A transition semigroup is a family of operators that describe the evolution of probability distributions over time in a Markov process, providing a powerful framework for analyzing stochastic processes. These semigroups capture the dynamics of the system by mapping initial distributions to their future states, and are fundamental in the study of continuous-time Markov processes and their generators.

Transition Semigroup

transition semigroup

family of operators

evolution of probability distributions

A Markov process is a stochastic process that satisfies the Markov property, meaning the future state is independent of the past given the present state. It is widely used in various fields to model random systems that evolve over time, where the next state depends only on the current state and not on the sequence of events that preceded it.

Markov process

framework for analyzing stochastic processes

dynamics of the system

initial distributions

future states

continuous-time Markov processes

Generators are special functions in programming that allow you to iterate over a sequence of values lazily, meaning they produce items one at a time and only as needed. This efficient handling of data makes them ideal for working with large datasets or streams where loading everything into memory at once is impractical.

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