The Whittaker 2d growth model is a triangular continuous Markov diffusion process that appears in many scientific contexts. It has been theoretically intriguing to establish a large deviation principle for this 2d process with a scaling factor. The main challenge is the spatiotemporal interactions and dynamics that may depend on potential sample-path intersections. We develop such a principle with a novel rate function. Our approach is mainly based on Schider's Theorem, contraction principle, and special treatment for intersecting sample paths.
We consider a discrete model for anisotropic (2+1)-dimensional growth of an interface height function. Owing to a connection with q-Whittaker functions, this system enjoys many explicit integral formulas. By considering certain Gaussian stochastic differential equation limits of the model we are able to prove a space-time limit to the (2+1)-dimensional additive stochastic heat equation (or Edwards-Wilkinson equation) along characteristic directions. In particular, the bulk height function converges to the Gaussian free field which evolves according to this stochastic PDE.
We study SDEs arising from limiting fluctuations in a $(2+1)$-dimensional surface growth model called the Whittaker driven particle system, which is believed to be in the anisotropic Kardar--Parisi--Zhang class. The main result of this paper proves an irrelevance of nonlinearity in the surface growth model in the continuum by weak convergence in a path space; the first instance of this irrelevance is obtained recently for this model in terms of the covariance functions along certain diverging characteristics. With the same limiting scheme, we prove that the derived SDEs converge in distributio…
We study some SDEs derived from the $q\to 1$ limit of a 2D surface growth model called the $q$-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that "come down from infinity": After rescaling and re-centering, convergence to the time-inverted stationary additive stochastic heat equation holds. The point of view in this paper is a probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations and the in-between slow decorrelation, all in particular integrated forms, explain the convergence of the re-cente…
We study the atypically large deviations of the height $H \sim {\cal O}(t)$ at the origin at late times in $1+1$-dimensional growth models belonging to the Kardar-Parisi-Zhang (KPZ) universality class. We present exact results for the rate functions for the discrete single step growth model, as well as for the continuum KPZ equation in a droplet geometry. Based on our exact calculation of the rate functions we argue that models in the KPZ class undergo a third order phase transition from a strong coupling to a weak coupling phase, at late times.
In this paper, employing the weak convergence method, based on a variational representation for expected values of positive functionals of a Brownian motion, we investigate moderate deviation %(CLT for abbreviation) for a class of stochastic differential delay equations with small noises, where the coefficients are allowed to be highly nonlinear growth with respect to the variables. Moreover, we obtain the central limit theorem for stochastic differential delay equations which the coefficients are polynomial growth with respect to the delay variables.
We study an inhomogeneous generalization of the classical corner growth in which the weights are exponentially distributed with random parameters. Our main interest is in the quenched and annealed large deviation properties of the last passage times. We derive variational representations of the rate functions for right tail large deviations. The quenched rate function can be computed explicitly for certain choices of the parameter distributions. We present a mechanism for rate n left tail annealed large deviations. In the quenched model these deviations have rate strictly greater than n. The a…
We study the maximal displacement and related population for a branching Brownian motion in Euclidean space in terms of the principal eigenvalue of an associated Schrödinger type operator. We first determine their growth rates on the survival event. We then establish the upper deviation for the maximal displacement under the possibility of extinction. Under the non-extinction condition, we further discuss the decay rate of the upper deviation probability and the population growth at the critical phase.
We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.
A moderate deviation principle for functionals, with at most quadratic growth, of moving average processes is established. The main assumptions on the moving average process are a Logarithmic Sobolev inequality for the driving random variables and the continuity, or weaker, of the spectral density of the moving average process. We also obtain the moderate deviations for the empirical spectral density, exhibiting an interesting new form of the rate function, i.e. with a correction term compared to the Gaussian rate functionnal.
Widely used closed product-form networks have emerged recently as a primary model of stochastic growth of sub-cellular structures, e.g., cellular filaments. In the baseline model, homogeneous monomers attach and detach stochastically to individual filaments from a common pool of monomers, resulting in seemingly explicit product-form solutions. However, due to the large-scale nature of such networks, computing the partition functions for these solutions is numerically infeasible. To this end, we develop a novel methodology, based on a probabilistic representation of product-form solutions and l…
We establish a moderate deviation principle for processes with independent increments under certain growth conditions for the characteristics of the process. Using this moderate deviation principle, we give a new proof for Strassen's functional law of the iterated logarithm. In particular, we show that any square-integrable Lévy process satisfies Strassen's law.
The continuous time Markov process considered in this paper belongs to a class of population models with linear growth and catastrophes. There, the catastrophes happen at the arrival times of a Poisson process, and at each catastrophe time, a randomly selected portion of the population is eliminated. For this population process, we derive an asymptotic upper bound for the maximum value and prove the local large deviation principle.
This is a brief survey of laws of large numbers, fluctuation results and large deviation principles for asymmetric interacting particle systems that represent moving interfaces on the plane. We discuss the exclusion process, the Hammersley process and the related last-passage growth models.
We consider discrete $β$-ensembles, as introduced by Borodin, Gorin and Guionnet in (Publications math{\' e}matiques de l'IH{\' E}S 125, 1-78, 2017). Under general assumptions, we establish a large deviation principle for the empirical (or spectral) measures corresponding to these models. Our results apply in the cases when the potential of the model depends on the number of particles, and/or has slow growth near infinity, leading to an equilibrium measure with infinite support.
