Martin Barlow, Ben Hambly
We consider the random walk on supercritical percolation clusters in the d-dimensional Euclidean lattice. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.
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We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and a near-diagonal heat kernel lower bound.
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.
Time-insensitive nonlocal parabolic Harnack estimates
本文针对非局部抛物方程建立不受等待时间现象影响的新型全局解 Harnack 估计，并证明局部解情形下等待时间不可避免。
The parabolic Harnack inequality for nonlocal equations
本文在变分框架下证明具有界可测系数的线性抛物非局部方程弱解的抛物 Harnack 不等式与 Hölder 估计，核心在于精细分析非局部尾项。
We consider difference equations in balanced, i.i.d. environments which are not necessary elliptic. In this setting we prove a parabolic Harnack inequality (PHI) for non-negative solutions to the discrete heat equation satisfying a (rather mild) growth condition, and we identify the optimal Harnack constant for the PHI. We show by way of an example that a growth condition is necessary and that our growth condition is sharp. Along the way we also prove a parabolic oscillation inequality and a (weak) quantitative homogenization result, which we believe to be of independent interest.
Harnack inequality for parabolic equations in double-divergence form with singular lower order coefficients
本文研究了双散度型二阶抛物方程非负解的Harnack不等式，设定主系数满足Dini平均振荡条件且低阶系数属于Morrey类，推进了相关方程及Fokker-Planck-Kolmogorov方程的理论基础。
We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.
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.
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.
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.
In this paper, we consider the following type of non-local (pseudo-differential) operators $\LL $ on $\R^d$: $$ \LL u(x) =\frac12 \sum_{i, j=1}^d \frac{\partial}{\partial x_i} (a_{ij}(x) \frac{\partial}{\partial x_j}) + \lim_{\eps \downarrow 0} \int_{\{y\in \R^d: |y-x|>\eps\}} (u(y)-u(x)) J(x, y) dy, $$ where $A(x)=(a_{ij}(x))_{1\leq i, j\leq d}$ is a measurable $d\times d$ matrix-valued function on $\R^d$ that is uniform elliptic and bounded and $J$ is a symmetric measurable non-trivial non-negative kernel on $\R^d\times \R^d$ satisfying certain conditions. Corresponding to $\LL$ is a symmetr…
We study various probabilistic and analytical properties of a class of degenerate diffusion operators arising in Population Genetics, the so-called generalized Kimura diffusion operators. Our main results is a stochastic representation of weak solutions to a degenerate parabolic equation with singular lower-order coefficients, and the proof of the scale-invariant Harnack inequality for nonnegative solutions to the Kimura parabolic equation. The stochastic representation of solutions that we establish is a considerable generalization of the classical results on Feynman-Kac formulas concerning t…
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.
We study master equations of the form $$(\partial_t+L)^su=f\quad\hbox{in}~\mathbb{R}\timesΩ$$ where $L$ is a divergence form elliptic operator and $Ω\subseteq\mathbb{R}^n$. These are nonlocal equations of order $2s$ in space and $s$ in time that take into account the values of $u$ everywhere in $Ω$ and for past times. We show parabolic interior and boundary Harnack inequalities and local parabolic Hölder continuity of solutions. To this end, we prove a characterization of fractional powers of parabolic operators $\partial_t+L$ with a degenerate parabolic extension problem.
We consider stochastic partial differential equations under minimal assumptions: the coefficients are merely bounded and measurable and satisfy the stochastic parabolicity condition. In particular, the diffusion term is allowed to be scaling-critical. We derive local supremum estimates with a stochastic adaptation of De Giorgi's iteration and establish a weak Harnack inequality for the solutions. The latter is then used to obtain pointwise almost sure continuity.
We establish elliptic and parabolic Harnack inequalities on graphs with unbounded weights. As an application we prove a local limit theorem for a continuous time random walk $X$ in an environment of ergodic random conductances taking values in $[0, \infty)$ satisfying some moment conditions.
Nonlocal operators related to nonsymmetric forms II: Harnack inequalities
本文针对与非对称型相关的非局部算子，证明了抛物和椭圆积分微分方程解的局部有界性及 Harnack 不等式，分别采用 De Giorgi 迭代和 Moser 迭代两种独立证明方法。
We characterize Gaussian estimates for transition probability of a discrete time Markov chain in terms of geometric properties of the underlying state space. In particular, we show that the following are equivalent: (1) Two sided Gaussian bounds on heat kernel (2) A scale invariant Parabolic Harnack inequality (3) Volume doubling property and a scale invariant Poincaré inequality. The underlying state space is a metric measure space, a setting that includes both manifolds and graphs as special cases. An important feature of our work is that our techniques are robust to small perturbations of t…
This paper introduces certain elliptic Harnack inequalities for harmonic functions in the setting of the product space $M \times X$, where $M$ is a (weighted) Riemannian Manifold and $X$ is a countable graph. Since some standard arguments for the elliptic case fail in this "mixed" setting, we adapt ideas introduced by Thierry Delmotte for the discrete parabolic case. We then present some useful applications of this inequality, namely, a kernel estimate for functions of the Laplacian on a graph. This application in turn provides sharp estimates for certain Markov kernels on graphs. We then clos…
We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases for nearest neighbor conductance models, the idea through parabolic Harnack inequalities does not work, since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theore…
We prove upper bounds on the transition probabilities of random walks with i.i.d. random conductances with a polynomial lower tail near $0$. We consider both constant and variable speed models. Our estimates are sharp. As a consequence, we derive local central limit theorems, parabolic Harnack inequalities and Gaussian bounds for the heat kernel. Some of the arguments are robust and applicable for random walks on general graphs. Such results are stated under a general setting.
