Self-similar co-ascent processes and Palm calculus

We discuss certain renormalised first passage bridges of self-similar processes. These processes generalise the Brownian co-ascent, a term recently introduced by Panzo [Panzo, H. (2018). "Scaled penalization of Brownian motion with drift and the Brownian ascent", arXiv preprint arXiv:1803.04157]. Our main result states that the co-ascent of a given process is the process under the Palm distribution of its record measure. We base our notion of Palm distribution on self-similarity, thereby complementing the more common approach of considering Palm distributions related to stationarity or stationarity of increments of the underlying processes.Comment: 14 page

Universality for persistence exponents of local times of self-similar processes with stationary increments

We show that P(ℓX(0,T]≤1)=(cX+o(1))T−(1−H), where ℓX is the local time measure at 0 of any recurrent H-self-similar real-valued process X with stationary increments that admits a sufficiently regular local time and cX is some constant depending only on X. A special case is the Gaussian setting, i.e. when the underlying process is fractional Brownian motion, in which our result settles a conjecture by Molchan [Commun. Math. Phys. 205, 97-111 (1999)] who obtained the upper bound 1−H on the decay exponent of P(ℓX(0,T]≤1). Our approach establishes a new connection between persistence probabilities and Palm theory for self-similar random measures, thereby providing a general framework which extends far beyond the Gaussian case

Quenched invariance principle for random walks on dynamically averaging random conductances

We prove a quenched invariance principle for continuous-time random walks in a dynamically averaging environment on Z. In the beginning, the conductances may fluctuate substantially, but we assume that as time proceeds, the fluctuations decrease according to a typical diffusive scaling and eventually approach constant unit conductances. The proof relies on a coupling with the standard continuous time simple random walk.publishedVersio

The emergence of a giant component in one-dimensional inhomogeneous networks with long-range effects

We study the weight-dependent random connection model, a class of sparse graphs featuring many real-world properties such as heavy-tailed degree distributions and clustering. We introduce a coefficient, (deltaf), measuring the effect of the degree-distribution on the occurrence of long edges. We identify a sharp phase transition in (deltaf) for the existence of a giant component in dimension (d=1)