If $B_t$ is the standard Brownian motion it is known that its generator is the operator $L$ defined as
$$Lf(x)=\frac{1}{2}\Delta f(x)$$
and that its the probability density $u(t, x)$ of the process satisfies the heat equation in the sense that
$$\frac{\partial}{\partial t}u(t, x)=\frac{1}{2}\Delta_x u(t, x)$$.
I would like to understand which is the relation between the generator of a Markov process and its probability density function. Could someone give me an hint of the proof for the previous case or suggest me a book in which it is explained?
Thank you very much
The relationship between the generator of a Markov process and the evolution of its density can be viewed as the relationship between an operator and its adjoint. (It is also known as the "forward and reverse" equations in literature (I never fully got those names)) The operator $\frac{1}{2}\Delta$ is quite special because it is self-adjoint.
It is not a book, but this explains it in an (advanced) undergraduate level
https://www.math.nyu.edu/faculty/goodman/teaching/StochCalc2013/notes/Week9.pdf