Bounded velocity conjecture for minimizers in random time-dependent potentials

Let

$$U(x,t)=\sum_{j=1}^N U_j(x)a_j^\omega(t),\quad x\in\mathbf{R}^d,\ t\in\mathbf{R},$$

where the $U_j(x)$ are fixed non-random potentials of class $C^1$ satisfying condition, and $(a_j^\omega(t),1\leq j\leq N)$ is a realization of a stationary vector-valued random process with exponentially decaying correlation. Assume that $\omega$ belongs to the corresponding probability space and

$$\sup_{j,t}|a_j^\omega(t)|\leq 1$$

for almost all $\omega$. For $t\leq -1$, let $\gamma^x_{t,0}:[t,0]\to\mathbf{R}^d$ be a minimizer with $\gamma^x_{t,0}(0)=x$. Bounded velocity conjecture. There exists a random constant $C^\omega(x)$ such that, uniformly for all $t\leq -1$,

$$|\dot\gamma^x_{t,0}(0)|\leq C^\omega(x).$$

The conjecture asserts bounded terminal velocity for minimizers in a class of random, bounded time-dependent potentials, in contrast with the accelerating examples constructed in the paper. The source does not provide a resolution of this conjecture.