[Paper Review] Rigidity phenomenons for an infinite dimension diffusion operator and cases of near equality in the Bakry--Ledoux isoperimetric comparison Theorem

Raphaël Bouyrie|arXiv (Cornell University)|Aug 23, 2017

Advanced Mathematical Modeling in Engineering3 citations

TL;DR

This paper establishes rigidity results for infinite-dimensional diffusion operators with positive curvature using semigroup interpolation, extending Obata's theorem to this setting. It further characterizes near-equality cases in the Bakry–Ledoux isoperimetric comparison theorem, deriving new quantitative bounds for spherical isoperimetry in high dimensions via semigroup methods inspired by Mossel and Neeman.

ABSTRACT

We study rigidity phenomenons for infinite dimension diffusion operators of positive curvature using semigroup interpolations. In particular, for such diffusions, an analogous statement of Obata's theorem is established. Moreover, the same rigidity holds for the Bakry--Ledoux isoperimetric comparison Theorem - a result due to Franck Morgan. Recently, Mossel and Neeman have exploited the semigroup proof of Bakry and Ledoux to derive dimension free bounds for the Gaussian isoperimetry. We extend theirs arguments to obtain in particular new quantitative bounds on the spherical isoperimetric inequality in large dimension.

Motivation & Objective

• To investigate rigidity phenomena in infinite-dimensional diffusion operators with positive curvature using semigroup interpolation.
• To extend Obata’s theorem to the infinite-dimensional setting, identifying conditions under which equality implies Gaussian-like structure.
• To characterize cases of near-equality in the Bakry–Ledoux isoperimetric comparison theorem.
• To derive dimension-free quantitative bounds for the spherical isoperimetric inequality in large dimensions.
• To extend Mossel and Neeman’s semigroup-based approach to obtain sharp estimates in high-dimensional isoperimetry.

Proposed method

• Utilizes semigroup interpolation techniques to analyze monotonicity of functional inequalities along the heat flow.
• Applies commutation relations between gradients and the semigroup to identify equality cases.
• Employs the Bakry–Émery framework of Markov triples $(E,\mu,\Gamma)$ to model infinite-dimensional diffusions.
• Derives pointwise estimates on the Hessian of the heat semigroup via bounds on the second derivatives of the transition density.
• Uses Jensen’s inequality and measure transformation to control weighted integrals of curvature-related terms.
• Establishes dimension-free bounds by showing convergence of Riemannian isoperimetric ratios to their Gaussian counterparts as dimension increases.

Experimental results

Research questions

• RQ1 Under what conditions does equality in the Bakry–Ledoux isoperimetric comparison theorem imply isometry to the Gaussian space in infinite dimensions?
• RQ2 Can semigroup methods detect rigidity in infinite-dimensional diffusions analogous to Obata’s theorem in finite dimensions?
• RQ3 What quantitative bounds can be obtained for the spherical isoperimetric inequality in high dimensions using semigroup techniques?
• RQ4 How do the constants in the isoperimetric estimates behave as dimension tends to infinity?
• RQ5 To what extent can the semigroup proof of Bakry and Ledoux be refined to capture near-equality cases?

Key findings

• A rigidity result analogous to Obata’s theorem holds for infinite-dimensional diffusions with positive curvature, where equality in functional inequalities implies the underlying space is Gaussian.
• The paper establishes that near-equality in the Bakry–Ledoux isoperimetric comparison theorem occurs only when the measure is close to Gaussian in a quantitative sense.
• New quantitative bounds on the spherical isoperimetric inequality are derived, with error terms decaying as $O(\sqrt{t}/n)$ for $t$ small and $n$ large.
• The dependence on dimension in the constants of the isoperimetric estimates can be removed, yielding dimension-free bounds via semigroup approximation.
• Pointwise estimates on the Hessian of the heat semigroup are shown to satisfy $\|\nabla^2 f_t\|_{HS}^2 \leq C f_t^2 / t^4$, leading to control of the $\Gamma_2 - \Gamma$ operator.
• The convergence of Riemannian isoperimetric ratios to their Gaussian counterparts is quantified, with error $O(n^{-1})$ uniformly in $u > 0$.