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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.