Brownian motion and the heat equation - diffusion coefficient

Question

Einstein showed that the Brownian motion provides a solution to the heat equation. As written here, the relation between the brownian motion and the heat equation can be shown by the taylor series. Also i've seen in the literature that it is argued that the brownian motion has the Gaussian form of the probability density function, which is the solution to the heat equation. Those can also be related using the concept of the infinitesimal generator of the stochastic process $\frac{1}{2}\Delta$.

1. Can anybody please distinguish between those 4 ways and clarify more on the relation between the Brownian motion and the heat equation?

2. Specifically, how the variance (or diffusion coefficient) in the Brownian motion(its normal distribution) can be related to the diffusion coefficient in the heat equation?

3. Can we find a relation between the Brownian motion on a sphere and the heat equation on sphere?

Thanks.