Is anyone aware of a time-dependent wave-function that is non-reversible? With non reversible, I mean that the initial state of some wavefunction as solution to some PDE, $\psi_0(x_1,\dots,x_n,t)$ in $\mathbb{R}^n$ subjected to some *random *perturbation $\hat{H}$ is not the same if you reverse the time direction from a final state $\psi_{fin}(x_1,\dots,x_n,t)$ (the final state of the wave-function after perturbation) backwards, hence it will not give $\psi_0(x_1,\dots,x_n,t)$ if you time-reverse it, and apply the perturbation as a completely random-perturbation.
Basically,
$$\psi_0 \xrightarrow{\hat{H}} \psi_{fin} $$
while
$$\psi_0 \not\xleftarrow{\hat{H}} \psi_{fin} $$
where $\hat{H}$ is a completely random perturbation.
Is there some model in physics known to the readers that has similar properties, i.e. in QM or QED?
If so, can some PDE's admit such "weird" solutions?
Thanks