How to construct a Wiener-like process on Tori?

Question

That means the process may be periodic by distribution. And in what condition could the process be present as a series of Fourier bases multiplied by random variables?

Answer 1

Brownian motion on a manifold starts with constructing the heat equation problem on it. One main reference is Stochastic Analysis On Manifolds.

As mentioned in Example 4.34 Heat Equation and Brownian Motion on Riemannian Manifolds, the transition density on the torus $\mathbb{T}^{n}:=\mathbb{R}^{n}/\mathbb{Z}^{n}$ is

$$\large p_{\mathbb{T}^{n}}(x,y,t):=\frac{1}{(4\pi t)^{n/2}}\sum_{m\in \mathbb{Z}^{n}}e^{-\|y'+m-x'\|^{2}/4t},$$

where for $\pi:\mathbb R^{n}\to\mathbb{T}^{n}$, we let $x',y'$ be two representatives such that $\pi(x')=x,\pi(y')=y$ and $\|\cdot\|$ is the Euclidean-norm.