Quantum Stochastic Cable Equation Acting on Functionals of Discrete-Time Normal Martingales

Abstract

Let M be a discrete-time normal martingale satisfying some mild conditions. Then Gel’fand triple S(M)⊂L2(M)⊂S⁎(M) can be constructed of functionals of M, where elements of S(M) are called testing functionals of M, while elements of S⁎(M) are called generalized functionals of M. In this paper, we consider a quantum stochastic cable equation in terms of operators from S(M) to S⁎(M). Mainly with the 2D-Fock transform as the tool, we establish the existence and uniqueness of a solution to the equation. We also examine the continuity of the solution and its continuous dependence on initial values.