Fractional Concentration of Measure for Geometric Functionals

You know how Max Fathi's 2024 paper uses concentration inequalities to get growth estimates on optimal transport maps? It's a fantastic example of applying concentration to a geometric object, not just a scalar. But what if we could get even finer control? That's where the work on fractional inequalities, like that by Wang et al. (2025) on super-quadratic processes and Afzal et al. (2025) on fractional interval-valued inequalities, comes in. My idea is to develop a new theoretical framework for "fractional concentration of measure." Instead of asking "what's the probability my random variable deviates from its mean by more than t?", we'd ask "what's the probability my random function deviates from its mean function in a fractional Sobolev norm by more than t?". This is a fundamentally different way to think about concentration. While Fathi's work provides polynomial growth bounds, a fractional approach could yield bounds on the roughness or regularity of the deviation itself. This could unlock new understanding in stochastic geometry, PDEs with random coefficients, and even in generative models where we care about the properties of the generated functions, not just their output values. It's a synthesis of Fathi's geometric focus with the novel analytical tools from the fractional calculus literature.

References:

1. Growth estimates on optimal transport maps via concentration inequalities. Max Fathi (2024).
2. Super-Quadratic Stochastic Processes with Fractional Inequalities and Their Applications. Yuanheng Wang, Usama Asif, Muhammad Zakria Javed, M. Awan, A. Kashuri, O. M. Alsalami (2025). Fractal and Fractional.
3. Some New Fractional Interval-Valued Inequalities for Set-Valued H(α, 1 − α)-Godunova-Levin Mappings with Applications. W. Afzal, Mujahid Abbas, J. Macías-Díaz, Armando Gallegos (2025). Contemporary Mathematics.