Ann-Kathrin Jarecki
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
This paper proves the strong parabolic Harnack inequality for local weak solutions to the heat equation associated with time-dependent (nonsymmetric) bilinear forms. The underlying metric measure Dirichlet space is assumed to satisfy the volume doubling condition, the strong Poincaré inequality, and a cutoff Sobolev inequality. The metric is not required to be geodesic. Further results include a weighted Poincaré inequality, as well as upper and lower bounds for non-symmetric heat kernels.
We consider the Dirichlet form given by \sE(f,f)&=&{1/2}\int_{\bR^d}\sum_{i,j=1}^d a_{ij}(x)\frac{\partial f(x)}{\partial x_i} \frac{\partial f(x)}{\partial x_j} dx &+&\int_{\bR^d\times \bR^d} (f(y)-f(x))^2J(x,y)dxdy. Under the assumption that the $\{a_{ij}\}$ are symmetric and uniformly elliptic and with suitable conditions on $J$, the nonlocal part, we obtain upper and lower bounds on the heat kernel of the Dirichlet form. We also prove a Harnack inequality and a regularity theorem for functions that are harmonic with respect to $\sE$.
We prove a scale-invariant boundary Harnack principle for inner uni- form domains in metric measure Dirichlet spaces. We assume that the Dirichlet form is symmetric, strongly local, regular, and that the volume doubling property and two-sided sub-Gaussian heat kernel bounds are satisfied. We make no assumptions on the pseudo-metric induced by the Dirichlet form, hence the underlying space can be a fractal space.
We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.
In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.
A tree-like fractal Dirichlet space lying between strong and weak elliptic Harnack inequalities
本文构建了一个自相似分形树状结构，并赋予其正则自相似Dirichlet形式，展示了均退出时间相对于典型度量球的异常行为，且在适当选择的自相似测度下，弱椭圆Harnack不等式成立而强不等式不成立。
In this paper, we establish stability of parabolic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces under general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincaré inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the Hölder regularity of parabolic functions for symmetric non-local Dirichlet forms.
We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of local regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.
In this paper, we consider the following symmetric Dirichlet forms on a metric measure space $(M,d,μ)$: $$\mathcal{E}(f,g) = \mathcal{E}(^{(c)}(f,g)+\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $\mathcal{E}(^{(c)}$ is a strongly local symmetric bilinear form and $J(dx,dy)$ is a symmetric Random measure on $M\times M$. Under general volume doubling condition on $(M,d,μ)$ and some mild assumptions on scaling functions, we establish stability results for upper bounds of heat kernel (resp.\ two-sided heat kernel estimates) in terms of the jumping kernels, the cut-off Sobolev inequali…
Let $(X,d,μ)$ be a metric measure space with a local regular Dirichlet form. We give necessary and sufficient conditions for a parabolic Harnack inequality with global space-time scaling exponent $β\ge 2$ to hold. We show that this parabolic Harnack inequality is stable under rough isometries. As a consequence, once such a Harnack inequality is established on a metric measure space, then it holds for any uniformly elliptic operator in divergence form on a manifold naturally defined from the graph approximation of the space.
We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes of the corresponding diffusions. In particular, we do not assume volume doubling property for the symmetric measure.
Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection
本文研究了退化随机偏微分方程的渐近对数Harnack不等式，揭示了其在热核估计、不变测度唯一性及渐近强Feller性质等方面的应用。
Spectral bounds and heat kernel upper estimates for Dirichlet forms
通过退出时间的Harnack型不等式和谱界，刻画无杀死部分的正则Dirichlet形式的热核上界，并证明该不等式在一致完美度量空间上拟对称度量变换下保持。
In classical PDE theory, Widder's theorem gives a representation for nonnegative solutions of the heat equation on $\mathbb{R}^n$. We show that an analogous theorem holds for local weak solutions of the canonical "heat equation" on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.
Large deviation principles are established for the two-parameter Poisson-Dirichlet distribution and two-parameter Dirichlet process when parameter $θ$ approaches infinity. The motivation for these results is to understand the differences in terms of large deviations between the two-parameter models and their one-parameter counterparts. New insight is obtained about the role of the second parameter $α$ through a comparison with the corresponding results for the one-parameter Poisson-Dirichlet distribution and Dirichlet process.
We obtain connectivity of annuli for a volume doubling metric measure Dirichlet space which satisfies a Poincaré inequality, a capacity estimate and a fast volume growth condition. This type of connectivity was introduced by Grigor'yan and Saloff-Coste in order to obtain stability results for Harnack inequalities and to study diffusions on manifolds with ends. As an application of our result, we obtain stability of the elliptic Harnack inequality under perturbations of the Dirichlet form with radial type weights.