For any graph having a suitable uniform Poincare inequality and volume growth regularity, we establish two-sided Gaussian transition density estimates and parabolic Harnack inequality, for constant speed continuous time random walks evolving via time varying, uniformly elliptic conductances, provided the vertex conductances (i.e. reversing measures), increase in time. Such transition density upper bounds apply for discrete time uniformly lazy walks, with the matching lower bounds holding once the parabolic Harnack inequality is proved.
We study the heat kernel and the Green's function on the infinite supercritical percolation cluster in dimension $d \geq 2$ and prove a quantitative homogenization theorem for these functions with an almost optimal rate of convergence. These results are a quantitative version of the local central limit theorem proved by Barlow and Hambly. The proof relies on a structure of renormalization for the infinite percolation cluster introduced by Armstrong and the first author, Gaussian bounds on the heat kernel established by Barlow and tools of the theory of quantitative stochastic homogenization. A…
Parabolic-hyperbolic dichotomy through half-plane coexistence
研究半平面共存对随机平面图的抛物性和双曲性影响，揭示URM的抛物-双曲分裂机制。
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…
In this paper characterizations of graphs satisfying heat kernel estimates for a wide class of space-time scaling functions are given. The equivalence of the two-sided heat kernel estimate and the parabolic Harnack inequality is also shown via the equivalence of the upper (lower) heat kernel estimate to the parabolic mean value (and super mean value) inequality.
We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.
We consider divergence-form parabolic equation with measurable uniformly elliptic matrix and the vector field in a large class containing, in particular, the vector fields in $L^p$, $p>d$, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$, $\varepsilon>0$. We establish Hölder continuity of the bounded soutions, sharp two-sided Gaussian bound on the heat kernel, Harnack inequality.
The following stochastic Cauchy initial-value problem is studied for the parabolic heat equation on a domain $ \mathbf{Q}\subset{\mathbf{R}}^{n}$ with random field initial data. \begin{align} &{\square}\widehat{u(x,t)} \equiv \bigg(\frac{\partial}{\partial t}-Δ_{x}\bigg)\widehat{u(x,t)}=0,~x\in\mathbf{Q},t> 0 \end{align} \begin{align} \widehat{u(x,0)}=ϕ(x)+\mathscr{J}(x),~x\in\mathbf{Q},t=0 \end{align} where $ϕ(x)\in C^{\infty}({\mathbf{Q}})$, and $\mathscr{J}(x)$ is a classical Gaussian random scalar field with expectation $\mathbb{E}[\![\mathscr{J}(x)]\!]=0 $ and with a regulated covariance …
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…
Let $\mathbb{M}$ be a smooth connected manifold endowed with a smooth measure $μ$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $μ$. We show that if $L$ satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in \cite{BG}, then the following properties hold: 1 The volume doubling property; 2 The Poincaré inequality; 3 The parabolic Harnack inequality. The key ingredient is the study of dimensional reverse log-Sobolev inequalities for the heat semi…
For a general class of percolation models with long-range correlations on $\mathbb Z^d$, $d\geq 2$, introduced in arXiv:1212.2885, we establish regularity conditions of Barlow arXiv:math/0302004 that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincaré inequality. As immediate corollaries, we deduce quenched heat kernel bounds, parabolic Harnack inequality, and finiteness of the dimension of harmonic functions with at most polynomial growth. Heat kernel bounds and the quenched invariance principle of arX…
We show that for infinite planar unimodular random rooted maps, many global geometric and probabilistic properties are equivalent, and are determined by a natural, local notion of average curvature. This dichotomy includes properties relating to amenability, conformal geometry, random walks, uniform and minimal spanning forests, and Bernoulli bond percolation. We also prove that every simply connected unimodular random rooted map is sofic, that is, a Benjamini-Schramm limit of finite maps.
In this paper, we are interested in the loop cluster model on $\mathbb{Z}^d$ for $d\geq 3$. It is a long range model with two parameters $α$ and $κ$, where the non-negative parameter $α$ measures the amount of loops, and $κ$ plays the role of killing on vertices penalizing ($κ\geq 0$) or favoring ($κ<0$) appearance of large loops. We consider the truncated loop cluster model formed by the Poisson point process $\mathcal{L}_{α,\leq m}$, which is the restriction of $\mathcal{L}_α$ on loops with at most $m$ jumps. We prove the existence of percolation in a $2$-dimensional slab for the truncated l…
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