# Stationary Quantum Markov processes as solutions of stochastic differential equations

Jürgen Hellmich; Claus Köstler; Burkhard Kümmerer

Banach Center Publications (1998)

- Volume: 43, Issue: 1, page 217-229
- ISSN: 0137-6934

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topHellmich, Jürgen, Köstler, Claus, and Kümmerer, Burkhard. "Stationary Quantum Markov processes as solutions of stochastic differential equations." Banach Center Publications 43.1 (1998): 217-229. <http://eudml.org/doc/208842>.

@article{Hellmich1998,

abstract = {From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.},

author = {Hellmich, Jürgen, Köstler, Claus, Kümmerer, Burkhard},

journal = {Banach Center Publications},

keywords = {quantum white noise; Brownian motion; stochastic integration; stochastic differential equations; quantum Markov process; stationary Markov processes; squeezed white noise; -white noise},

language = {eng},

number = {1},

pages = {217-229},

title = {Stationary Quantum Markov processes as solutions of stochastic differential equations},

url = {http://eudml.org/doc/208842},

volume = {43},

year = {1998},

}

TY - JOUR

AU - Hellmich, Jürgen

AU - Köstler, Claus

AU - Kümmerer, Burkhard

TI - Stationary Quantum Markov processes as solutions of stochastic differential equations

JO - Banach Center Publications

PY - 1998

VL - 43

IS - 1

SP - 217

EP - 229

AB - From the operator algebraic approach to stationary (quantum) Markov processes there has emerged an axiomatic definition of quantum white noise. The role of Brownian motion is played by an additive cocycle with respect to its time evolution. In this report we describe some recent work, showing that this general structure already allows a rich theory of stochastic integration and stochastic differential equations. In particular, if a quantum Markov process is represented by a unitary cocycle, we can reconstruct an additive cocycle ('quantum Brownian motion') and the unitary cocycle ('quantum Markov process') appears as the solution of a certain stochastic differential equation. This establishes a one-to-one correspondence between multiplicative and additive adapted cocycles. As an application of this result we construct stationary Markov processes, driven by squeezed white noise and q-white noise.

LA - eng

KW - quantum white noise; Brownian motion; stochastic integration; stochastic differential equations; quantum Markov process; stationary Markov processes; squeezed white noise; -white noise

UR - http://eudml.org/doc/208842

ER -

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