Aim of this paper is to take the first steps in the study of the Schrodinger problem for lattice gases (SPLG), which we formulate relying on classical results in large deviations theory. Our main contributions are a dynamical characterization of optimizers through a coupled system of PDEs and a precise study of the evolution and convexity of the quasi-potential along Schrodinger bridges. In particular, our computations show that, although SPLG does not admit a variational interpretation through Otto calculus, the fundamental geometric properties of the classical Schrodinger problem for independent particles still admit a natural generalization. These observations motivate the development of a Riemannian calculus on the space of probability measures associated with the class of geodesic distances studied in [3]. All our computations are formal, further efforts are needed to turn them into rigorous results.
Chiarini, A., Conforti, G., Tamanini, L., Schrödinger Problem for Lattice Gases: A Heuristic Point of View, Contributed paper, in Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) - 5th International Conference on Geometric Science of Information, GSI 2021, (PARIGI, 21-23 July 2021), SPRINGER INTERNATIONAL PUBLISHING AG, GEWERBESTRASSE 11, CHAM, CH-6330, SWITZERLAND 2021:12829 LNCS 891-899. 10.1007/978-3-030-80209-7_95 [https://hdl.handle.net/10807/232491]
Schrödinger Problem for Lattice Gases: A Heuristic Point of View
Tamanini, Luca
2021
Abstract
Aim of this paper is to take the first steps in the study of the Schrodinger problem for lattice gases (SPLG), which we formulate relying on classical results in large deviations theory. Our main contributions are a dynamical characterization of optimizers through a coupled system of PDEs and a precise study of the evolution and convexity of the quasi-potential along Schrodinger bridges. In particular, our computations show that, although SPLG does not admit a variational interpretation through Otto calculus, the fundamental geometric properties of the classical Schrodinger problem for independent particles still admit a natural generalization. These observations motivate the development of a Riemannian calculus on the space of probability measures associated with the class of geodesic distances studied in [3]. All our computations are formal, further efforts are needed to turn them into rigorous results.
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