Let us first recall how Brownian motion is described in pure mathematics. For all $t\in [0,\infty)$, we have a probability distribution $\mathscr D_t:= \text{Normal}(0,t)$, and random variables $B_t$ drawn according to $\mathscr D_t$, s.t. almost surely, $[t\mapsto B_t]:[0,\infty) \to \mathbb R$ is a continuous function. (And of course other properties relating the value of Bronwian motion at different times $0\leq s<t<\infty$, like independent increments, Markovian/memorylessness, etc.). In this way, we get continuous sample paths of Brownian motion, all starting at $0$, and s.t. studying ("measuring") a bunch of paths at a fixed time $t=t_0$ we see a Gaussian probability distribution.
In quantum mechanics as far as I understand it (I am a complete novice to physics, but have a background in math), we have the position say of a particle at a time $t$ as a random variable distributed according the the probability distribution $\mathscr D_t := |\Psi(t)|^2$ (the Born rule).
Is it possible to define a stochastic process $X_t(\omega): [0,\infty)\times \Omega \to \mathbb R$ with $X_0 \equiv 0$ s.t. $X_t \sim \mathscr D_t$, and has continuous sample paths? In this way, we get particles that do have a definite position at all times (resolving the question of "measurements"), and also obey the exact same statistics that Schrodinger's equation predicts.
(Although I phrase it as a question relating to the measurement problem, this really is a completely mathematical/rigorous question, about constructing a stochastic process satisfying certain mathematical properties. I was thinking of posting to some of the math SE sites, but I thought maybe physicists might know better if this idea has appeared/been discussed before.)
