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

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.