Markov semigroup (continuous time)

A Markov semigroup is the continuous-time analogue of a Markov chain and is the natural language for equilibrium/non-equilibrium dynamics, including jump processes and diffusions.

Prerequisites: probability measure , expectation , master equation .

Definition (Markov semigroup)

A Markov semigroup is a family of linear operators $(P_t)_{t\ge 0}$ acting on bounded measurable functions $f$ on a state space $S$ such that:

1. (Semigroup) $P_0=\mathrm{Id}$ and $P_{t+s}=P_t P_s$ for all $t,s\ge 0$.
2. (Positivity) If $f\ge 0$ then $P_t f\ge 0$.
3. (Mass preservation) $P_t \mathbf{1}=\mathbf{1}$.

If $X_t$ is a time-homogeneous Markov process, then

Dual action on measures

For a probability measure $\mu$ on $S$, define $\mu_t=\mu P_t$ by

A measure $\pi$ is stationary if $\pi P_t=\pi$ for all $t\ge 0$.

Generator

The (infinitesimal) generator $L$ is defined on a suitable domain by

whenever the limit exists.

The semigroup then formally solves the backward equation

Forward equation and the master equation

On measures (or densities), the evolution is governed by the adjoint generator $L^*$:

For pure jump processes on a countable state space, this becomes the master equation with rate matrix $Q$.

Detailed balance

A stationary $\pi$ satisfies detailed balance (reversibility) when $L$ is self-adjoint in $L^2(\pi)$, equivalently when stationary probability currents vanish.