Ann-Kathrin Jarecki
For a Markov process associated with a diffusion type Dirichlet form an upper bound is shown for the law of the finite dimensional distributions of the process. Under some more assumptions on the underlaying space this is also shown for the law of the Markov process itself. In the last section we want to give an application to the Wasserstein diffusion.
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The large deviations principle for the empirical measure for both continuous and discrete time Markov processes is well known. Various expressions are available for the rate function, but these expressions are usually as the solution to a variational problem, and in this sense not explicit. An interesting class of continuous time, reversible processes was identified in the original work of Donsker and Varadhan for which an explicit expression is possible. While this class includes many (reversible) processes of interest, it excludes the case of continuous time pure jump processes, such as a re…
Consider a sequence of continuous-time Markov chains $(X^{(n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{(n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{(0)} \,+\, \sum_{1\le p\le q} (1/θ^{(p)}_n) \, I^{(p)}$ for some rate functionals $I^{(p)}$. The weights $θ^{(p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{(n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{(p)}$ identify the me…
The classical Metropolis-Hastings (MH) algorithm can be extended to generate non-reversible Markov chains. This is achieved by means of a modification of the acceptance probability, using the notion of vorticity matrix. The resulting Markov chain is non-reversible. Results from the literature on asymptotic variance, large deviations theory and mixing time are mentioned, and in the case of a large deviations result, adapted, to explain how non-reversible Markov chains have favorable properties in these respects. We provide an application of NRMH in a continuous setting by developing the necessa…
Exclusion processes with non-reversible boundary: hydrodynamics and large deviations
该研究针对一维排斥动力学与边界库的温和接触问题，揭示了扩散尺度下粒子密度遵循带非线性Robin边界条件的热方程演化规律，证明了其动力学大偏差原理，且边界速率适当选择时偏微分方程存在多个平稳解。
In the framework of Harnack type Dirichlet forms, we prove a large deviation principle for the asymptotics of reversible Markov processes with rate function given by the energy of the paths.
We build optimal exponential bounds for the probabilities of large deviations of sums \sum_{k=1}^nf(X_k) where (X_k) is a finite reversible Markov chain and f is an arbitrary bounded function. These bounds depend only on the stationary mean E_πf, the end-points of the support of f, the sample size n and the second largest eigenvalue λof the transition matrix.
Let $S_N$ be the sum of vector-valued functions defined on a finite Markov chain. An analogue of the Bernstein--Hoeffding inequality is derived for the probability of large deviations of $S_N$ and relates the probability to the spectral gap of the Markov chain. Examples suggest that this inequality is better than alternative inequalities if the chain has a sufficiently large spectral gap and the function is high-dimensional.
In this paper, we investigate the functional central limit theorem for stochastic processes associated to partial sums of additive functionals of reversible Markov chains with general spate space, under the normalization standard deviation of partial sums. For this case, we show that the functional central limit theorem is equivalent to the fact that the variance of partial sums is regularly varying with exponent 1 and the partial sums satisfy the CLT. It is also equivalent to the conditional CLT.
$Γ$-expansion of the measure-current large deviations rate functional of non-reversible finite-state Markov chains
本研究针对非可逆有限状态连续时间马尔可夫链的测度-流大偏差率泛函，证明其可按亚稳态演化时间尺度分解为$Γ$-展开形式，完善了层级亚稳态行为的刻画，还给出率泛函决定原动力学的最优条件并推广了i.i.d.情形的导数结果。
A variational formula for the asymptotic variance of general Markov processes is obtained. As application, we get a upper bound of the mean exit time of reversible Markov processes, and some comparison theorems between the reversible and non-reversible diffusion processes.
In Monte-Carlo methods the Markov processes used to sample a given target distribution usually satisfy detailed balance, i.e. they are time-reversible. However, relatively recent results have demonstrated that appropriate reversible and irreversible perturbations can accelerate convergence to equilibrium. In this paper we present some general design principles which apply to general Markov processes. Working with the generator of Markov processes, we prove that for some of the most commonly used performance criteria, i.e., spectral gap, asymptotic variance and large deviation functionals, samp…
Full $Γ-$expansion for the level-two large deviation rate functionals of non-reversible one-dimensional diffusions with periodic boundary conditions
针对一维环面上非可逆扩散过程，在温度趋于零时，将其二级大偏差率泛函展开为与亚稳态时间尺度相关的Γ级数，得到各阶率泛函。
Precise upper deviation estimates for the maximum of a branching random walk
本文研究了分支随机游动最大位移的精确上大偏差估计，描述了该大偏差事件条件下的极值过程，并引入了类似分支布朗运动装饰测度的点测度族。
This paper presents a new, short proof of the computation of the upper tail large deviation rate function for the Brownian directed percolation model. Through a distributional equivalence between the last passage time in this model and the largest eigenvalue in a random matrix drawn from the Gaussian Unitary Ensemble, this provides a new proof a previously known result. The method leads to associated results for the stationary Brownian directed percolation model which have not been observed before.
