  We provide two applications of an elementary (yet seemingly unknown)
probabilistic representation of matrix ordered exponentials, which generalizes
the Feynman-Kac formula in finite dimensions and the change of measure formula
between two continuous-time Markov processes on a finite state space.
  Our first and main application consists of a new Feynman-Kac formula for a
class of vector-valued Schr\"odinger operators on the line, which is driven by
two sources of randomness: The usual Brownian motion, and a continuous-time
Markov process on a finite state space. An important feature of these formulas
-- which is at the core of our motivation -- is that they enable the
calculation of the joint moments of the semigroup kernels when the matrix
potential function contains a continuous Gaussian noise. In particular, our
moment formulas shed new light on what the joint moments of the Feynman-Kac
kernels of the multivariate stochastic Airy operators of Bloemendal and Vir\'ag
(Ann. Probab., 44(4):2726--2769, 2016.) should be; we state a precise
conjecture to that effect, which we pursue in a forthcoming paper.
  Our second application consists of Feynman-Kac formulas for the expected
square modulus $\mathbf E\big[|\Psi(t,x)|^2\big]$ of the solutions of the
Schr\"odinger equation $\partial_t\Psi=-\mathsf i\mathcal H(t)\Psi$ with a
time-dependent Hamiltonian $\mathcal H(t)$. Using this, we show that when we
take $\mathcal H(t)=-\Delta+q(t,x)$ restricted to a finite box within $\mathbb
Z^d$, where $q(t,x)$ is a possibly time-dependent Gaussian process, $\mathbf
E\big[|\Psi(t,x)|^2\big]$ can be written as a relatively simple expectation
that involves self- and mutual-intersections of random walks. In particular,
this formula hints at a unified combinatorial mechanism that explains the
occurrence of localization for both time-dependent and time-independent noises.
