Uniform Height Bounds for Brownian Excursions with an Exact Area Constraint

For a standard Brownian excursion of duration \(T\), exact area \(\rho T\), and maximum \(M_T\), with fixed \(\rho>0\), we prove

\[\mathbb P(M_T>H\mid A_T=\rho T)\le C_\rho T^{3/2}e^{-c_\rho H}\]

for large \(T\) and \(H\ge1\), and logarithmic bounds for every positive maximum moment. A positivity-preserving Gaussian bridge shift compares increasing functionals under exact-area conditioning with an exponential area tilt; scalar Airy-area asymptotics and kernel and reflection estimates complete the proof.

For the literal duration-\(2N\), area-\(N\) normalization in AIM item 2.18, the entire curve divided by \(N^{1/3}\) converges uniformly to zero. Thus an \(N^{2/3}\) time window cannot yield a nondegenerate random limit at that height scale. No changed-area, moving-wall, multiline or constant-scale process-limit theorem is claimed.