Elton P. Hsu, Cheng Ouyang
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.
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We prove that the joint distribution of the occupation time ratios for ergodic transformations preserving an infinite measure converges to a multidimensional version of Lamperti's generalized arcsine distribution, in the sense of strong distributional convergence. Our results can be applied to interval maps and Markov chains. We adopt the double Laplace transform method, which has been utilized in the study of occupation times of diffusions on multiray. We also discuss the inverse problem.
Beyond the Arcsine Law: Exact Two-Time Statistics of the Occupation Time in Jump Processes
针对一维跳跃过程双时间停留统计难以精确刻画的问题，采用广义Wiener-Hopf方法得到联合分布、老化定律与自相关函数，发现连续时间标度极限下的普适特征，为复杂环境老化输运研究提供基础。
Large deviations related to Dynkin--Lamperti arcsine laws for last visit times of Markov processes
本文建立了从属子的Dynkin-Lamperti反正弦律相关大偏差估计，并将其应用于Hunt过程最后访问时间的大偏差研究，覆盖了多类过程并拓展了相关概率理论结果。
We establish a functional limit theorem for the joint-law of occupations near and away from indifferent fixed points of interval maps, and of waits for the occupations away from these points, in the sense of strong distributional convergence. It is a functional and joint-distributional extension of Darling--Kac type limit theorem, of Lamperti type generalized arcsine laws for occupation times, and of Dynkin and Lamperti type generalized arcsine laws for waiting times, at the same time.
Classical arcsine law states that fraction of occupation time on the positive or the negative side in Brownian motion does not converge to a constant but converges in distribution to the arcsine distribution. Here, we consider how a preparation of the system affects the arcsine law, i.e., aging of the arcsine law. We derive aging distributional theorem for occupation time statistics in Brownian motion, where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution. Furthermore, we show that this result can be general…
Large deviations for occupation and waiting times of infinite ergodic transformations
本文研究了保持无限测度的遍历变换的occupation和waiting时间的大偏差，采用双Laplace变换方法建立了与Darling-Kac定理和广义正弦定律相关的新估计。
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.
State-dependent inverse-subordinator time changes of regenerative processes: Excursion structure and multiscale occupation-time limits
本文通过将不同独立子序分配给可测游程类，构造状态依赖逆子序时间变换的再生过程，并证明变换后的游程点过程仍为泊松过程，进而推导出多尺度联合占据时间极限定理。
The occupation time of a random walk generated by the uniform random permutation--a GPT-6 Astra proof
本文借助GPT-6 Astra证明了均匀排列随机游走中零上边数服从离散反正弦律的猜想，并评论了LLM在数学证明中的应用。
An open problem of interest, first infused into the applied probability community in the work of Bingham and Doney in 1988, (see \cite{Bingham}) is stated as follows: find the distribution of the quadrant occupation time of planar Brownian motion. In this short communication, we study an alternate formulation of this longstanding open problem: let $X(t), Y(t), t \geq 0$ be standard Brownian motions starting at $x,y$ respectively. Find the distribution of the total time $T=Leb\{t \in [0,1]: X(t) \times Y(t) >0\}$, when $x=y=0$, i.e., the occupation time of the union of the first and third quadr…
Occupation times and areas derived from random sampling
本文利用随机抽样与组合视角证明球面（分数）Brown运动正占据面积均匀分布，并初等证明 Lévy 第二反正弦定律，进而刻画所有 Lévy 过程的占据时间分布。
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.
Nonequilibrium fluctuations and moderate deviations for the occupation time of the SSEP with Glauber dynamics
研究SSEP与Glauber动力学的非平衡测度起始下的occupation时间波动，利用鞅方法和相对熵方法证明二维中心极限定理和一维样本路径中等偏差原理。
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.
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.
