Probabilistic Representations of Ordered Exponentials: Vector-Valued Schrödinger 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 Schrödinger 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 Virág (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 $\mathbf{E}[\mathrm{\mid }\mathrm{\Psi }(t,x){\mathrm{\mid }}^2]$ of the solutions of the Schrödinger equation ${\mathrm{\partial }}_t\mathrm{\Psi }=-\mathsf{i}\mathcal{H}(t)\mathrm{\Psi }$ with a time-dependent Hamiltonian $\mathcal{H}(t)$. Using this, we show that when we take $\mathcal{H}(t)=-\mathrm{\Delta }+q(t,x)$ restricted to a finite box within ${\mathbb{Z}}^d$, where $q(t,x)$ is a possibly time-dependent Gaussian process, $\mathbf{E}[\mathrm{\mid }\mathrm{\Psi }(t,x){\mathrm{\mid }}^2]$ 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.