Randomized Hamiltonian Feynman integrals and Schrödinger-Itô stochastic equations

Abstract

In this paper, we consider stochastic Schrödinger equations with twodimensional white noise. Such equations are used to describe the evolution of an open quantum system undergoing a process of continuous measurement. Representations are obtained for solutions of such equations using a generalization to the stochastic case of the classical construction of Feynman path integrals over trajectories in the phase space.

Adiabatic theorem for a class of stochastic differential equations on a Hilbert space

“…Remark 3 In a Banach space version of the proposition, the condition iL 2 is self-adjoint should be replaced by L 2 is a generator of isometries. The proof is technically more involved and requires a version of Trotter-Kato formula that does not seem to be available in the literature ( [12] assumes compact state space, while [8,6] assume the Hilbert space structure). In particular that if dL j , j = 1, 2 is the generator of a propagator U j (s, s ′ ), then the propagator U(s, s ′ ) generated by a sum dL 1 + dL 2 can be expressed as…”

Chernoff Approximation for Semigroups Generated by Killed Feller Processes and Feynman Formulae for Time-Fractional Fokker–Planck–Kolmogorov Equations

“…It is worth to mention that the method of Chernoff approximation has a wide range of applications. For example, this method has been used to investigate Schrödinger type evolution equations in [71,66,74,41,30,84,81,83]; stochastic Schrödinger type equations have been studied in [58,57,59,34]. Second order parabolic equations related to diffusions in different geometrical structures (e.g., in Eucliean spaces and their subdomains, Riemannian manifolds and their subdomains, metric graphs, Hilbert spaces) have been studied, e.g., in [19,15,69,14,67,82,70,7,20,90,18,89,17,13,12,86,11,10,85,56].…”

On a Stochastic Trotter Formula with Application to Spontaneous Localization Models

“…Stochastic versions of the Trotter formula have been considered since long ( [16]), but the underlying state space has been assumed to be locally compact, which excludes infinite-dimensional Hilbert spaces. More recently, a version of the Trotter formula for a class of stochastic Schrödinger evolutions with deterministic jump times and bounded collapse operators has been established ( [10]).…”