We prove a Freidlin-Wentzell result for stochastic differential equations in infinite-dimensional Hilbert spaces perturbed by a cylindrical Wiener process. We do not assume the drift to be Lipschitz continuous, but only continuous with at most linear growth. Our result applies, in particular, to a large class of nonlinear fractional diffusion equations perturbed by a space-time white noise.
We consider the branching process in random environment $\{Z_n\}_{n\geq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $\mathbb{P}[Z_n > e^{ρn}]$. Moreover in the subcritical case, under the Cramér condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.
We give a proof of a result on the growth of the number of particles along chosen paths in a branching Brownian motion. The work follows the approach of classical large deviations results, in which paths in $C[0,1]$ are rescaled onto $C[0,T]$ for large $T$. The methods used are probabilistic and take advantage of modern spine techniques.
Functionals of spatial point process often satisfy a weak spatial dependence condition known as stabilization. In this paper we prove process level moderate deviation principles (MDP) for such functionals, which are a level-3 result for empirical point fields as well as a level-2 result for empirical point measures. The level-3 rate function coincides with the so-called specific information. We show that the general result can be applied to prove MDPs for various particular functionals, including random sequential packing, birth-growth models, germ-grain models and nearest neighbor graphs.
We study the stochastic growth process in discrete time $x_{i+1} = (1 + μ_i) x_i$ with growth rate $μ_i = ρe^{Z_i - \frac12 var(Z_i)}$ proportional to the exponential of an Ornstein-Uhlenbeck (O-U) process $dZ_t = - γZ_t dt + σdW_t$ sampled on a grid of uniformly spaced times $\{t_i\}_{i=0}^n$ with time step $τ$. Using large deviation theory methods we compute the asymptotic growth rate (Lyapunov exponent) $λ= \lim_{n\to \infty} \frac{1}{n} \log \mathbb{E}[x_n]$. We show that this limit exists, under appropriate scaling of the O-U parameters, and can be expressed as the solution of a variation…
In this paper we study several aspects of the growth of a supercritical Galton-Watson process {Z_n:n\ge1}, and bring out some criticality phenomena determined by the Schroder constant. We develop the local limit theory of Z_n, that is, the behavior of P(Z_n=v_n) as v_n\nearrow \infty, and use this to study conditional large deviations of {Y_{Z_n}:n\ge1}, where Y_n satisfies an LDP, particularly of {Z_n^{-1}Z_{n+1}:n\ge1} conditioned on Z_n\ge v_n.
We prove the large deviation principle for the law of the solutions to a class of parabolic semilinear stochastic partial differential equations driven by multiplicative noise, in $C\big([0,T]:L^ρ(D)\big)$, where $D\subset {\mathbb R}^d$ with $d\geqslant 1$ is a bounded convex domain with smooth boundary and $ρ$ is any real, positive and large enough number. The equation has nonlinearities of polynomial growth of any order, the space variable is of any dimension, and the proof is based on the weak convergence method.
We consider a branching particle system where each particle moves as an independent Brownian motion and breeds at a rate proportional to its distance from the origin raised to the power $p$, for $p\in[0,2)$. The asymptotic behaviour of the right-most particle for this system is already known; in this article we give large deviations probabilities for particles following "difficult" paths, growth rates along "easy" paths, the total population growth rate, and we derive the optimal paths which particles must follow to achieve this growth rate.
This paper studies the effect of truncation on the large deviations behavior of the partial sum of a triangular array coming from a truncated power law model. Each row of the triangular array consists of i.i.d. random vectors, whose distribution matches a power law on a ball of radius going to infinity, and outside that it has a light-tailed modification. The random vectors are assumed to be R^d-valued. It turns out that there are two regimes depending on the growth rate of the truncating threshold, so that in one regime, much of the heavy tailedness is retained, while in the other regime, the…
This paper studies a portfolio optimization problem in a discrete-time Markovian model of a financial market, in which asset price dynamics depend on an external process of economic factors. There are transaction costs with a structure that covers, in particular, the case of fixed plus proportional costs. We prove that there exists a self-financing trading strategy maximizing the average growth rate of the portfolio wealth. We show that this strategy has a Markovian form. Our result is obtained by large deviations estimates on empirical measures of the price process and by a generalization of …
We utilize the weak convergence method to establish the Freidlin--Wentzell large deviations principle (LDP) for stochastic delay differential equations (SDDEs) with super-linearly growing coefficients, which covers a large class of cases with non-globally Lipschitz coefficients. The key ingredient in our proof is the uniform moment estimate of the controlled equation, where we handle the super-linear growth of the coefficients by an iterative argument. Our results allow both the drift and diffusion coefficients of the considered equations to super-linearly grow not only with respect to the del…
A branching process in random environment $(Z_n, n \in \N)$ is a generalization of Galton Watson processes where at each generation the reproduction law is picked randomly. In this paper we give several results which belong to the class of {\it large deviations}. By contrast to the Galton-Watson case, here random environments and the branching process can conspire to achieve atypical events such as $Z_n \le e^{cn}$ when $c$ is smaller than the typical geometric growth rate $\bar L$ and $ Z_n \ge e^{cn}$ when $c > \bar L$. One way to obtain such an atypical rate of growth is to have a typical r…
