Growth rate of a stochastic growth process driven by an exponential Ornstein-Uhlenbeck process

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Abstract

We study the stochastic growth process in discrete time xi+1 = (1 + μi)xi with the growth rate μi=ρeZi-12Var(Zi) proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process dZt = -γZtdt + σdWt sampled on a grid of uniformly spaced times with time step τ. Using large deviation theory, we compute the asymptotic growth rate (Lyapunov exponent) λ=limn→∞1nlogE[xn]. We show that this limit exists, under appropriate scaling of the O-U parameters, and is expressed as the solution of a variational problem. The asymptotic growth rate is equal to the thermodynamical pressure of a one-dimensional lattice gas with attractive exponential potentials. For Zt, a stationary O-U process of the lattice gas coincides with a model considered previously by Kac and Helfand. We derive upper and lower bounds on λ. In the large mean-reversion limit γnτ ≫ 1, the two bounds converge and the growth rate is given by a lattice version of the van der Waals equation of state. The predictions are tested against numerical simulations of the stochastic growth model.

Original languageEnglish
Article number083303
JournalJournal of Mathematical Physics
Volume63
Issue number8
DOIs
StatePublished - 1 Aug 2022

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