Extremal and global-time statistics of sluggish random walks with stochastic resetting
We study the extremal and global-time statistics of a one-dimensional sluggish random walk, whose diffusivity decays as a power law of the distance from the origin, under Poissonian resetting to the origin. A change of variable maps the process onto a signed Bessel process, so that renewal theory applies. We obtain the exact Laplace transform of the distribution of the maximum over a finite window. At long times the centered maximum of the transformed process follows the Gumbel law, and in physical space the typical maximum grows as a power of the logarithm of the window length. The far-tail rate function does not depend on $α$ in the Bessel variable, so the second-order dynamical transition known for Brownian resetting, between maxima produced by a single anomalously long inter-reset interval and maxima produced by an interval occupying essentially the whole window, holds for every $α$. The time of the maximum is uniformly distributed in the bulk of the window, with boundary layers that shift its mean above the midpoint logarithmically in the window length. The last-passage time through the origin has an exact density, and the time elapsed since it converges to a Gamma law. Reflection symmetry makes the occupation time of the positive half-line and the last-passage time uncorrelated. The results reduce to the Brownian case and agree with simulations.