[Paper Review] Quantitative inequalities for the expected lifetime of Brownian motion

Daesung Kim|arXiv (Cornell University)|Apr 21, 2019

Advanced Mathematical Modeling in Engineering30 references4 citations

TL;DR

This paper establishes quantitative stability estimates for the isoperimetric inequality governing the expected lifetime of Brownian motion in bounded domains. By introducing deficit bounds in terms of Fraenkel asymmetry and level set deviations, it proves that the expected lifetime is maximized for balls, with improvements quantified via dimensional constants and geometric asymmetry measures.

ABSTRACT

The isoperimetric inequalities for the expected lifetime of Brownian motion state that the $L^p$-norms of the expected lifetime in a bounded domain for $1\leq p\leq \infty$ are maximized when the region is a ball with the same volume. In this paper, we prove quantitative improvements of the inequalities. Since the isoperimetric properties hold for a wide class of Lévy processes, many questions arise from these improvements.

Motivation & Objective

• To provide quantitative improvements of the isoperimetric inequality for the expected lifetime of Brownian motion in bounded domains.
• To measure the deviation of a domain and a point from optimality (i.e., the ball and center) when the deficit in the inequality is small.
• To establish lower bounds on the deficit in terms of geometric quantities: Fraenkel asymmetry and the measure of super-level sets.
• To extend the analysis to fractional and stable processes, suggesting broader applicability to Lévy processes and related functional inequalities.

Proposed method

• Uses the symmetric decreasing rearrangement of the expected lifetime function and applies the sharp quantitative isoperimetric inequality.
• Introduces a critical level $ t_* $ based on the measure of super-level sets $ \mu(t) = |\{y \in D : u_D(y) > t\}| $, defined via a threshold involving the Fraenkel asymmetry $ A(D) $.
• Applies the layer cake representation and coarea formula to relate the $ L^p $-norms of the expected lifetime to the fractional perimeter of super-level sets.
• Employs the fractional Pólya–Szegö inequality with a remainder term involving $ A(D)^{2/\alpha} $, derived from the quantitative isoperimetric inequality for fractional perimeter.
• Uses the inequality $ \int_0^{t_*} \mu(t)^{1/r} dt \geq \|u \wedge t_*\|_r $ for $ r = \frac{2n}{2n - \alpha} > 1 $, to control the $ L^p $-norm deviation.
• Establishes a lower bound on the deficit $ \delta(x,D) $ as $ |D|^{-2/n} \left( \mu(u_D(x))^{2/n} + \mathsf{C}_n (u_D(x) \wedge t_*) A(D)^2 \right) $, where $ \mathsf{C}_n $ is a dimensional constant.

Experimental results

Research questions

• RQ1 How can the deficit in the isoperimetric inequality for the expected lifetime of Brownian motion be quantitatively bounded below in terms of geometric deviation from a ball?
• RQ2 What role does the Fraenkel asymmetry $ A(D) $ play in measuring the deviation of a domain from optimality?
• RQ3 Can the level set measure $ \mu(t) = |\{y \in D : u_D(y) > t\}| $ be used to refine the stability estimate for the expected lifetime?
• RQ4 To what extent can the stability estimates for Brownian motion be extended to $ \alpha $-stable processes and fractional Sobolev norms?
• RQ5 Is there a quantitative version of the fractional Pólya–Szegö inequality that incorporates asymmetry and yields a Saint-Venant-type inequality with remainder?

Key findings

• The deficit $ \delta(x,D) = 1 - \frac{u_D(x)}{u_B(0)} $ is bounded below by $ |D|^{-2/n} \left( \mu(u_D(x))^{2/n} + \mathsf{C}_n (u_D(x) \wedge t_*) A(D)^2 \right) $, where $ t_* $ is defined via the super-level set measure.
• The bound incorporates both the deviation of the point $ x $ (via $ \mu(u_D(x)) $) and the domain $ D $ (via $ A(D) $), with $ \mathsf{C}_n = \beta_n \omega_n^{1/n} $ a dimensional constant.
• The result improves upon pointwise rearrangement estimates by incorporating asymmetry and level set geometry, providing a stronger quantitative stability than prior results.
• The method extends to fractional Sobolev norms and suggests a path toward quantitative Saint-Venant inequalities via $ [u]_{\alpha,2} \geq [u^*]_{\alpha,2} + \Phi(t_*, A(D)) $, though this remains an open problem.
• For $ \alpha = 2 $, the inequality $ \|u_B\|_p \geq \|u_D\|_p $ is quantitatively strengthened using the same asymmetry and level set framework.
• The analysis suggests that quantitative improvements for $ \alpha $-stable processes may require symmetrization at the function level $ U $, not the seminorm, and that the $ \alpha = 1 $ case may be more tractable due to connections to the Cauchy process.