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, $⟨A^n{⟩}_c\approx c_{n}t$, 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.

21 pages, 6 figures

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