Does the wave function decay? Does the wave function decay? Is the wave function subject to entropy? Does a particle decay while it's in superposition or does decay start when a measurement occurs?
 A: Decay of an initial density of particles governed by stochastic differential equation (SDE) dynamics to the invariant equilibrium density is a classic topic in statistical mechanics, for which you may see the book by Risken. Given a stochastic process $x_s \sim p(s,x)$, for all $0 \leq s$, with the SDE dynamics $d x_s = b(x_s) ds + \sigma dw_s$ and the corresponding linear Fokker-Planck (FP) partial differential equation (PDE) dynamics $\partial_t p = \mathcal{L}^\dagger p$, the question posed is: will any initial density decay to the invariant density $p^\infty(x)$ which solves the stationary FP equation and at what rate? Such analyses have been explored for certain systems, namely, Ornstein-Uhlenbeck and Langevin systems. The key idea is to consider the initial density as a perturbation from the equilibrium given as $p(0,x) = p^\infty(x) + p^\infty(x) \tilde{p}(0,x)$ and prove that $\tilde{p}(s,x)$ vanishes under the limit $s \rightarrow +\infty$, by using the eigen properties of the generator $\mathcal{L}^\dagger$ of the SDE which is indeed the adjoint of the forward FP operator $\mathcal{L}$. Therefore, properties of the SDE dynamics determine whether or not we can show that the equilibrium density is indeed attained by particles with any arbitrary initial density. This property is, in fact, the reversibility of the dynamics of the stochastic process.
Analogously for the linear Schrödinger equation, eigen properties and corresponding stability results have been analyzed, in particular see the comprehensive book by Berezen on this topic, with extensive exploration for various classes of Schrödinger potentials. Recently, these ideas have found applications in the theory of mean field games (MFGs) starting with the seminal work in the linear or integrator dynamics regime (similar to the Ornstein-Uhlenbeck case in the physics literature) and more recently in the case of Langevin dynamics. The latter work uses the imaginary-time linear Schrödinger equation formulation of the original problem, thus relying on eigen properties of the Schrödinger operator.
