What is a semigroup in stochastic analysis?

Question

I just read a paper, in which the author wrote

we use the same notation for the Brownian semigroup on $C_b(\mathbb{R}_{+})$

Apart from this the paper made no other reference to semigroup. I'm wondering what is a semigroup and why we need it in stochastic analysis, is there any good/short introduction to semigroup in stochastic analysis? I have a good grasp on the general theory of stochastic calculus, for example, the content in Bernt Oksendal.

Answer 1

It is the semi-group of operators in $C_b(\mathbb R_{+})$ defined by $(T_t)f(x)=E(f(B(t)+x))$ where $(B_t)$ is standard Brownian motion.