TY - JOUR
T1 - Growth rate of a stochastic growth process driven by an exponential Ornstein-Uhlenbeck process
AU - Pirjol, Dan
N1 - Publisher Copyright:
© 2022 Author(s).
PY - 2022/8/1
Y1 - 2022/8/1
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=85137281336&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85137281336&partnerID=8YFLogxK
U2 - 10.1063/5.0065342
DO - 10.1063/5.0065342
M3 - Article
AN - SCOPUS:85137281336
SN - 0022-2488
VL - 63
JO - Journal of Mathematical Physics
JF - Journal of Mathematical Physics
IS - 8
M1 - 083303
ER -