MarkovSemigruppen

Markov semigroups are a fundamental concept in probability theory and functional analysis, closely related to Markov processes. A Markov semigroup is a family of linear operators on a Banach space that satisfies certain properties. Specifically, for a given Banach space $X$ and a parameter $t \ge 0$, a Markov semigroup is a family of operators $(T_t)_{t \ge 0}$ such that: 1. $T_0$ is the identity operator. 2. For all $s, t \ge 0$, $T_{s+t} = T_s T_t$ (the semigroup property). 3. For all $t \ge 0$, $T_t$ is a bounded linear operator. 4. For all $t \ge 0$, $T_t$ maps the set of positive elements of $X$ to itself, and $\|T_t\| \le 1$ if $X$ is a space of functions where positive functions are mapped to positive functions.

These operators are often used to describe the evolution of probability distributions over time. If $P_t(x, dy)$ is the transition probability kernel of a Markov process, then the action of the semigroup operator $T_t$ on a function $f(x)$ often corresponds to the expected value of $f$ at time $t$, given the state at time 0. This can be expressed as $(T_t f)(x) = \int f(y) P_t(x, dy)$.

The study of Markov semigroups is crucial for understanding the long-term behavior of stochastic systems, including convergence to stationary distributions and the analysis of diffusion processes and other continuous-time Markov chains. The infinitesimal generator of a Markov semigroup provides a differential characterization of the semigroup, linking it to partial differential equations.