A factorization formula for the partition function in the semi-discrete parabolic Anderson model

We consider a continuous-time simple symmetric random walk on the integer lattice ${\mathbb{Z}}^d$ in dimension $d\ge 3$, subject to a random potential given by a field of two-sided Wiener processes. In the high-temperature regime, we prove the existence of the $L^2$- and almost sure limits of the partition function as time $t\to \pm \mathrm{\infty }$, and show that these limiting partition functions are positive almost surely. Our main result is a factorization formula for the point-to-point partition function, which is shown to be valid up to any sub-ballistic scale.