In this paper we use methods from Stochastic Analysis to establish Li-Yau type estimates for positive solutions of the heat equation. In particular, we want to emphasize that Stochastic Analysis provides natural tools to derive local estimates in the sense that the gradient bound at given point depends only on universal constants and the geometry of the Riemannian manifold locally about this point.
The Harnack and log Harnack inequalities for stochastic differential equation driven by $G$-Brownian motion with multiplicative noise are derived by means of coupling by change of mesure. All of the above results extend the existing ones in the linear expectation setting. Moreover, the gradient estimate generalize the nonlinear results appeared in [11].
In this paper, the coupling by change of measure is constructed for a class of SDEs with integrable drift and additive noise, from which the Harnack and shift Harnack inequalities are derived. Finally, as applications, the gradient estimate, the regularity of the heat kernel and the distribution properties of the associated transition probability are also obtained. The important tool is Krylov's estimate.
In the paper, Harnack inequalities are established for stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H<1/2$. As applications, strong Feller property, log-Harnack inequality and entropy-cost inequality are given.
We develop connections between Harnack inequalities for the heat flow of diffusion operators with curvature bounded from below and optimal transportation. Through heat kernel inequalities, a new isoperimetric-type Harnack inequality is emphasized. Commutation properties between the heat and Hopf-Lax semigroups are developed consequently, providing direct access to the heat flow contraction property along Wasserstein distances.
Dimension-independent Harnack inequalities are derived for a class of subordinate semigroups. In particular, for a diffusion satisfying the Bakry-Emery curvature condition, the subordinate semigroup with power $α$ satisfies a dimension-free Harnack inequality provided $α\in(1/2, 1)$, and it satisfies the log-Harnack inequality for all $α\in (0,1).$ Some infinite-dimensional examples are also presented.
Using coupling by change of measure and an approximation technique, Wang's Harnack inequalities are established for a class of functional SDEs driven by subordinate Brownian motions. The results cover the corresponding ones in the case without delay.
By the coupling method, we establish the Harnack inequalities, derivative formula and Driver's integration by parts formula for the stochastic Klein-Gordon type equations in the interval. We provide a detailed discussion about the nonlinear term. Some applications are given.
The existence and uniqueness of the mild solutions for a class of degenerate functional SPDEs are obtained, where the drift is assumed to be Hölder-Dini continuous. Moreover, the non-explosion of the solution is proved under some reasonable conditions. In addition, the Harnack is derived by the coupling by change of measure. Finally, the shift Harnack inequality is obtained for the equations without delay, which is new even in the non-degenerate case.
The existence and uniqueness are established for McKean-Vlasov SDEs driven by Lévy processes. By using an approximation technique and coupling by change of measures, Harnack inequalities are investigated for McKean-Vlasov SDEs driven by subordinate Brownian motions.
We establish transportation cost inequalities, with respect to the uniform and $L_2$-metric, on the path space of continuous functions, for laws of solutions of stochastic differential equations with reflections. We also consider the case of stochastic differential equations involving local times. Harnack inequalities for the associated semigroups are also established.
By using the existing sharp estimates of density function for rotationally invariant symmetric $α$-stable Lévy processes and rotationally invariant symmetric truncated $α$-stable Lévy processes, we obtain that Harnack inequalities hold for rotationally invariant symmetric $α$-stable Lévy processes with $α\in(0,2)$ and Ornstein-Uhlenbeck processes driven by rotationally invariant symmetric $α$-stable Lévy process, while logarithmic Harnack inequalities are satisfied for rotationally invariant symmetric truncated $α$-stable Lévy processes.
By using the coupling argument, we establish the Harnack and log-Harnack inequalites for stochastic differential equations with non-Lipschitz drifts and driven by additive anisotropic subordinated Brownian motions (in particular, cylindrical $α$-stable processes). Moreover, the gradient estimate is also derived when the drift is Lipschitz continuous.
Based on coupling in two steps and the regularization approximations of the underlying subordinators, we establish log-Harnack inequalities for Markov semigroups generated by a class of non-local Gruschin type operators. Some concrete examples are also presented.
