On quantum stochastic differential equations

TL;DR

This paper establishes existence, uniqueness, and conditions for quantum stochastic differential equations with nontrivial initial conditions, advancing the mathematical framework for quantum stochastic processes.

Contribution

It provides new theorems for solutions of quantum stochastic differential equations with nontrivial initial conditions and characterizes conjugate pairs of quantum stochastic cocycles.

Findings

Proved existence and uniqueness theorems for quantum stochastic differential equations.

Derived necessary and sufficient conditions for quantum stochastic cocycles.

Presented an alternative approach to quantum stochastic convolution cocycles.

Abstract

Existence and uniqueness theorems for quantum stochastic differential equations with nontrivial initial conditions are proved for coefficients with completely bounded columns. Applications are given for the case of finite-dimensional initial space or, more generally, for coefficients satisfying a finite localisability condition. Necessary and sufficient conditions are obtained for a conjugate pair of quantum stochastic cocycles on a finite-dimensional operator space to strongly satisfy such a quantum stochastic differential equation. This gives an alternative approach to quantum stochastic convolution cocycles on a coalgebra.