We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give examples in which we compare with a famous exponential inequality for sums of correlated indicators, the Janson inequality. Here our bound sometimes proves to be superior to Janson's bound.
In this paper we consider the first passage percolation with identical and independent exponentially distributions, called the Eden growth model, and we study the upper tail large deviations for the first passage time ${\rm T}$. Our main results prove that for any $ξ>0$ and $x\neq 0$, $\mathbb{P}({\rm T}(0,nx)>n(μ+ξ))$ decays as $\exp{(-(2dξ+o(1))n)}$ with a time constant $μ$ and a dimension $d$. Moreover, we extend the result to stretched exponential distributions. On the contrary, we construct a continuous distribution with a finite exponential moment where the rate function does not exist.
Full $Γ$-expansion of reversible Markov chains level two large deviations rate functionals
本文提出了一种基于马尔可夫链metastable行为的工具，推导出可逆马尔可夫链二级大偏差率函数的全展开方法，并应用于随机游走问题。
Upper moderate deviation probabilities for the maximum of branching Brownian motion
本文研究分支布朗运动极大值的上中等偏差概率，利用修正的第二矩方法给出其渐近等价，并刻画了贡献粒子的典型行为。
We provide a general approach to obtain upper bounds for small deviations $ \mathbb{P}(\Vert y \Vert \le ε)$ in different norms, namely the supremum and $β$- Hölder norms. The large class of processes $y$ under consideration takes the form $y_t= X_t + \int_0^t a_s d s$, where $X$ and $a$ are two possibly dependent stochastic processes. Our approach provides an upper bound for small deviations whenever upper bounds for the \textit{concentration of measures} of $L^p$- norm of random vectors built from increments of the process $X$ and \textit{large deviation} estimates for the process $a$ are av…
Large deviations of the empirical measures of a strong-Feller Markov process inside a subset and quasi-ergodic distribution
本文针对强Feller过程建立了子区域内不退出条件下占用测度的大偏差原理，确定速率函数仅在拟遍历分布处消失且可逆时为狄利克雷形式，并将结果应用于椭圆随机微分方程解等多个随机过程。
A supersymmetric quantum perspective on the explicit large deviations for reversible Markov jump processes, with applications to pure and random spin chains
本文从超对称量子哈密顿量视角重构可逆马尔可夫跳过程的多层级显式大偏差速率函数，并将其应用于纯/随机自旋链系统，建立了统计物理大偏差理论与量子自旋模型的对应关系。
Branching Processes in a Random Environment (BPREs) $(Z_n:n\geq0)$ are a generalization of Galton Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. We determine here the upper large deviation of the process when the reproduction law may have heavy tails. The behavior of BPREs is related to the associated random walk of the environment, whose increments are distributed like the logarithmic mean of the offspring distributions. We obtain an expression of the upper rate function of $(Z_n:n\geq0)$, that is the limit of $-\log \mathbb{P}(Z_n\geq e…
Conditional exponential directed last passage percolation under a one-point upper large deviation event
探讨有向最后通过渗流模型在单点上大偏差事件下最后通过时间场的变化，通过分析显式多点分布，证明条件大数定律并计算极限涨落。
We develop Markov chain mixing time estimates for a class of Markov chains with restricted transitions. We assume transitions may occur along a cycle of $n$ nodes and on $n^γ$ additional edges, where $γ< 1$. We find that the mixing times of reversible Markov chains properly interpolate between the mixing times of the cycle with no added edges and of the cycle with $cn$ added edges (which is in turn a Small World Network model). In the case of non-reversible Markov-chains, a considerable gap remains between lower and upper bounds, but simulations give hope to experience a significant speedup co…
Upper large deviations for power-weighted edge lengths in spatial random networks
本文研究基于 Poisson 的空间随机网络中幂加权边长和的上大偏差，证明在 $\alpha>d$ 时其由凝聚机制主导，并给出可通过优化问题表示的速率函数，适用于多种近邻图与 $\beta$-骨架。
Reversible Imprecise Markov Chains
本文为可逆不精确马尔可夫链建立理论框架，引入基于两步联合分布矩阵信任集的对称表示，将可逆性简化为矩阵对称性，并用于计算路径依赖泛函的期望。
Large deviations of the stochastic area for linear diffusions
本文计算了线性随机微分方程（SDEs）中随机面积的生成函数，提取了长时间极限下概率密度主导部分的大偏差函数及有效SDE，并求得与概率流相关的渐近均值和用于判断可逆性的渐近方差。
We prove precise deviations results in the sense of Cramér and Petrov for the upper tail of the distribution of the maximal value for a special class of determinantal point processes that play an important role in random matrix theory. Here we cover all three regimes of moderate, large and superlarge deviations for which we determine the leading order description of the tail probabilities. As a corollary of our results we identify the region within the regime of moderate deviations for which the limiting Tracy-Widom law still predicts the correct leading order behavior. Our proofs use that the…
