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The Theory of Open Quantum Systems$
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Heinz-Peter Breuer and Francesco Petruccione

Print publication date: 2007

Print ISBN-13: 9780199213900

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780199213900.001.0001

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Classical Probability Theory and Stochastic Processes

Classical Probability Theory and Stochastic Processes

The Theory of Open Quantum Systems

Heinz-Peter Breuer

Francesco Petruccione

Oxford University Press

This chapter contains a survey of classical probability theory and stochastic processes. It starts with a description of the fundamental concepts of probability space and Kolmogorov axioms. These concepts are then used to define random variables and stochastic processes. The mathematical formulation of the special class of Markov processes through classical master equations is given, including deterministic processes (Liouville equation), jump processes (Pauli master equation), and diffusion processes (Fokker–Planck equation). Special stochastic processes which play an important role in the developments of the following chapters, such as piecewise deterministic processes and Lévy processes, are described in detail together with their basic physical properties and various mathematical formulations in terms of master equations, path integral representation, and stochastic differential equations.

Keywords:   probability space, Kolmogorov axioms, random variables, stochastic processes, Markov processes, Chapman–Kolmogorov equation, Fokker–Planck equation, piecewise deterministic processes, Lévy processes

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