Deviation inequalities for convex functions motivated by the Talagrand conjecture (1706.08688v1)

Abstract: Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.

• The paper establishes sharp deviation inequalities for convex functions under the Gaussian measure by eliminating extraneous logarithmic factors.
• It employs convexity properties and semi-convexity lemmas, along with Ehrhard’s inequality, to derive robust bounds for function deviations.
• The results extend to probability measures via convex transformations, offering a methodological blueprint for advanced analysis in stochastic processes and related fields.

Deviation Inequalities for Convex Functions: Analysis and Implications

Introduction

The paper entitled "Deviation Inequalities for Convex Functions Motivated by the Talagrand Conjecture" explores a sophisticated mathematical landscape. The paper explores deviation inequalities for log-semiconvex functions under the Gaussian measure, inspired by Talagrand’s conjecture on the hypercontractivity phenomena of semigroups. This research primarily seeks to simplify the approach to deriving deviation inequalities, particularly for log-convex functions, expanding on the foundational works concerning the Ornstein-Uhlenbeck semigroup.

Talagrand's Conjecture and Ornstein-Uhlenbeck Semigroup

Talagrand conjectured that a form of hypercontractivity exists on the Boolean hypercube through the convolution of a biased coin. An analogue of this conjecture applies to the Ornstein-Uhlenbeck semigroup in the continuous domain, which operates on integrable functions under Gaussian measures. The key result is a deviation inequality for such functions, suggesting a regularization effect even when hypercontractivity in its classical sense is absent. Notably, Lehec's resolution of this conjecture provides an optimal bound on the deviation inequality involving a factor of $\sqrt{\log t}$, where the parameters are independent of the dimensional context (1706.08688).

Core Results and Their Implications

The paper highlights several critical findings. A pivotal result is the deviation inequality for convex functions under the Gaussian measure. The bound established—$\gamma_n(f \geq t) \leq \overline{\Phi}(\sqrt{2t})$, where $\overline{\Phi}$ denotes the Gaussian tail function—remains sharp and applies specifically to convex functions. This finding sharpens the earlier results by removing the additional logarithmic factors prevalent in prior works. The preservation of log-convexity by the Ornstein-Uhlenbeck semigroup underlines this improvement of Lehec’s results.

Moreover, new deviation inequalities arise for probability measures that are pushforwards of Gaussian measures via convex transformations. This generalization highlights the broader applicability of the theoretical framework developed in this research.

Methodological Innovations

The methodological backbone of this investigation is rooted in exploiting the convexity properties of the function $\varphi(s) = \Phi^{-1}(\gamma_n(A_s))$, where $A_s$ denotes the sublevel sets of the function of interest. The concavity of this function, coupled with Ehrhard's inequality, serves as the technical foundation for deriving deviation inequalities.

The authors also propose a novel lemma regarding semi-convexity, leading to a secondary derivation of the main results. This multi-angle approach underscores the robustness and resilience of the theoretical contributions.

Future Directions and Speculations

The implications of these deviation inequalities are far-reaching, extending into fields reliant on Gaussian measure understanding and semigroup actions. Furthermore, the paper's exploration of semi-convex functions potentially opens avenues for addressing broader classes of functions under similar frameworks. The conjecture proposed concerning the semi-convexity of the transport map $T_f$ highlights an area ripe for further exploration, albeit the presented counterexamples caution against overly simplistic assumptions.

This research represents a significant step in understanding the interplay between convexity, measure theory, and functional inequalities. Further exploration could lead to new insights in stochastic processes, statistical physics, and advanced probability theory.

Conclusion

The elucidation of deviation inequalities for convex functions within Gaussian spaces not only addresses longstanding conjectures but also propels forward the theoretical understanding of hypercontractive phenomena. By confirming and refining classical conjectures, this paper lays a methodological blueprint for future research in mathematical analysis and probability theory, tethering rigorous mathematical inquiry with potential practical applications. As researchers continue to traverse these theoretical terrains, the foundational insights from this work will undoubtedly serve as a guiding compass.