I've read in the literature, this process has the Langevin equation

$\frac{d}{dt} n_{i}(t) = \sqrt{2 D} (\delta_{i j} - n_{i}(t) n_{j}(t)) \xi_{j}(t) - 2 D n_{i}(t)$,

where $\boldsymbol{\xi}(t)$ is again Gaussian noise which we can take to be uncorrelated.

What is the characteristic functional for this process? Is it Gaussian? If it is not Gaussian and the formula is unobtainable, can I justify approximating it by a Gaussian?

Edit: For full information, the quantity that I need to compute is $\bigg\langle e^{i \mathbf{a} \cdot \int_{0}^{t} \mathbf{n}(t') \,dt'} \frac{d}{d t} \mathbf{n}(t) \bigg\rangle$ for arbitrary $\mathbf{a}$. If there is an easier way to do this, that would be great. I am particularly interested in the long time limit.

Edit 2: Okay, I believe the characteristic functional is definitely not Gaussian. For rotational diffusion, the probability distribution is a sum of spherical harmonics. That should be equivalent if we set the $\mathbf{k}(t) = \mathbf{k} \delta(t)$ and Fourier invert. Anyway, is there an easy way to derive this functional from a higher dimensional Brownian motion or justify the Gaussian approximation?

• I am not familiar with the physicists' notation&terminology for stochastic differential equations (= Langevin equations, right?). But if $n(t)$ is BM on a sphere, then why do you think that $\frac{\partial}{\partial t} n(t)$ exists? Or in what sense? Also, if $n(t)$ is BM on a sphere, then its distribution should be supported on that sphere, right? So it can't be Gaussian (at least not in the usual sense). Rogers&Williams, Diffusions, Markov Processes and Martingales, Vol 2, might be helpful. See Section 31 "Brownian motion on a submanifold of $R^n$". Example (31.33) treats BM in $S^2$. – UwF Jan 15 '14 at 13:55
• Thanks for the reply! Yes, Langevin equations are the physicist term for stochastic differential equations. When we write $\frac{d}{dt} n(t) = \mathrm{stuff}$, we mean that it should be implicitly interpreted as an integral over stuff. I had a look at that book you suggested, but I don't think it is getting me any closer to what I want. – user2333829 Jan 15 '14 at 17:59
• Locally BM on a sphere will look just like BM in the plane, so I don't think it's trajectories can be differentiable. That's why I asked in what sense you want to take the derivative $\frac{d}{dt}n(t)$. If you want the velocity to exist, e.g., if you want something like a Brownian particle constrained to move in a sphere, then I think you have to include a friction term, like in the Ornstein-Uhlenbeck equations. – UwF Jan 16 '14 at 11:16