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Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges

Giovanni Conforti

Abstract

We investigate the quadratic Schrödinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schrödinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schrödinger bridges satisfy a logarithmic Sobolev inequality on the product space. Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.

Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges

Abstract

We investigate the quadratic Schrödinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schrödinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schrödinger bridges satisfy a logarithmic Sobolev inequality on the product space. Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.
Paper Structure (13 sections, 12 theorems, 130 equations)

This paper contains 13 sections, 12 theorems, 130 equations.

Table of Contents

  1. Mathematics Subject Classification (2020)
  2. Introduction and statement of the main results
  3. Organization
  4. The Schrödinger system
  5. Weak semiconvexity and semiconcavity bounds for Schrödinger potentials
  6. Invariant sets of weakly convex functions for the HJB flow
  7. Second order bounds for Schrödinger potentials
  8. Weak semiconvexity of $\psi$ implies weak semiconcavity of $\varphi$
  9. Weak semiconcavity of $\varphi$ implies weak semiconvexity of $\psi$
  10. Proof of Theorem \ref{['thm:semiconvexity_estimate_Schr_pot']}
  11. Logarithmic Sobolev inequality for Schrödinger bridges
  12. Local LSIs and gradient estimates
  13. Appendix

Key Result

Theorem 1.2

Let Assumption ass:marginals hold and $(\varphi,\psi)$ be solutions of the Schrödinger system. Then $\varphi,\psi$ are twice differentiable and for all $r>0$ we have where $\alpha_{\psi}>\alpha_{\nu}-1/T$ can be taken to be the smallest solution of the fixed point equation where for all $\alpha\geq\alpha_{\nu}-1/T$:

Theorems & Definitions (31)

  • Remark 1.1
  • Remark 1.2
  • Theorem 1.2
  • Corollary 1.1
  • Remark 1.3
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.3
  • Remark 1.6
  • Remark 1.7
  • ...and 21 more