We study the decay rate of large deviation probabilities of occupation times, up to time $t$, for the voter model $η\colon\Z^2\times[0,\infty)\ra\{0,1\}$ with simple random walk transition kernel, starting from a Bernoulli product distribution with density $ρ\in(0,1)$. Bramson, Cox and Griffeath (1988) showed that the decay rate order lies in $[\log(t),\log^2(t)]$.
In this paper, we establish the true decay rates depending on the level. We show that the decay rates are $\log^2(t)$ when the deviation from $ρ$ is maximal (i.e., $η\equiv 0$ or 1), and $\log(t)$ in all other situations.
We prove a geometric extension of Lévy's occupation time arcsine law near a hypersurface on a Riemannian manifold. The deviation from the classic arcsine law is of the order of the square-root of the time horizon and is expressed explicitly in terms of the mean curvature of the hypersurface and the local time of the underlying standard Brownian motion.
We shall establish a large deviation principle for some occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. As a consequence, we obtain the exact rate of exponential ergodicity of the stochastic system under $τ$-topology.
Using the hyper-exponential recurrence criterion, a large deviation principle for the occupation measure is derived for a class of non-linear monotone stochastic partial differential equations. The main results are applied to many concrete SPDEs such as stochastic $p$-Laplace equation, stochastic porous medium equation, stochastic fast-diffusion equation, and even stochastic real Ginzburg-Landau equation driven by $α$-stable noises.
We establish a large deviation principle for the occupation measure of the stochastic real Ginzburg-Landau equation driven by $α$-stable noises. The proof is based on a hyper-exponential recurrence criterion. Our result indicates a phenomenon that strong dissipation beats heavy tailed noises to produce a large deviation, it seems to us that this phenomenon has not been reported in the known literatures.
The present work is concerned about two-dimensional stochastic convective Brinkman-Forchheimer (2D SCBF) equations perturbed by a white noise (non degenerate) in smooth bounded domains in $\R^{2}$. We establish two important properties of the Markov semigroup associated with the solutions of 2D SCBF equations (for the absorption exponent $r=1,2,3$), that is, irreducibility and strong Feller property. These two properties implies the uniqueness of invariant measures and ergodicity also. Then, we discuss about the ergodic behavior of 2D SCBF equations by providing a Large Deviation Principle (LD…
Our aim is to unify and extend the large deviation upper and lower bounds for the occupation times of a Markov process with $L_2$ semigroups under minimal conditions on the state space and the process trajectories; for example, no strong Markov property is needed. The methods used here apply in both continuous and discrete time. We present the proofs for continuous time only because of the inherent technical difficulties in that situation; the proofs can be adapted for discrete time in a straightforward manner.
The present work deals with the global solvability as well as asymptotic analysis of stochastic generalized Burgers-Huxley (SGBH) equation perturbed by space-time white noise in a bounded interval of $\mathbb{R}$. We first prove the existence of unique mild as well as strong solution to SGBH equation and then obtain the existence of an invariant measure. Later, we establish two major properties of the Markovian semigroup associated with the solutions of SGBH equation, that is, irreducibility and strong Feller property. These two properties guarantees the uniqueness of invariant measures and er…
In this paper we consider two related stochastic models. The first one is a branching system consisting of particles moving according to a Markov family in R^d and undergoing subcritical branching with a constant rate of V>0. New particles immigrate to the system according to a homogeneous space time Poisson random field. The second model is the superprocess corresponding to the branching particle system. We study rescaled occupation time process and the process of its fluctuations with very mild assumptions on the Markov family. In the general setting a functional central limit theorem as wel…
For the one-dimensional telegraph process, we obtain explicit distribution of the occupation time of the positive half-line. The long-term limiting distribution is then derived when the initial location of the process is in the range of sub-normal or normal deviations from the origin; in the former case, the limit is given by the arcsine law. These limit theorems are also extended to the case of more general occupation-type functionals.
