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

Probabilistic Representations of Ordered Exponentials: Vector-Valued Schr\"odinger Semigroups and the Combinatorics of Anderson Localization