proof of existence of the Brownian motion

Question

I just come about with the definition of Brownian Motion and I'd like to know whether it is not trivial that such a process exists.

Earlier this year, I read about Kolmogorov's theorem and I thought that maybe this could prove the existence of Brownian motion, but unfort. I didn't catch the idea of Kolmogorv's theorem.

So, can one set properties "randomly" on a stochastic process and expect it to exists ?

I also add the folowing question : is such a process unique?

I wasn't that shocked reading about poisson's process. Perhaps, because of the jumps, it feels more natural. On the other hand, Brownian motion seems pretty restrictive and I have no idea how such a process could potentially exists.

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My definition of Brownian Motion is the following :

1. $$ B_0 = 0$$
2. $$ \text{ stationary and independant increment } $$
3. $$ B_t \sim N(0,t) $$
4. $$ \text{It has continuous sample paths.} $$