Brownian motion motivation of construction

Question

I have read Stochastic Differential Equations by Bernt Oksendal
It constructs Brownian motion by Kolmogorov extension theorem by consider $p(t,x,y)=(2\pi t)^{-n/2} e^{- \frac{|x-y|^{2}}{2t}}$
Why Brownian motion is construct in such way? Why such $p(t,x,y)$ is considered. Why this construction can model Brownian motion.
I had asked the same question but somebdy migrate my question to physics SE.I can't delet it so I modify my question and ask again.Please don't move it to somewhere else. I do ask these in mathematics meaning.

Answer 1

The easiest way to the density is by trying to model Brownian motion as the limit of random walks. I will keep it informal about the kind of convergence we get and for which kinds of steps these are valid.

Let's suppose you have a random walk that is given by $W_{k\Delta t} = \sum_{i=1}^k X_i$ where every step $X_i$ is independent of each other and has variance $\sigma^2$. That means that $$ \text{Var} \,W_{k\Delta t} = k \,\sigma^2 $$ which means that: $$ \text{Var} \frac{W_{k\Delta t}}{\sqrt{k}} = \sigma^2 $$

Now we take the limit $k\to\infty$, and $k\Delta t \to s$, we get a variable: $$ B_s = \lim_{k\to \infty} \frac{W_{k\Delta t}}{\sqrt{k}} = \lim_{k\to \infty} \frac{\sum_{i=1}^k X_i}{\sqrt{k}} $$ By the central limit theorem this converges to a gaussian variable, from where you get form of the law on your book.

For the technical detais, look the Wikipedia article referenced.