Proposition: White support-noise generator

In the white Gaussian displacement model, averaging the stochastic Hamiltonian gives

For $\kappa =\pm M{\omega }^2$, its coefficient scales as $M^2{\omega }^4{S}_u/(2{\hslash }^2)$. It is zero when $\kappa =0$.

Proof. Represent the force term by $-\kappa x\sqrt{S_u}\circ \mathrm{d}{W}_t$ in the stochastic Hamiltonian. The Stratonovich density equation has stochastic superoperator

Its Itô correction is $\frac{1}{2}{B}^2\rho =-{\kappa }^2{S}_u[x,[x,\rho ]]/(2{\hslash }^2)$. The stochastic integral has zero mean. The moment-domain or random-unitary realization in Assumption ass:admitted justifies the averaging.