Characterization of unitary processes with independent and stationary increments

TL;DR

This paper extends previous work by characterizing stationary unitary increment Gaussian processes under weaker continuity assumptions, proving their equivalence to solutions of Hudson-Parthasarathy equations.

Contribution

It replaces the assumption of uniform continuity with weak continuity and establishes unitary equivalence to Hudson-Parthasarathy solutions under new technical conditions.

Findings

Proved unitary equivalence to Hudson-Parthasarathy solutions

Replaced uniform continuity with weak continuity

Extended characterization to broader class of processes

Abstract

This is a continuation of the earlier work \cite{SSS} to characterize stationary unitary increment Gaussian processes. The earlier assumption of uniform continuity is replaced by weak continuity and with a technical assumption on the domain of the generator, unitary equivalence of the processes to the solution of Hudson-Parthasarathy equation is proved.