Equilibrium moderate deviations for occupation times of SSEP on regula trees
本文针对正则树上对称简单排他过程，利用与两类随机游走的对偶关系建立替换引理，从而证明占位时间的中偏差原理。
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…
Generalized arcsine laws for a sluggish random walker with subdiffusive growth
本研究精确计算了亚扩散迟缓随机游走者三个可观测量的分布，推广了Lévy反正弦定律至一般α≥0，发现α>0时分布各异且形状依赖α。
Central limit theorems and moderate deviations for additive functionals of SSEP on regular trees
本文证明了正则树上对称简单排他过程加性泛函的中心极限定理和中等偏差原理，推广了占位时间的结果，并利用图形表示扩展了鞅分解公式。
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…
Large deviations for long-time occupation measures of stochastic evolution equations with small, asymptotically rough noise
研究可分希尔伯特空间中一类耗散随机演化方程长时间小噪声行为，建立占据测度大偏差原理，涵盖多种无穷维方程。通过解析半群等技术，结合弱收敛方法，给出速率函数显式公式并构建近最优控制。
Statistics of the number of renewals, occupation times and correlation in ordinary, equilibrium and aging alternating renewal processes
本文研究交替更新过程中更新次数与占据时间的统计性质，通过解析计算给出矩、关联函数及极限定理，揭示间隔时间矩发散时的反常涨落与Mittag-Leffler分布和广义反正弦定律。
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 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.
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…
Asymptotics of Brownian occupation measures with unusually large intersections
研究布朗运动在大交集条件下占用测度的弱收敛性，通过大偏差原理证明其密度为Gagliardo-Nirenberg不等式优化解的平方，推导了无条件布朗运动占用测度的紧致大偏差原理，并证明了倾斜后的大偏差原理。
Markov-bridge representation of ergodic large-deviation principles
本文将Donsker-Varadhan占位测度和经验通量的遍历大偏差原理嵌入一般离散时间框架，并通过条件化与混合论证，用马尔可夫桥给出速率泛函的新表达式。
In this paper, we study small noise asymptotics of Markov-modulated diffusion processes in the regime that the modulating Markov chain is rapidly switching. We prove the joint sample-path large deviations principle for the Markov-modulated diffusion process and the occupation measure of the Markov chain (which evidently also yields the large deviations principle for each of them separately by applying the contraction principle). The structure of the proof is such that we first prove exponential tightness, and then establish a local large deviations principle (where the latter part is split int…
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 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 …
Long-Time Large Deviation Asymptotics for Mean-Field Granular Media Equations
研究颗粒介质扩散的长时平均场大偏差，在耗散性假设下建立大偏差原理，并发现二次势情形下变分问题存在狄拉克与非狄拉克极小值的转变。
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…
Occupation time of a system of Brownian particles on the line with steplike initial condition
本文研究阶梯状初始条件下非相互作用布朗粒子在正半轴占据时间的统计特性，发现系统具有长期记忆效应，并通过解析计算与重要性采样蒙特卡洛模拟验证了退火和淬火分布的大偏差行为。
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 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 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 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.
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…
Level 2.5 large deviations and uncertainty relations for self-interacting jump processes: tilting constructions and the emergence of time-scale separation
本文研究了自相互作用跳跃过程的大偏差原理，通过倾斜构造揭示微观动力学与记忆驱动演化的时间尺度分离，并推导了动能和热力学不确定性关系，扩展了经典马尔可夫界限至非马尔可夫系统。
This thesis is devoted to the study of ergodicity and large deviations for the stochastic nonlinear wave (NLW) equation with smooth white noise in 3D. Under some standard growth and dissipativity assumptions on the nonlinearity, we show that the Markov process associated with the flow of NLW equation has a unique stationary measure that attracts the law of any solution with exponential rate. This result implies, in particular, the strong law of large numbers as well as the central limit theorem for the trajectories. We next consider the problem of small noise asymptotics for the family of stat…
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…
Large deviations for 2D stochastic Navier-Stokes Equations driven by a periodic force and a degenerate noise
针对周期力与退化噪声驱动的二维随机 Navier-Stokes 方程，建立时间非齐次 Feynman-Kac 算子的 RPF 定理，并据此证明占据测度的 Donsker-Varadhan 型大偏差原理。
Lower bounds for bulk deviations for the simple random walk on $\mathbb{Z}^d$, $d\geq 3$
本文针对 $\mathbb{Z}^d$（$d\geq 3$）上的简单随机游走，利用倾斜随机游走方法推导了占位时间局部函数平均值超过给定正值的概率的主指数衰减率渐近下界，该下界与 Sznitman 的上界在首阶匹配，并用 Dirichlet 能量的约束最小值刻画。
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