Members working in this area: Ana Bela Cruzeiro, Léonard Monsaingeon, Jean-Claude Zambrini

Some recent students and postdocs: Neeraj Bhauryal, Xin Chen, Qiao Huang, Alexandra Symeonides

Our members working in this area focus mostly on the following topics:

- Feynman integrals
- Stochastic deformation
- Stochastic geometric mechanics
- Optimal transport

although other topics in stochastic analysis, such as stochastic optimal control problems and stochastic partial differential equations, are also studied by the group.

Stochastic analysis is the analytical study of random phenomena. It is based on works of Wiener, Itô, Cameron-Martin, Girsanov and others. Traditionally we have been developing Malliavin calculus, a theory of differentiation on (flat as well as curved) path spaces equipped with Wiener measure, which is adapted to very irregular functionals generated by Itô calculus.

Recently a large part of our activity in this area is devoted to stochastic geometric mechanics. Geometric mechanics is a field applying geometric methods to various mechanical systems, from mechanics of particles to fluid dynamics. We develop new stochastic geometric mechanics approaches, extending geometric mechanics of classical (deterministic) dynamical systems to the case of random ones, using Lie group-valued diffusion processes. Randomness may be explained by a suitable noisy perturbation of the system or, more fundamentally, as being at the heart of the phenomena usually described by deterministic equations. Symmetry reduction yields deterministic as well as stochastic variational principles for dissipative equations of motion. In infinite dimensions hydrodynamical equations such as incompressible (or compressible, by considering advected quantities formulated via semidirect products) Navier-Stokes are derived, without appealing to thermodynamics. Also a second-order (Meyer-Schwartz) differential geometry can be elaborated, this time motivated by quantum mechanical analogies.

Stochastic deformation was started in 1984-6, founded on a forgotten variational problem of Schrödinger (1931), and motivated by the puzzling probabilistic content of quantum mechanics. Part of the results obtained there provided a mathematical re-interpretation of the Feynman path integral approach, in terms of two adjoint parabolic equations. The method, however, is much more general and can be regarded as a systematic way to deform classical structures into probabilistic ones, along the sample paths of appropriate stochastic processes. Such processes have been referred to as Bernstein reciprocal, variational, or even local Markov or two-sided Markov. Feynman considered informally only diffusion and jump processes but our Stochastic deformation is applicable to any kind of processes. All associated probability measures are intrinsically invariant under time reversal in a more general sense than the one traditionally considered by probabilists.

This program can also be regarded from the perspective of statistical mechanics, more precisely quantum statistical physics, where the concept of entropy is natural. The community of mass transportation has recently re-discovered “Schrödinger’s problem” and used it profitably as a regularization tool. It is now intensively used and developed.

Feynman’s path integral theory has a curious status in mathematical physics. On one hand, it is hard to find new ideas of quantum or statistical physics which cannot be formulated more elegantly in these terms. On the other hand, it still basically lacks a rigorous mathematical foundation. There are, however, mathematical counterparts of some aspects of Feynman’s approach. The first one, due to M. Kac, is regarded as the “Euclidean” version of Feynman’s original representation of the solution of Schrödinger equation by a path integral where, instead, the associated heat equation (with potential) is considered. The familiar relation between the free heat equation and Wiener measure is the key of this approach. Our solution of Schrödinger’s problem is a completely different Euclidean framework, with a richer dynamical content.

The relation between Schrödinger original variational principle and mass transportation theory is due to Mikami in 2004 (cf. T. Mikami, “Stochastic Optimal Transportation”, Springer, 2021, for its history). A way to interpret this principle is as a problem of entropy minimization (H.Föllmer) converging toward Benamou-Brenier fluid mechanical interpretation of deterministic optimal transport. This provides deep insights into fundamental questions such as curvature, geometry, functional inequalities, to name a few, and extends to more general situations than independent Brownian particles in Euclidean spaces. Moreover, such a stochastic entropic deformation of deterministic optimal transport can be leveraged to derive fast, parallelizable algorithms to accurately approximate otherwise dimension-cursed and thus computationally unfeasible optimal transport problems.

Some recent key publications:

- M. Arnaudon, A. B. Cruzeiro and J. C. Zambrini, An entropic interpolation problem for incompressible viscous fluids,
*Ann. Inst. Henri Poincaré Probab. Stat.*56 (2020), 2211-2235

URL: https://doi.org/10.1214/19-AIHP1036 - X. Chen, A. B. Cruzeiro and T. S. Ratiu, Stochastic Variational Principles for Dissipative Equations with Advected Quantities, J
*. Nonlinear Sci.*33 (2023), 5, 62 pp.

URL: https://doi.org/10.1007/s00332-022-09846-1 - A. B. Cruzeiro, D. D. Holm and T. Ratiu, Momentum Maps and Stochastic Clebsch Action Principles,
*Comm. Math. Phys.*357 (2018), 873-912

URL: https://doi.org/10.1007/s00220-017-3048-x - Q. Huang and J. C. Zambrini, From second order differential geometry to stochastic geometric mechanics, J
*. Nonlinear Sci.*33 (2023), 6, 127 pp.

URL: https://doi.org/10.1007/s00332-023-09917-x - L. Monsaingeon, L. Tamanini and D. Vorotnikov, The dynamical Schrödinger problem in abstract metric spaces,
*Adv. Math.*426 (2023), 109100

URL: https://doi.org/10.1016/j.aim.2023.109100