Boundary Harnack principle for diffusion with jumps on metric measure spaces
针对度量测度空间上与另一Hunt过程在开集退出分布估计、格林函数界及跳跃密度函数可比较性等条件下弱对偶的一大类不连续Hunt过程，本文得到了任意开集上非尺度不变的边界哈纳克原理（BHP），并通过实例说明主要结果与新贡献。
The Poisson--Dirichlet distribution arises in many different areas. The parameter $θ$ in the distribution is the scaled mutation rate of a population in the context of population genetics. The limiting case of $θ$ approaching infinity is practically motivated and has led to new, interesting mathematical structures. Laws of large numbers, fluctuation theorems and large-deviation results have been established. In this paper, moderate-deviation principles are established for the Poisson--Dirichlet distribution, the GEM distribution, the homozygosity, and the Dirichlet process when the parameter $…
In this paper we continue our investigation of the potential theory of Markov processes with jump kernels decaying at the boundary. To be more precise, we consider processes in ${\mathbb R}^d_+$ with jump kernels of the form ${\mathcal B}(x,y) |x-y|^{-d-α}$ and killing potentials $κ(x)=cx_d^{-α}$, $0<α<2$. The boundary part ${\mathcal B}(x,y)$ is comparable to the product of three terms with parameters $β_1, β_2$, $β_3$ and $β_4$ appearing as exponents in these terms. The constant $c$ in the killing term can be written as a function of $α$, ${\mathcal B}$ and a parameter $p\in ((α-1)_+, α+β_1)…
In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-α}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works [6,7] where we imposed critical killing, here we assume that the killing potential is identically zero. In case $α\in (1,2)$ we first show that the corresponding Hunt process has finite lifetime and dies at the boundary. Then, as our main contribution, we prove the boundary Harnack principle and establish sharp two-sided Green function estimates. Our results cover t…
The Boundary Harnack Principle and the 3G Principle in Fractal-Type Spaces
本文证明了一大类 Dirichlet 空间中有界内一致域上 Green 函数的广义 3G 原理，结果适用于高维分形及无明确 Hausdorff 维数的广义分形型空间，并探讨了其在 Schrödinger 算子上的应用。
In this paper, we consider the following symmetric non-local Dirichlet forms of pure jump type on metric measure space $(M,d,μ)$: $$\mathcal{E}(f,g)=\int_{M\times M} (f(x)-f(y))(g(x)-g(y))\,J(dx,dy),$$ where $J(dx,dy)$ is a symmetric Radon measure on $M\times M\setminus {\rm diag}$ that may have different scalings for small jumps and large jumps. Under general volume doubling condition on $(M,d,μ)$ and some mild quantitative assumptions on $J(dx, dy)$ that are allowed to have light tails of polynomial decay at infinity, we establish stability results for two-sided heat kernel estimates as well…
This paper continues the analysis, started in [2, 3], of a class of degenerate elliptic operators defined on manifolds with corners, which arise in Population Biology. Using techniques pioneered by J. Moser, and extended and refined by L. Saloff-Coste, Grigoryan, and Sturm, we show that weak solutions to the parabolic problem defined by a sub-class of these operators, which consists of those that can be defined by Dirichlet forms and have non-vanishing transverse vector field, satisfy a Harnack inequality. This allows us to conclude that the solutions to these equations belong, for positive ti…
Perturbed Kullback-Leibler Deviation Bounds for Dirichlet Processes
本研究针对Dirichlet过程，通过引入受控扰动的方法，依托超加性技术和约简为Beta分布两种证明思路，得到了采用KL散度构建的改进型非渐近偏差界，拓展了Beta分布的同类不等式。
Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals
针对带次高斯热核估计的Dirichlet空间上的非局部方程，本文借助适配任意Dirichlet空间的Caffarelli-Silvestre延拓方法，证明生成元纯幂次齐次方程弱解的Hölder连续性与Harnack不等式，补充了分形等空间上偏微分方程的相关研究。
Several results of large deviations are obtained for distributions that are associated with the Poisson--Dirichlet distribution and the Ewens sampling formula when the parameter $θ$ approaches infinity. The motivation for these results comes from a desire of understanding the exact meaning of $θ$ going to infinity. In terms of the law of large numbers and the central limit theorem, the limiting procedure of $θ$ going to infinity in a Poisson--Dirichlet distribution corresponds to a finite allele model where the mutation rate per individual is fixed and the number of alleles going to infinity. …
We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which…
