Condensation transition in large deviations of self-similar Gaussian processes with stochastic resetting

Abstract

AbstractWe study the fluctuations of the areaA(t)=∫0tx(τ)dτunder a self-similar Gaussian processx(τ) with Hurst exponentH> 0 (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rater. Typical fluctuations ofA(t) scale as∼tfor largetand on this scale the distribution is Gaussian, as one would expect from the central limit theorem. Here our main focus is on atypically large fluctuations ofA(t). In the long-time limitt→ ∞, we find that the full distribution of the area takes the formPrA|t∼exp−tαΦA/tβwith anomalous exponentsα= 1/(2H+ 2) andβ= (2H+ 3)/(4H+ 4) in the regime of moderately large fluctuations, and a different anomalous scaling formPrA|t∼exp−tΨA/t2H+3/2in the regime of very large fluctuations. The associated rate functions Φ(y) and Ψ(w) depend onHand are found exactly. Remarkably, Φ(y) has a singularity that we interpret as a first-order dynamical condensation transition, while Ψ(w) exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of Φ(y) around the originy= 0 correctly describes the typical, Gaussian fluctuations ofA(t). Despite these anomalous scalings, we find that all of the cumulants of the distributionPrA|tgrow linearly in time,⟨An⟩c≈cnt, in the long-time limit. For the case of reset Brownian motion (corresponding toH= 1/2), we develop a recursive scheme to calculate the coefficientscnexactly and use it to calculate the first six nonvanishing cumulants.