Let B(t) be standard Brownian motion, let M(t)=sup_0≤ s≤ tB(s), and let A=int_0^1 M(t),dt. Write f_ A for the density function of A. Tail-density conjecture. The density satisfies f_ A(x)sim frac2· 3^1/6Gamma(2/3)x^1/3e^-3x^2/2,qquad x→∞. The preceding tail estimate gives only P( A>x)=exp-3x^2/2+o(x^2), so this conjecture proposes the sharper prefactor and polynomial correction. The source presents the precise density asymptotic as a natural conjecture; its resolution is not stated.

Conjecture on the tail density of the integrated Brownian supremum

The problem asks for the precise large-value asymptotic of the density of the area under Brownian motion’s running maximum. It was posed as Conjecture 3.2 in a 2007 paper.

Known results

The 2007 paper established only the logarithmic tail estimate $\mathbb{P}(\mathcal A>x)=\exp\{-3x^2/2+o(x^2)\}$ and stated the sharper density asymptotic as a conjecture.
A 2010 paper gave an explicit density in terms of confluent hypergeometric functions.

2010 theorem

The 2010 paper states the conjectured asymptotic as Theorem 1.9 and derives it from the explicit density formula and the asymptotics of $U$ :

f_{\mathcal A}(x)\sim\frac{2\cdot 3^{1/6}}{\Gamma(2/3)}x^{1/3}e^{-3x^2/2}.

Current status (as of August 2026): The conjecture is resolved in the 2010 paper, which states and derives the asserted density asymptotic; no narrower issue remains recorded in the retrieved sources.

Conjecture on the tail density of the integrated Brownian supremum

Primary source

Progress summary

Solutions 0