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

Naftali R. Smith, Satya N. Majumdar

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15 Scopus citations


We study the fluctuations of the area A(t)=0tx(τ)dτ under a self-similar Gaussian process x(τ) 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 1/4t 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 → ∞, we find that the full distribution of the area takes the form PrA|t 1/4exp-tαφA/tβ with anomalous exponents α = 1/(2H + 2) and β = (2H + 3)/(4H + 4) in the regime of moderately large fluctuations, and a different anomalous scaling form PrA|t 1/4exp-tψA/t2H+3/2 in the regime of very large fluctuations. The associated rate functions φ(y) and ψ(w) depend on H and 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 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 PrA|t grow linearly in time, An c≈cnt, 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 six nonvanishing cumulants.

Original languageEnglish
Article number053212
JournalJournal of Statistical Mechanics: Theory and Experiment
Issue number5
StatePublished - 1 May 2022


  • Brownian motion
  • dynamical processes
  • large deviations in non-equilibrium systems

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Statistics, Probability and Uncertainty


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