By the method of coupling and Girsanov transformation, Harnack inequalities [F.-Y. Wang, 1997] and strong Feller property are proved for the transition semigroup associated with the multivalued stochastic evolution equation on a Gelfand triple. The concentration property of the invariant measure for the semigroup is investigated. As applications of Harnack inequalities, explicit upper bounds of the $L^p$-norm of the density, contractivity, compactness and entropy-cost inequality for the semigroup are also presented.
By using coupling arguments, Harnack type inequalities are established for a class of stochastic (functional) differential equations with multiplicative noises and non-Lipschitzian coefficients. To construct the required couplings, two results on existence and uniqueness of solutions on an open domain are presented.
By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semi-linear…
We establish Harnack inequalities for stochastic differential equations (SDEs) driven by a time-changed fractional Brownian motion with Hurst parameter $H\in(0,1/2)$. The Harnack inequality is dimension-free if the SDE has a drift which satisfies a one-sided Lipschitz condition, otherwise we still get Harnack-type estimates, but the constants will, in general, depend on the space dimension. Our proof is based on a coupling argument and a regularization argument for the time-change.
For stochastic differential equation driven by fractional Brownian motion with Hurst parameter $H>1/2$, Harnack type inequalities are established by constructing a coupling with unbounded time-dependent drift. These inequalities are applied to the study of existence and uniqueness of invariant measure for a discrete Markov semigroup constructed in terms of the distribution of the solution. Furthermore, we show that entropy-cost inequality holds for the invariant measure.
In this paper, the Harnack inequalities and super-Poincaré inequality for generalized CIR model are obtained. Since the noise is degenerate, the intrinsic metric has been introduced to construct the coupling by change of measure. By using isopermetric constant, the optimal estimate of the rate function in the super Poincaré inequality for the associated Dirichlet form is also obtained.
We give Harnack inequalities for the hitting distributions of a large family of symmetric random walks on $\Z^2$, and their projections onto the lattice torus $\Z^2_K$. This extends a framework for the simple random walk in Dembo, et al. (2006), and generalizes the results in Bass, Rosen (2007) to the toral projection.
By constructing a new coupling, the log-Harnack inequality is established for the functional solution of a delay stochastic differential equation with multiplicative noise. As applications, the strong Feller property and heat kernel estimates w.r.t. quasi-invariant probability measures are derived for the associated transition semigroup of the solution. The dimension-free Harnack inequality in the sense of \cite{W97} is also investigated.
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…
This work examines a class of switching jump diffusion processes. The main effort is devoted to proving the maximum principle and obtaining the Harnack inequalities. Compared with the diffusions and switching diffusions, the associated operators for switching jump diffusions are non-local, resulting in more difficulty in treating such systems. Our study is carried out by taking into consideration of the interplay of stochastic processes and the associated systems of integro-differential equations.
The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers \cite{Flandoli, Zhang11}. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric s…
In this article, we study a class of semilinear stochastic partial differential equations driven by an additive space time white noise. We establish Harnack inequalities for the semigroup associated with the solution by using coupling method, which implies the strong Feller property. Our results generalize the results of Zhang [Potential Analysis 33 (2010), no. 2, 137-151.] and can be applied to some types of SPDE such as reaction-diffusion equation and transport-diffusion equation perturbed by space time white noise.
We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking κ(x,y)/|x-y|^{n+α}dxdy, x,y\in G as the jumping measure. When G is a C^{1,β-1} open set, 1<α<β\leq 2 and κ\in C^{1}(\overline{G}\times \overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\partial G)^{α-1} order decay for harmonic functions near the boundary. For a C^{1,β-1} open set D\subset \overline{D}\subset G, 0<α\leq (1\veeα)<β\leq 2, we prove a boundary Harnack inequality giving dist(x,\partial …
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.
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.
The dimension free Harnack inequality for the heat semigroup is established on the $\RCD(K,\infty)$ space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott-Sturm-Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.
By a new approximate method, dimensional free Harnack inequalities are established for a class of semilinear stochastic differential equations in Hilbert space with multiplicative noise. These inequalities are applied to study the strong Feller property for the semigroup and some properties of invariant measure.
By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.
