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

TL;DR

This paper investigates the large deviation properties of the area under self-similar Gaussian processes with stochastic resetting, revealing phase transitions and anomalous scaling behaviors in the distribution of fluctuations.

Contribution

It provides exact rate functions for large deviations, uncovers dynamical phase transitions, and develops a recursive scheme for cumulant calculations in reset Gaussian processes.

Findings

Identifies a first-order dynamical condensation transition in the rate function.

Discovers a second-order phase transition related to resetting events.

Derives exact asymptotic forms for the distribution of the area in large-time limit.

Abstract

We study the fluctuations of the area $A(t)= \int_0^t x(\tau)\, d\tau$ under a self-similar Gaussian process (SGP) $x(\tau)$ with Hurst exponent $H>0$ (e.g., standard or fractional Brownian motion, or the random acceleration process) that stochastically resets to the origin at rate $r$. Typical fluctuations of $A(t)$ scale as $\sim \sqrt{t}$ for large $t$ and 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 of $A(t)$. In the long-time limit $t\to\infty$, we find that the full distribution of the area takes the form $P_{r}\left(A|t\right)\sim\exp\left[-t^{\alpha}\Phi\left(A/t^{\beta}\right)\right]$ with anomalous exponents $\alpha=1/(2H+2)$ and $\beta = (2H+3)/(4H+4)$ in the regime of moderately large fluctuations, and a different anomalous scaling form $P_{r}\left(A|t\right)\sim\exp\left[-t\Psi\left(A/t^{\left(2H+3\right)/2}\right)\right]$ in the regime of very large fluctuations. The associated rate functions $\Phi(y)$ and $\Psi(w)$ depend on $H$ and are found exactly. Remarkably, $\Phi(y)$ has a singularity that we interpret as a first-order dynamical condensation transition, while $\Psi(w)$ exhibits a second-order dynamical phase transition above which the number of resetting events ceases to be extensive. The parabolic behavior of $\Phi(y)$ around the origin $y=0$ correctly describes the typical, Gaussian fluctuations of $A(t)$. Despite these anomalous scalings, we find that all of the cumulants of the distribution $P_{r}\left(A|t\right)$ grow linearly in time, $\langle A^n\rangle_c\approx c_n \, t$, in the long-time limit. For the case of reset Brownian motion (corresponding to $H=1/2$), we develop a recursive scheme to calculate the coefficients $c_n$ exactly and use it to calculate the first 6 nonvanishing cumulants.