The mode of classical and free Brownian motion at large time

Let μ be a probability measure on R with ∫ eε x²\,dμ(x)<∞ for some ε>0. Hasebe and Ueda proved that μ*N(0,t) is unimodal for large t and asked where its mode is, and the same question for free Brownian motion μ S(0,t). We show that for large t both densities have a unique global maximum, that both maxima converge to the mean m of μ at rate $1/t$ with explicit constants depending only on ε and ∫ eε x²\,dμ, and that \[ M_c(t)=m-μ_3/2t+O(t⁻²), M_f(t)=m-3μ_3/t+O(t⁻²), \] where μ₃ is the third central moment; second-order terms are also computed. In the free case uniqueness of the global maximum needs only a finite variance.