The large deviation principle is established for the Poisson--Dirichlet distribution when the parameter $θ$ approaches infinity. The result is then used to study the asymptotic behavior of the homozygosity and the Poisson--Dirichlet distribution with selection. A phase transition occurs depending on the growth rate of the selection intensity. If the selection intensity grows sublinearly in $θ$, then the large deviation rate function is the same as the neutral model; if the selection intensity grows at a linear or greater rate in $θ$, then the large deviation rate function includes an additiona…
Stability of EHI and regularity of MMD spaces
本文证明了在满足度量倍增性质的完备局部紧可分度量空间中，强局部狄利克雷型的椭圆哈纳克不等式在有界型扰动下保持稳定。
On the Resistance Conjecture
本文通过证明体积倍增、上容量界和Poincaré不等式可推出截止Sobolev不等式，从而解决电阻猜想，并统一了度量空间、分形、图和流形上所有p∈(1,∞)的Harnack不等式和稳定性现象的证明。
Small noise fluctuations and large deviations of conservative SPDEs with Dirichlet boundary conditions
本文为广义Dean--Kawasaki随机偏微分方程建立中心极限定理与大偏差原理，刻画其小噪声波动，并推广至一般$C^2$有界域上的Dirichlet边界条件，对应粒子边界吸收或注入。
Potential theory of Dirichlet forms with jump kernels blowing up at the boundary
该文研究半空间上带边界爆破跳核与杀伤位势的Dirichlet形式位势理论，证明了全参数范围与任意维度下边界Harnack原理成立，给出对应过程Green函数的精确双边估计。
Dirichlet metric measure spaces: spectrum, irreducibility, and small deviations
本文研究超压缩不可约 Dirichlet 度量测度空间，证明其在任意连通开集上的 Dirichlet 谱离散，并给出 Hunt 过程小偏差与广义热含量长时间渐近应用。
We consider the symmetric non-local Dirichlet form $(E, F)$ given by \[ E (f,f)=\int_{R^d} \int_{R^d} (f(y)-f(x))^2 J(x,y) dx dy \] with $F$ the closure of the set of $C^1$ functions on $R^d$ with compact support with respect to $E_1$, where $E_1 (f, f):=E (f, f)+\int_{R^d} f(x)^2 dx$, and where the jump kernel $J$ satisfies \[ κ_1|y-x|^{-d-α} \leq J(x,y) \leq κ_2|y-x|^{-d-β} \] for $0<α< β<2, |x-y|<1$. This assumption allows the corresponding jump process to have jump intensities whose size depends on the position of the process and the direction of the jump. We prove upper and lower estimate…
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.
The two-parameter Poisson-Dirichlet distribution is the law of a sequence of decreasing nonnegative random variables with total sum one. It can be constructed from stable and Gamma subordinators with the two-parameters, $α$ and $θ$, corresponding to the stable component and Gamma component respectively. The moderate deviation principles are established for the two-parameter Poisson-Dirichlet distribution and the corresponding homozygosity when $θ$ approaches infinity, and the large deviation principle is established for the two-parameter Poisson-Dirichlet distribution when both $α$ and $θ$ app…
The behavior of the Poisson-Dirichlet distribution with small mutation rate is studied through large deviations. The structure of the rate function indicates that the number of alleles is finite at the instant when mutation appears. The large deviation results are then used to study the asymptotic behavior of the homozygosity, and the Poisson-Dirichlet distribution with symmetric selection. The latter shows that several alleles can coexist when selection intensity goes to infinity in a particular way as the mutation rate approaches zero.
Consider a population of individuals belonging to an infinity number of types, and assume that type proportions follow the two-parameter Poisson-Dirichlet distribution. A sample of size n is selected from the population. The total number of different types and the number of types appearing in the sample with a fixed frequency are important statistics. In this paper we establish the moderate deviation principles for these quantities. The corresponding rate functions are explicitly identified, which help revealing a critical scale and understanding the exact role of the parameters. Conditional, …
Robust estimates for elliptic nonlocal operators on doubling spaces
本文针对满足体积加倍条件的度量测度空间上的对称非局部狄利克雷形式，证明了调和函数的弱哈纳克不等式和先验赫尔德正则性，并建立了估计常数的鲁棒性，同时给出了局部能量可比性条件的充分条件。
Sharp Deviations Bounds for Dirichlet Weighted Sums with Application to analysis of Bayesian algorithms
本文推导Dirichlet加权和的尖锐非渐近偏差界，基于新颖积分表示与复分析，推广Beta分布结果，并应用于贝叶斯bootstrap及多臂老虎机TS算法，使遗憾界独立于支撑大小。
By using the dimension-free Harnack inequality and the integration by parts formula for the associated diffusion semigroup, we prove the central limit theorem, the moderate deviation principle, and the logarithmic iteration law for the sample entropy production rate of stochastic differential equations with Lipschitz continuous and dissipative drifts.
