I have a Wiener process in 3 dimensions, with time-independent drift and diffusion, ie x \in \mathbb{R}^3,

where \sigma(x) is diagonal. I also have an initial distribution/density p_0(x) at t = 0. Importantly, the domain is infinite (ie x is not bounded).

I would like to obtain an approximation of the distributions, eg densities p_t(0). I have already simulated this and everything is nice and smooth, so I am wondering if I could be using spectral methods for this.

Reading up about the problem, almost all texts that I have seen recommend pseudospectral methods where they convert to an ODE and solve that. But I would like to understand *why* one cannot treat the whole domain (ie \mathbb{R}^3 \times [0, T]) with spectral methods. The only package I have seen that seems to do this is ApproxFun.jl.

Is there a numerical problem with using a spectral representation for every dimension, **including time**, that the approach of Olver and Townsend somehow avoids? Or is the approximation is OK, I could be using eg a rationally transformed Chebyshev basis as well?

(Note: answers can assume that I am clueless about numerical solutions to PDEs and recommend nice intro texts. I have Boyd (2000)).