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 this paper we consider Harnack inequalities with respect to a symmetric $α$-stable Lévy process $X$ in $\mathbb{R}^d$, $α\in (0,2)$, $d\geq 2$. We study the example from the article \cite{bg-sz-1}. There, the authors have associated the Harnack inequality with the relative Kato condition, which is a condition on the Lévy measure. By checking the condition, in the case $α\in (0,1)$, they have established that the Harnack inequality does not hold. We give an alternative proof of this fact, using the setting of \cite{bg-sz-1}. We define the harmonic functions explicitly. For a given starting p…
By constructing successful couplings for degenerate diffusion processes, explicit derivative formula and Harnack type inequalities are presented for solutions to a class of degenerate Fokker-Planck equations on $\R^m\times\R^{d}$. The main results are also applied to the study of gradient estimate, entropy/transportation-cost inequality and heat kernel inequalities.
We consider operators of the form $L u(x) = \sum_{y \in \mathbb{Z}} k(x-y) \big( u(y) - u(x)\big)$ on the one-dimensional lattice with symmetric, integrable kernel $k$. We prove several results stating that under certain conditions on the kernel the operator $L$ satisfies the curvature-dimension condition $CD_Υ(0,F)$ (recently introduced by two of the authors) with some $CD$-function $F$, where attention is also paid to the asymptotic properties of $F$ (exponential growth at infinity and power-type behaviour near zero). We show that $CD_Υ(0,F)$ implies a Li-Yau inequality for positive solution…
We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.
The Harnack inequality established in [13] for generalized Mehler semigroup is improved and generalized. As applications, the log-Harnack inequality, the strong Feller property, the hyper-bounded property, and some heat kernel inequalities are presented for a class of O-U type semigroups with jump. These inequalities and semigroup properties are indeed equivalent, and thus sharp, for the Gaussian case. As an application of the log-Harnack inequality, the HWI inequality is established for the Gaussian case. Perturbations with linear growth are also investigated.
In this paper, we consider transient subordinate Brownian motion X in R^d, d \geq 1, where the Laplace exponent ϕof the corresponding subordinator satisfies some mild conditions. The scaleinvariant Harnack inequality is proved for X. We first give new forms of asymptotical properties of the Levy and potential density of the subordinator near zero. Using these results we find asymptotics of the Levy density and potential density of X near the origin, which is essential to our approach. The examples which are covered by our results include geometric stable processes and relativistic geometric st…
We show that shift Harnack type inequalities (in the sense of F.-Y.~Wang \cite{Wan14}) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general Lévy processes.
By the approximation method introduced in \cite{FYW}, the existence and uniqueness are proved for a class of distribution-dependent stochastic functional differential equations (DDSFDEs). Moreover, combining the Harnack and shift-Harnack inequalities for classical stochastic functional differential equations with Girsanov's theorem, Harnack and shift-Harnack inequalities are obtained for the non-linear semigroup $P_t^*$ associated to the functional solution.
By constructing successful couplings, the derivative formula, gradient estimates and Harnack inequalities are established for the semigroup associated with a class of degenerate functional stochastic differential equations.
We consider the Harnack inequality for harmonic functions with respect to three types of infinite dimensional operators. For the infinite dimensional Laplacian, we show no Harnack inequality is possible. We also show that the Harnack inequality fails for a large class of Ornstein-Uhlenbeck processes, although functions that are harmonic with respect to these processes do satisfy an a priori modulus of continuity. Many of these processes also have a coupling property. The third type of operator considered is the infinite dimensional analog of operators in Hörmander's form. In this case a Harnac…
By using a coupling method, an explicit log-Harnack inequality with local geometry quantities is established for (sub-Markovian) diffusion semigroups on a Riemannian manifold (possibly with boundary). This inequality as well as the consequent $L^2$-gradient inequality, are proved to be equivalent to the pointwise curvature lower bound condition together with the convexity or absence of the boundary. Some applications of the log-Harnack inequality are also introduced.
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.
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 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…
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…
In this article, new curvature conditions are introduced to establish functional inequalities including gradient estimates, Harnack inequalities and transportation-cost inequalities on manifolds with non-convex boundary.
On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $μ$ and for $K$ a constant, the curvature condition $\Ric-\nn Z\ge K$ together with the convexity of the boundary is equivalent to the heat kernel entropy inequality $$\int_M p_t(x,z)\log \ff{p_t(x,z)}{p_t(y,z)} μ(\d z)\le \ff{K\rr(x,y)^2}{2(\e^{2Kt}-1)}, t>0, x,y\in M,$$ where $\rr$ is the Riemannian dist…
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…