On the upper tail large deviation rate function for chemical distance in supercritical percolation
本文研究了超临界键渗流中化学距离的上尾大偏差问题，证明了当 $d\ge 3$ 且 $\varepsilon>0$ 足够小时速率函数的存在性，并揭示该大偏差事件由时空切点迫使测地线绕行引发。
We prove a moderate deviation principle for subgraph count statistics of Erdos-Renyi random graphs. This is equivalent in showing a moderate deviation principle for the trace of a power of a Bernoulli random matrix. It is done via an estimation of the log-Laplace transform and the Gaertner-Ellis theorem. We obtain upper bounds on the upper tail probabilities of the number of occurrences of small subgraphs. The method of proof is used to show supplemental moderate deviation principles for a class of symmetric statistics, including non-degenerate U-statistics with independent or Markovian entrie…
We consider the standard first passage percolation in $\mathbb{Z}^{d}$ for $d\geq 2$ and we denote by $ϕ_{n^{d-1},h(n)}$ the maximal flow through the cylinder $]0,n]^{d-1} \times ]0,h(n)]$ from its bottom to its top. Kesten proved a law of large numbers for the maximal flow in dimension three: under some assumptions, $ϕ_{n^{d-1},h(n)} / n^{d-1}$ converges towards a constant $ν$. We look now at the probability that $ϕ_{n^{d-1},h(n)} / n^{d-1}$ is greater than $ν+ ε$ for some $ε>0$, and we show under some assumptions that this probability decays exponentially fast with the volume of the cylinder…
Let M_n denote the number of sites in the largest cluster in critical site percolation on the triangular lattice inside a box side length n. We give lower and upper bounds on the probability that M_n / E(M_n) > x of the form exp(- C x^(2/alpha)) for x > 1 and large n with alpha = 5/48 and C > 0. Our results extend to other two dimensional lattices and strengthen the previously known exponential upper bound derived by Borgs, Chayes, Kesten and Spencer [BCKS99]. Furthermore, under some general assumptions similar to those in [BCKS99], we derive a similar upper bound in dimensions d > 2.
Upper moderate deviation probabilities for the maximum of a branching random walk
本文研究超临界分支随机游走最大值的上适度偏差概率，基于Bramson等人策略，得到渐近等价并分析粒子行为。
Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h}(t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h}_N(t,ξ) := \frac{1}{N}\mathsf{h}(Nt,Nξ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp(-O(N)) $, referred to as speed-$ N $; while the other with probability $ \ex…
We consider two approaches to study non-reversible Markov processes, namely the Hypocoercivity Theory (HT) and GENERIC (General Equations for Non-Equilibrium Reversible-Irreversible Coupling); the basic idea behind both of them is to split the process into a reversible component and a non-reversible one, and then quantify the way in which they interact. We compare such theories and provide explicit formulas to pass from one formulation to the other; as a bi-product we give a simple proof of the link between reversibility of the dynamics and gradient flow structure of the associated Fokker-Plan…
We consider the standard first passage percolation model in the rescaled graph $\mathbb {Z}^d/n$ for $d\geq2$ and a domain $Ω$ of boundary $Γ$ in $\mathbb {R}^d$. Let $Γ^1$ and $Γ^2$ be two disjoint open subsets of $Γ$ representing the parts of $Γ$ through which some water can enter and escape from $Ω$. We investigate the asymptotic behavior of the flow $ϕ_n$ through a discrete version $Ω_n$ of $Ω$ between the corresponding discrete sets $Γ^1_n$ and $Γ^2_n$. We prove that under some conditions on the regularity of the domain and on the law of the capacity of the edges, the upper large deviatio…
The Gamma Expansion of the Level Two Large Deviation Rate Functional for Reversible Diffusion Processes
本文研究了可逆扩散过程 Donsker--Varadhan 第二层大偏差率泛函的多尺度 Gamma 展开，揭示了扩散过程在不同亚稳态时间尺度下于势阱间演化的 Markov 链大偏差结构。
We introduce the concept of matrix liberation process, a random matrix counterpart of the liberation process in free probability, and prove a large deviation upper bound for its empirical distribution with several properties on its rate function. As a simple consequence we obtain the almost sure convergence of the empirical distribution of the matrix liberation process to that of the corresponding liberation process as continuous processes in large $N$ limit.
