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

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.