Extremal and global-time statistics of sluggish random walks with stochastic resetting
We study a sluggish random walk, whose diffusivity decreases as a power law with distance from the origin, under Poissonian resetting to the origin. Using a mapping to a signed Bessel process and renewal theory, we obtain exact results for the maximum trajectory value in a finite window, the time at which it occurs, and the time of the last visit to the origin. Resetting changes the growth of the typical maximum from a power law in time to a power of its logarithm. The centered maximum approaches a Gumbel law, whose width shrinks in physical space for any positive sluggishness exponent, unlike the Brownian case. In the Bessel coordinate, the probability that the maximum exceeds a value growing in proportion to the observation window length decays exponentially, with a rate that has a second-order dynamical transition and is independent of sluggishness. The time of the maximum is uniform in the bulk, with boundary layers that shift its mean above the midpoint logarithmically with window length. The time since the last visit to the origin follows a Gamma law with a sluggishness-dependent shape. Reflection symmetry makes the last-visit time uncorrelated with the occupation time of the positive half-line.