In this work, we investigate links between the formulation of the flow of marginals of reversible diffusion processes as gradient flows in the space of probability measures and path wise large deviation principles for sequences of such processes. An equivalence between the LDP principle and Gamma-convergence for a sequence of functionals appearing in the gradient flow formulation is proved. As an application, we study large deviations from the hydrodynamic limit for two variants of the Ginzburg-Landau model endowed with Kawasaki dynamics.
We formulate two types of extensions of the large deviation theory initiated by Bahadur in a non-regular setting. One can be regarded as a bound of the point estimation, the other can be regarded as the limit of a bound of the interval estimation. Both coincide in the regular case, but do not necessarily coincide in a non-regular case. Using the limits of relative Rényi entropies, we derive their upper bounds and give a necessary and sufficient condition for the coincidence of the two upper bounds. We also discuss the attainability of these two bounds in several non-regular location shift fami…
Upper deviation probabilities for level sets of a supercritical branching random walk
本文研究超临界分枝随机游走水平集粒子数的上偏差概率，给出其指数收敛速率，补充了下偏差结果并完善了分枝布朗运动的相关结论。
In this paper, we investigate the asymptotic behaviors of the critical branching process with immigration $\{Z_n, n\ge 0\}$. First we get some estimation for the probability generating function of $Z_n$. Based on it, we get a large deviation for $Z_{n+1}/Z_n$. Lower and upper deviations for $Z_n$ are also studied. As a by-product, an upper deviation for $\max_{1\le i\le n} Z_i$ is obtained.
Upper deviation probabilities for the range of a supercritical super-Brownian motion
本文研究了$d$维超临界超布朗运动游程半径的上偏差概率，通过理论分析证实了Engländer关于$R_t$上偏差概率渐近公式的猜想。
Failure of ambient closed-set large-deviation upper bounds in entropic optimal transport
研究熵最优传输中闭集大偏差上界失效问题，通过构造连续成本模型展示闭集上界失效现象，并确定闭集到紧集过渡的精确尾准则。
Upper tail large deviation for the first passage time in the frog model on $Z^d$ with $d\geq 2$
该论文研究d≥2时Z^d上青蛙模型首达时间的上尾大偏差，推导了大偏差渐近公式，揭示终点附近的减速机制，通过修正探索过程、重正化与覆盖论证完成机制分析。
Second Time Scale of the Metastability of Reversible Inclusion Processes
本文扩展可逆包含过程的亚稳态研究，通过分析转移速率地貌并估计平衡势与容量上下界，证明第二时间尺度为 $N/d_N^2$。
Bernstein-type Inequalities for Markov Chains and Markov Processes: A Simple and Robust Proof
该研究用初等方法提出了适用于一般非可逆离散时间马尔可夫链的新Bernstein型偏差不等式，仅需满足迭代Poincaré不等式，鲁棒性更强且方法可推广至连续时间马尔可夫过程。
In this work, we analyse the metastability of non-reversible diffusion processes $$dX_t=\boldsymbol{b}(X_t)dt+\sqrt h\,dB_t$$ on a bounded domain $Ω$ when $\mathbf{b}$ admits the decomposition $\mathbf{b}=-(\nabla f+\mathbf{\ell})$ and $\nabla f \cdot \mathbf{\ell}=0$. In this setting, we first show that, when $h\to 0$, the principal eigenvalue of the generator of $(X_t)_{t\ge 0}$ with Dirichlet boundary conditions on the boundary $\partialΩ$ of $Ω$ is exponentially close to the inverse of the mean exit time from $Ω$, uniformly in the initial conditions $X_0=x$ within the compacts of $Ω$. The …
Upper tail large deviations of the directed landscape
从单点尾部界出发，在度量层面建立有向景观的上尾大偏差原理，速率函数由测度的 Kruzhkov 熵给出，并应用于证明有向测地线的大偏差原理。
The Density Formula Approach for Non-reversible Isomorphism Theorems, with Applications
本文通过密度公式方法证明了非可逆同构定理，并将其应用于永久过程的比较不等式推导以及非可逆马尔可夫链覆盖时间的上界估计。
We prove an upper bound on the total variation mixing time of a finite Markov chain in terms of the absolute spectral gap and the number of elements in the state space. Unlike results requiring reversibility or irreducibility, this bound is finite whenever the chain converges. The dependence on the number of elements means that the new bound cannot capture the behavior of rapidly mixing chains; but an example shows that such dependence is necessary. We also provide a sharpened result for reversible chains that are not necessarily irreducible. The proof of the general bound exploits a connectio…
