The law of large numbers for quantum stochastic filtering and control of many-particle systems

Theoretical and Mathematical Physics - There is an extensive literature on the dynamic law of large numbers for systems of quantum particles, that is, on the derivation of an equation describing...     

Complex Measures on Path Space: An Introduction to the Feynman Integral Applied to the Schro¨dinger Equation

A simple general approach to the construction of measures on path space is developed. It is used for the path integral representation of evolutionary equations including Feller processes, the Schro¨dinger equation, and dissipative Schro¨dinger equations. At the end of the paper we give a short guide to the immense literature on path integration sketching the main known approaches to the construction of the Feynman integral and indicating possible generalizations.     

The L��vy�CKhintchine type operators with variable Lipschitz continuous coefficients generate linear or nonlinear Markov processes and semigroups

Ito??s construction of Markovian solutions to stochastic equations driven by a Lévy noise is extended to nonlinear distribution dependent integrands aiming at the effective construction of linear and nonlinear Markov semigroups and the corresponding processes with a given pseudo-differential generator. It is shown that a conditionally positive integro-differential operator (of the Lévy?CKhintchine type) with variable coefficients (diffusion, drift and Lévy measure) depending Lipschitz continuously on its parameters (position and/or its distribution) generates a linear or nonlinear Markov semigroup, where the measures are metricized by the Wasserstein?CKantorovich metrics. This is a non-trivial but natural extension to general Markov processes of a long known fact for ordinary diffusions.