On the Existence Theory of Hilbert Space valued Diffusion Processes

TL;DR

This paper establishes a method for constructing solutions to specific Hilbert space-valued stochastic differential equations by splitting them into deterministic and stochastic parts, using an iterative approach inspired by semigroup theory.

Contribution

It introduces a novel splitting method for solving certain linear stochastic differential equations in Hilbert spaces, extending the Lie-Trotter product formula to this context.

Findings

Solution construction via splitting method demonstrated

Applicable to bounded operator stochastic equations and Schrödinger equations

Provides a new iterative approach for Hilbert space SDEs

Abstract

We prove that the solution of certain linear stochastic differential equations in Hilbert spaces, namely those with bounded operators as well as the conservative stochastic Schr\"odinger equations, can be obtained - along the lines of the Lie-Trotter product formula from semigroup theory - by splitting the equation into a ``deterministic'' and a ``stochastic'' part and alternately applying the corresponding solution flows in an iterative manner to the initial value.