We study a large class of reversible Markov chains with discrete state space and transition matrix $P_N$. We define the notion of a set of {\it metastable points} as a subset of the state space $\G_N$ such that (i) this set is reached from any point $x\in \G_N$ without return to x with probability at least $b_N$, while (ii) for any two point x,y in the metastable set, the probability $T^{-1}_{x,y}$ to reach y from x without return to x is smaller than $a_N^{-1}\ll b_N$. Under some additional non-degeneracy assumption, we show that in such a situation: \item{(i)} To each metastable point corres…
We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$ started from step initial data and obtain the exact Lyapunov exponents for $H_0(t)$, the integrated current of ASEP. As a corollary, we derive an explicit formula for the upper-tail large deviation rate function for $-H_0(t)$. Our result matches with the rate function for the integrated current of the totally asymmetric simple exclusion process (TASEP) obtained in [Johansson 00](arXiv:math/9903134).
A large deviation principle for the empirical measures of Metropolis-Hastings chains
本文针对一般状态空间上的 Metropolis-Hastings 链，首次证明其经验测度满足大偏差原理，并依据接受与拒绝动力学刻画速率函数，为分析该类 MCMC 方法的效率提供了新的理论基础。
Using martingale methods, we obtain some upper bounds for large and moderate deviations of products of independent and identically distributed elements of GL d (R). We investigate all the possible moment conditions, from super-exponential moments to weak moments of order p \textgreater{} 1, to get a complete picture of the situation. We also prove a moderate deviation principle under an appropriate tail condition.
For a $Δ$-regular connected graph ${\sf H}$ the problem of determining the upper tail large deviation for the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$, an Erdős-Rényi graph on $n$ vertices with edge probability $p$, has generated significant interests. For $p\ll 1$ and $np^{Δ/2} \gg (\log n)^{1/(v_{\sf H}-2)}$, where $v_{\sf H}$ is the number of vertices in ${\sf H}$, the upper tail large deviation event is believed to occur due to the presence of localized structures. In this regime the large deviation event that the number of copies of ${\sf H}$ in $\mathbb{G}(n,p)$ exceeds its exp…
Large deviations for triangles in scale-free random graphs
本文研究了无标度非均匀随机图中三角形数量上尾的大偏差估计，发现其概率在 $α<4/3$、$α>4/3$ 和 $α=4/3$ 时分别由多个度数为 $n$ 的顶点、单个特定度数枢纽及有限个枢纽引起，并呈现半指数或多项式衰减的相变现象。
Upper tail large deviations for Brownian motions with one-sided collisions
本文针对属于KPZ普适类的单侧反射相互作用布朗运动系统，推导了三种基本初始条件下的上尾大偏差原理并给出速率函数显式形式，发现周期情形下速率函数含Fredholm行列式公式中已有的Lambert-W函数。
In this paper we are concerned with hitting times of a family of density-dependent Markov chains. A moderate deviation principle of the hitting time is given. The proof of the main theorem relies heavily on moderate deviations of density-dependent Markov chains given in \cite{Xue2021} and upper bounds of large deviations of Markov processes given in \cite{Dupuis1991}. An analogue moderate deviation of the hitting time of the diffusion approximation of the density-dependent Markov chain introduced in \cite{Ethier1986} is also given.
In order to sample from a given target distribution (often of Gibbs type), the Monte Carlo Markov chain method consists in constructing an ergodic Markov process whose invariant measure is the target distribution. By sampling the Markov process one can then compute, approximately, expectations of observables with respect to the target distribution. Often the Markov processes used in practice are time-reversible (i.e., they satisfy detailed balance), but our main goal here is to assess and quantify how the addition of a non-reversible part to the process can be used to improve the sampling prop…
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