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

Pierre Yves Gaudreau Lamarre
Nov 2023
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.
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