What Does K-Dimensional Brownian Motion Represent in a Fokker-Planck Equation?

[coolnessitself]

I'm trying to write a fokker Planck equation for a particular SDE, but I'm caught up on an aside by the author I'm following.
He has a SDE with drift $$b \in \mathbb{R}^n$$, a dispersion matrix $$\sigma \in \mathbb{R}^{n\times k}$$, and k-dimensional brownian motion $$W_t$$, resulting in something like this
$$dX_t &=& b(X_t)dt + \sigma(X_t)dW_t$$
My confusion comes from this k-dimensional brownian motion. What is this k-th dimension? I'm guessing that X_t will take a value in n-d,and W_t is k-by-1, so that would make sigma like a covariance matrix. But what do these k components actually mean physically?

Thanks

 

[Evalesco]

It sounds like you're trying to understand what the k-dimensional Brownian motion means physically in the Fokker-Planck equation. The k-dimensional Brownian motion is basically a collection of stochastic processes, each of which has its own independent Wiener process. Each component can be thought of as representing a different dimension of the system, so that the noise in each dimension is independent of the others. The covariance matrix is then used to describe how the different components are correlated. Hope this helps!