Invariant Measure for Stochastic Schrödinger Equations

Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they defined as solutions so-called Stochastic Schrödinger Equations, which nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to study invariant measures trajectories. Particularly, we prove measure unique under an ergodicity condition on mean evolution, “purification” generator evolution. We further show converge in law exponentially fast toward this measure. illustrate our results with examples where can derive explicit expressions for