In proving large deviation estimates, the lower bound for open sets and upper bound for compact sets are essentially local estimates. On the other hand, the upper bound for closed sets is global and compactness of space or an exponential tightness estimate is needed to establish it. In dealing with the occupation measure $L_t(A)=\frac{1}{t}\int_0^t{\1}_A(W_s) \d s$ of the $d$ dimensional Brownian motion, which is not positive recurrent, there is no possibility of exponential tightness. The space of probability distributions $\mathcal {M}_1(\R^d)$ can be compactified by replacing the usual topo…
We derive a large deviation principle for the density profile of occupation times of random interlacements at a fixed level in a large box of Z^d, with d bigger or equal to 3. As an application, we analyze the asymptotic behavior of the probability that atypically high values of the density profile insulate a macroscopic body in a large box. As a step in this program, we obtain a similar large deviation principle for the occupation-time measure of Brownian interlacements at a fixed level in a large box of R^d, and we derive a new identity for the Laplace transform of the occupation-time measur…
We consider the damped nonlinear wave (NLW) equation driven by a spatially regular white noise. Assuming that the noise is non-degenerate in all Fourier modes, we establish a large deviations principle (LDP) for the occupation measures of the trajectories. The lower bound in the LDP is of a local type, which is related to the weakly dissipative nature of the equation and seems to be new in the context of randomly forced PDE's. The proof is based on an extension of methods developed in \cite{JNPS-2012} and \cite{JNPS-2014} in the case of kick forced dissipative PDE's with parabolic regularisati…
We prove an energy estimate for the polar empirical measure of the two-dimensional symmetric simple exclusion process. We deduce from this estimate and from results in reference [2] large deviations principles for the polar empirical measure and for the occupation time of the origin.
We prove large deviations principles in large time, for the Brownian occupation time in random scenery. The random scenery is constant on unit cubes, and consist of i.i.d. bounded variables, independent of the Brownian motion. This model is a time-continuous version of Kesten and Spitzer's random walk in random scenery. We prove large deviations principles in ``quenched'' and ``annealed'' settings.
In this paper we study intermittency for the parabolic Anderson equation $\partial u/\partial t=κΔu+γξu$ with $u:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$, where $κ\in[0,\infty)$ is the diffusion constant, $Δ$ is the discrete Laplacian, $γ\in(0,\infty)$ is the coupling constant, and $ξ:\mathbb{Z}^d\times[0,\infty)\to\mathbb{R}$ is a space--time random medium. The solution of this equation describes the evolution of a ``reactant'' $u$ under the influence of a ``catalyst'' $ξ$. We focus on the case where $ξ$ is the voter model with opinions 0 and 1 that are updated according to a random walk tr…
We formulate large deviations principle (LDP) for diffusion pair $(X^ε,ξ^ε)=(X_t^ε,ξ_t^ε)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time. More exactly, the LDP is established for $(X^ε,ν^ε)$ with $ν^ε(dt,dz)$ being an occupation type measure corresponding to $ξ_t^ε$. In some sense we obtain a combination of Freidlin-Wentzell's and Donsker-Varadhan's results. Our approach relies the concept of the exponential tightness and Puhalskii's theorem.
We consider discrete-time Markov chains and study large deviations of the pair empirical occupation measure, which is useful to compute fluctuations of pure-additive and jump-type observables. We provide an exact expression for the finite-time moment generating function, which is split in cycles and paths contributions, and scaled cumulant generating function of the pair empirical occupation measure via a graph-combinatorial approach. The expression obtained allows us to give a physical interpretation of interaction and entropic terms, and of the Lagrange multipliers, and may serve as a starti…
We study the problem of non-asymptotic deviations between a reference measure and its empirical version, in the 1-Wasserstein metric, under the standing assumption that the measure satisfies a transport-entropy inequality. We extend some results of F. Bolley, A. Guillin and C. Villani with simple proofs. Our methods are based on concentration inequalities and extend to the general setting of measures on a Polish space. Deviation bounds for the occupation measure of a Markov chain are also given, under the assumption that the chain is contractive on the space of Lipschitz functions. Throughout …
