I have taken a course in path integrals and the teacher introduced an example with a Markovian motion.
Let $x$ and $x'$ two coordinates of a ball following a markovian motion.
The probability to go from $x$ to $x'$ is $p(x-x')$ : we don't need to know where the particle was before $x$ to be more accurate.
We have then :
$$ p(x_0,x_n)=\int dx_1...dx_n p(x_1-x_0)...p(x_n-x_{n-1})$$
Then he wrote : $n \rightarrow \infty$ and $n \epsilon = T$.
And : $p(x-x') \rightarrow e^{\frac{-(x-x')^2}{2D}}$
In fact I don't understand what he precisely mean.
Indeed here we have the integral of a product of probability, is there a theorem that tells us that it behaves like a gaussian ?
My problem is probably not well posed but I hope that you will understand what the teacher meant.
I am not looking for precise proof but I just want to get the globally what he does and which theorem he uses.