Pascal Mittenbühler | Searcharxiv

arXiv2025

On the Optimal Rate of Convergence for Translation-Invariant 1D Quantum Walks

We study the convergence rate of translation-invariant discrete-time quantum dynamics on a one-dimensional lattice. We prove that the cumulative distributions function of the ballistically scaled position $X(n)/{n}$ after $n$ steps converges at a rate of $n^{-1/3}$ in the L\'evy metric as $n\to\infty$. In the special case of step-coin quantum walks with two-dimensional coin space, we recover the same convergence rate for the supremum distance and prove optimality.

math-ph↗

arXiv2023

Persistence probabilities of a smooth self-similar anomalous diffusion process

We consider the persistence probability of a certain fractional Gaussian process $M^H$ that appears in the Mandelbrot-van Ness representation of fractional Brownian motion. This process is self-similar and smooth. We show that the persistence exponent of $M^H$ exists and is continuous in the Hurst parameter $H$. Further, the asymptotic behaviour of the persistence exponent for $H\downarrow0$ and $H\uparrow1$, respectively, is studied. Finally, for $H\to 1/2$, the suitably renormalized process converges to a non-trivial limit with non-vanishing persistence exponent, contrary to the fact that $M^{1/2}$ vanishes.

math.PR↗