We prove a Large Deviations Principle for the number of intersections of two independent infinite-time ranges in dimension five and more, improving upon the moment bounds of Khanin, Mazel, Shlosman and Sina{ï} [KMSS94]. This settles, in the discrete setting, a conjecture of van den Berg, Bolthausen and den Hollander [BBH04], who analyzed this question for the Wiener sausage in finite-time horizon. The proof builds on their result (which was resumed in the discrete setting by Phetpradap [Phet12]), and combines it with a series of tools that were developed in recent works of the authors [AS17, A…
This paper studies large deviations of a ``fully coupled" finite state mean-field interacting particle system in a fast varying environment. The empirical measure of the particles evolves in the slow time scale and the random environment evolves in the fast time scale. Our main result is the path-space large deviation principle for the joint law of the empirical measure process of the particles and the occupation measure process of the fast environment. This extends previous results known for two time scale diffusions to two time scale mean-field models with jumps. Our proof is based on the me…
We provide non-asymptotic $L^1$ bounds to the normal for four well-known models in statistical physics and particle systems in $\mathbb{Z}^d$; the ferromagnetic nearest-neighbor Ising model, the supercritical bond percolation model, the voter model and the contact process. In the Ising model, we obtain an $L^1$ distance bound between the total magnetization and the normal distribution at any temperature when the magnetic moment parameter is nonzero, and when the inverse temperature is below critical and the magnetic moment parameter is zero. In the percolation model we obtain such a bound for …
We consider a jump-diffusion process on a bounded domain with reflection at the boundary, and establish long-term results for a general additive process of its path. This includes the long-term behaviour of its occupation time in the interior and on the boundaries. We derive a characterization of the large deviation rate function which quantifies the rate of exponential decay of probabilities of rare events for paths of the process. The characterization relies on a solution of a partial integro-differential equation (PIDE) with boundary constraints. We develop a practical implementation of our…
The voter model is a simple agent-based model to mimic opinion dynamics in social networks: a randomly chosen agent adopts the opinion of a randomly chosen neighbour. This process is repeated until a consensus emerges. Although the basic voter model is theoretically intriguing, it misses an important feature of real opinion dynamics: it does not distinguish between an agent's publicly expressed opinion and her inner conviction. A person may not feel comfortable declaring her conviction if her social circle appears to hold an opposing view. Here we introduce the Concealed Voter Model where we a…
Consider a large system of $N$ Brownian motions in $\mathbb{R}^d$ with some non-degenerate initial measure on some fixed time interval $[0,β]$ with symmetrised initial-terminal condition. That is, for any $i$, the terminal location of the $i$-th motion is affixed to the initial point of the $σ(i)$-th motion, where $σ$ is a uniformly distributed random permutation of $1,...,N$. Such systems play an important role in quantum physics in the description of Boson systems at positive temperature $1/β$. In this paper, we describe the large-N behaviour of the empirical path measure (the mean of the Di…
Neutral models aspire to explain biodiversity patterns in ecosystems where species difference can be neglected, as it might occur at a specific trophic level, and perfect symmetry is assumed between species. Voter-like models capture the essential ingredients of the neutral hypothesis and represent a paradigm for other disciplines like social studies and chemical reactions. In a system where each individual can interact with all the other members of the community, the typical time to reach an absorbing state with a single species scales linearly with the community size. Here we show, by using …
We derive the large deviation principle for radial Schramm-Loewner evolution ($\operatorname{SLE}$) on the unit disk with parameter $κ\rightarrow \infty$. Restricting to the time interval $[0,1]$, the good rate function is finite only on a certain family of Loewner chains driven by absolutely continuous probability measures $\{ϕ_t^2 (ζ)\, dζ\}_{t \in [0,1]}$ on the unit circle and equals $\int_0^1 \int_{S^1} |ϕ_t'|^2/2\,dζ\,dt$. Our proof relies on the large deviation principle for the long-time average of the Brownian occupation measure by Donsker and Varadhan.