Boundary driven weakly asymmetric Blume-Capel model: Large deviations for mixed Dirichlet-Neumann boundary conditions
本文研究了带有混合Dirichlet-Neumann边界条件的弱非对称Blume-Capel模型，通过建立水动力极限和证明动态大偏差原理，揭示了该模型在宏观行为和大偏差性质上的特征。
Upper Bounds on Large Deviations of Dirichlet $L$-functions in the $q$-aspect
本文证明了给定模数的偶本原 Dirichlet $L$-函数中心值在 $q$-方面的大偏差上界，得出了分数阶矩的精确上界，并进一步给出了 $V=o(\log\log q)$ 情况下的界。
We provide a stochastic interpretation of non-commutative Dirichlet forms in the context of quantum filtering. For stochastic processes motivated by quantum optics experiments, we derive an optimal finite time deviation bound expressed in terms of the non-commutative Dirichlet form. Introducing and developing new non-commutative functional inequalities, we deduce concentration inequalities for these processes. Examples satisfying our bounds include tensor products of quantum Markov semigroups as well as Gibbs samplers above a threshold temperature.
Spectral gap bounds on H-type groups
该研究利用概率技术（尤其是亚椭圆布朗运动的小偏差性质），推导了H型群上Dirichlet次拉普拉斯算子的谱间隙界。
Fractional Macroscopic Fluctuation Theory for a Superdiffusive Ginzburg-Landau dynamics
研究长程相互作用边界驱动的吉布斯-兰道动力学，建立宏观演化方程和稳态分布，推导大偏离原理并验证其与准势函数的一致性。
We prove that the Beta random walk has second order cubic fluctuations from the large deviation principle of the GUE Tracy-Widom type for arbitrary values $\upalpha>0$ and $\upbeta>0$ of the parameters of the Beta distribution, removing previous restrictions on their values. Furthermore, we prove that the GUE Tracy-Widom fluctuations still hold in the intermediate disorder regime. We also show that any random walk in space-time random environment that matches certain moments with the Beta random walk also has GUE Tracy-Widom fluctuations in the intermediate disorder regime. As a corollary we s…
This paper explores large sample properties of the two-parameter $(α,θ)$ Poisson--Dirichlet Process in two contexts. In a Bayesian context of estimating an unknown probability measure, viewing this process as a natural extension of the Dirichlet process, we explore the consistency and weak convergence of the the two-parameter Poisson--Dirichlet posterior process. We also establish the weak convergence of properly centered two-parameter Poisson--Dirichlet processes for large $θ+nα.$ This latter result complements large $θ$ results for the Dirichlet process and Poisson--Dirichlet sequences, and …
Large deviations principle for invariant measures of stochastic Burgers equations
针对带Dirichlet边界条件的(0,1)上随机Burgers方程，研究小噪声渐近性，考虑比时空白噪声更奇异的噪声，证明解的大偏差原理（关于初值一致），并在特定Q_ε收敛条件下证明不变测度的大偏差原理。
Large deviations for invariant measure of stochastic Allen-Cahn equation with inhomogeneous boundary conditions and multiplicative noise
本文研究了带有非均匀Dirichlet边界条件和乘性噪声的一维随机Allen-Cahn方程的不变测度的大偏差原理，通过Simon定理证明了噪声消失时的动力学收敛到Ginzburg-Landau能量泛函的唯一极小值，得出不变测度在Sobolev空间上的估计。
Large deviations and additivity principle for the open harmonic process
本文研究边界驱动开可积XXX链的调和模型，刻画其平稳测度并确定混合分布，基于非平衡稳态严格推导了宏观涨落理论的相关公式，为无界态空间能量输运模型的宏观性质提供首个微观层面的严格证明。
Sharp small-deviation inequalities for sums of independent nonnegative random variables
研究独立非负随机变量和\(S\)小于其期望加\(\delta\)的概率下界，核心方法是结合精确狄利克雷校准定理等结果，主要贡献是给出精确小偏差不等式，肯定了Feige猜想在\(\delta\ge 1\)时的正确性。
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in $\mathbb{Z}^d$. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, th…
Curvature bound of Dyson Brownian Motion
本文在 sine β 系综构型空间上构造满足 BE(0,∞) 曲率界的对称 Dirichlet 型，证明局部泛函不等式与半群正则性，并刻画对偶半群为熵意义下的唯一梯度流，同时推广到一维 (β,s)-circular Riesz 气体。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。