[Paper Review] The mean field Schrödinger problem: ergodic behavior, entropy estimates and functional inequalities

Julio Backhoff‐Veraguas, Giovani Conforti+2 more|University of Twente Research Information|May 7, 2019

Stochastic processes and financial applications25 references4 citations

TL;DR

This paper introduces and analyzes the Mean Field Schrödinger Problem (MFSP), a large-deviation-based framework for modeling the most likely evolution of interacting Brownian particles conditioned on initial and final configurations. By formulating the problem via a McKean-Vlasov control system and linking it to forward-backward stochastic differential equations (FBSDEs), the authors establish exponential convergence to equilibrium, derive novel functional inequalities involving the mean field entropic cost, and reveal a deep connection to Otto's Riemannian calculus on the space of probability measures.

ABSTRACT

We study the mean field Schrödinger problem (MFSP), that is the problem of finding the most likely evolution of a cloud of interacting Brownian particles conditionally on the observation of their initial and final configuration. Its rigorous formulation is in terms of an optimization problem with marginal constraints whose objective function is the large deviation rate function associated with a system of weakly dependent Brownian particles. We undertake a fine study of the dynamics of its solutions, including quantitative energy dissipation estimates yielding the exponential convergence to equilibrium as the the time between observations grows larger and larger, as well as a novel class of functional inequalities involving the mean field entropic cost (i.e. the optimal value in (MFSP)). Our strategy unveils an interesting connection between forward backward stochastic differential equations and the Riemannian calculus on the space of probability measures introduced by Otto, which is of independent interest.

Motivation & Objective

• To formalize the Mean Field Schrödinger Problem as a large deviation minimization problem under marginal constraints.
• To establish the existence and ergodic properties of mean field Schrödinger bridges (MFSBs) in the context of weakly interacting particle systems.
• To derive quantitative energy dissipation estimates and prove exponential convergence to equilibrium as the time horizon increases.
• To introduce and prove a new class of functional inequalities involving the mean field entropic cost, generalizing classical results in optimal transport and information theory.
• To uncover a novel connection between forward-backward stochastic differential equations and Otto's Riemannian calculus on the space of probability measures.

Proposed method

• Formulates the MFSP as a large deviation rate minimization problem for weakly interacting Brownian particles with drift interactions governed by a pair potential W.
• Establishes equivalence between the MFSP and a McKean-Vlasov control problem, where the control is the drift of the particle system.
• Derives a forward-backward stochastic differential equation (FBSDE) system that characterizes the optimal dynamics of the MFSB.
• Applies Otto's Riemannian calculus on the space of probability measures to interpret the dynamics in geometric terms.
• Uses time reversal and Girsanov-type transformations to derive relative entropy estimates and control the corrector term in the dynamics.
• Employs differential inequalities and Lyapunov-type arguments to prove exponential convergence to equilibrium and derive functional inequalities for the entropic cost.

Experimental results

Research questions

• RQ1 How does the mean field Schrödinger problem behave dynamically as the time horizon T increases, particularly regarding convergence to equilibrium?
• RQ2 What functional inequalities govern the mean field entropic cost, and how do they generalize classical inequalities in optimal transport and information theory?
• RQ3 What is the precise connection between the solution dynamics of the MFSP and forward-backward stochastic differential equations (FBSDEs)?
• RQ4 How can Otto's Riemannian calculus on the space of probability measures be used to understand the geometry of the MFSP solution?
• RQ5 What are the quantitative energy dissipation and convergence rates in the MFSP, and how do they depend on the interaction potential W?

Key findings

• The solution to the MFSP, known as the mean field Schrödinger bridge (MFSB), exhibits exponential convergence to equilibrium as the time horizon T grows, with a rate that depends on the interaction potential W.
• The paper establishes a novel class of functional inequalities involving the mean field entropic cost, which generalize the classical HWI and log-Sobolev inequalities.
• A quantitative upper bound is derived for the corrector term in the Girsanov transformation, crucial for proving relative entropy estimates and convergence.
• The dynamics of the MFSB are characterized by a forward-backward stochastic differential equation (FBSDE) system, linking the optimal drift to the Schrödinger potentials.
• The authors prove that the relative entropy H(P|Γ(P)) is finite if and only if H(P|Rµin) is finite, under appropriate regularity conditions on the drift.
• A new connection is revealed between the MFSP and Otto's Riemannian calculus on P2(Rd), providing a geometric interpretation of the optimal path measure.