We analyse optimal voting weights in two-tier voting systems. In our model, the overall population (or union) is split in groups (or member states) of different sizes. The individuals comprising the overall population constitute the first tier, and the council is the second tier. Each group has a representative in the council that casts votes on their behalf. By "optimal weights", we mean voting weights in the council which minimise the democracy deficit, i.e. the expected deviation of the council vote from a (hypothetical) popular vote. We assume that the voters within each group interact via…
We investigate percolation of the vacant set of random interlacements on $\mathbb{Z}^d$, $d\geq 3$, in the strongly percolative regime. We consider the event that the interlacement set at level $u$ disconnects the discrete blow-up of a compact set $A\subseteq \mathbb{R}^d$ from the boundary of an enclosing box. We derive asymptotic large deviation upper bounds on the probability that the local averages of the occupation times deviate from a specific function depending on the harmonic potential of $A$, when disconnection occurs. If certain critical levels coincide, which is plausible but open a…
We consider $p$ independent Brownian motions in $\R^d$. We assume that $p\geq 2$ and $p(d-2)<d$. Let $\ell_t$ denote the intersection measure of the $p$ paths by time $t$, i.e., the random measure on $\R^d$ that assigns to any measurable set $A\subset \R^d$ the amount of intersection local time of the motions spent in $A$ by time $t$. Earlier results of Chen \cite{Ch09} derived the logarithmic asymptotics of the upper tails of the total mass $\ell_t(\R^d)$ as $t\to\infty$. In this paper, we derive a large-deviation principle for the normalised intersection measure $t^{-p}\ell_t$ on the set of …
We study a class of dissipative PDE's perturbed by a bounded random kick force. It is assumed that the random force is non-degenerate, so that the Markov process obtained by the restriction of solutions to integer times has a unique stationary measure. The main result of the paper is a large deviation principle for occupation measures of the Markov process in question. The proof is based on Kifer's large deviation criterion, a Lyapunov-Schmidt type reduction, and an abstract result on large-time asymptotic for generalised Markov semigroups.
We study the distribution of additive functionals of reset Brownian motion, a variation of normal Brownian motion in which the path is interrupted at a given rate and placed back to a given reset position. Our goal is two-fold: (1) For general functionals, we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting. (2) For three examples of functionals (positive occupation time, area and absolute area), we investigate the effect of resetting by com…
We introduce two probabilistic models for $N$ interacting Brownian motions moving in a trap in $\mathbb {R}^d$ under mutually repellent forces. The two models are defined in terms of transformed path measures on finite time intervals under a trap Hamiltonian and two respective pair-interaction Hamiltonians. The first pair interaction exhibits a particle repellency, while the second one imposes a path repellency. We analyze both models in the limit of diverging time with fixed number $N$ of Brownian motions. In particular, we prove large deviations principles for the normalized occupation measu…
Level 2.5 large deviations and uncertainty relations for self-interacting jump processes: tilting constructions and the emergence of time-scale separation
University of Nottingham（诺丁汉大学）；Centre for the Mathematics and Theoretical Physics of Quantum Non-Equilibrium Systems, University of Nottingham（诺丁汉大学量子非平衡系统数学与理论物理中心）
In this paper, we first show the existence of solutions to the following system of nonlinear equations \begin{eqnarray*}\left\{\begin{array}{l} a_{11}x_1+a_{12}x_2+a_{13}x_3+\cdots+a_{1n}x_{n} = b_{11}\frac{1}{x_1}+b_{12}\frac{1}{x_2}+b_{13}\frac{1}{x_3}+\cdots+b_{1n}\frac{1}{x_{n}},\\ a_{21}\frac{1}{x_1}+a_{22}\frac{x_2}{x_1}+a_{23}\frac{x_3}{x_1}+\cdots+a_{2n}\frac{x_{n}}{x _1}=b_{21}x_1+b_{22}\frac{x_1}{x_2}+b_{23}\frac{x_1}{x_3}+\cdots+b_{2n}\frac{x_1}{x_{n}},\\ a_{31}\frac{x_1}{x_2}+a_{32}\frac{1}{x_2}+a_{33}\frac{x_3}{x_2}+\cdots+a_{3n}\frac{x_{n}}{x _2}=b_{31}\frac{x_2}{x_1}+b_{32}x_2+b…
We study large deviations principles for $ N $ random processes on the lattice $ \Z^d $ with finite time horizon $ [0,β] $ under a symmetrised measure where all initial and terminal points are uniformly given by a random permutation. That is, given a permutation $ σ$ of $ N $ elements and a vector $ (x_1,...,x_N) $ of $ N $ initial points we let the random processes terminate in the points $ (x_{σ(1)},...,x_{σ(N)}) $ and then sum over all possible permutations and initial points, weighted with an initial distribution. There is a two-level random mechanism and we prove two-level large deviation…
