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Given a symmetric diffusion process and a jump process on the same underlying space, is there a subordinator such that the jump process and the subordinated diffusion process are comparable? We address this question when the diffusion satisfies a sub-Gaussian heat kernel estimate and the jump process satisfies a polynomial-type jump kernel bounds. Under these assumptions, we obtain necessary and sufficient conditions on the jump kernel estimate for such a subordinator to exist. As an application of our results and the recent stability results of Chen, Kumagai and Wang, we obtain parabolic Harnack inequality for a large family of jump processes. In particular, we show that any jump process with polynomial-type jump kernel bounds on such a space satisfy the parabolic Harnack inequality. [ABSTRACT FROM AUTHOR]

Consider a symmetric Markovian jump process {Xt} on a metric measure space (M, d, μ). Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming (M, d, μ) satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if (M, d, μ) is a graph (or more generally, if M contains any atoms x such that μ(x) > 0) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an "auxiliary space" that smooths out the atoms. We show that many properties transfer from (M, d, μ) to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations. [ABSTRACT FROM AUTHOR]

We study the Li–Yau inequality for the heat equation corresponding to the Dunkl harmonic oscillator, which is a nonlocal Schrödinger operator parameterized by reflections and multiplicity functions. In the particular case when the reflection group is isomorphic to ℤ 2 d , the result is sharp in the sense that equality is achieved by the heat kernel of the classic harmonic oscillator. We also provide the application on parabolic Harnack inequalities. [ABSTRACT FROM AUTHOR]

In this paper, we establish sharp two-sided estimates for transition densities of a large class of subordinate Markov processes. As applications, we show that the parabolic Harnack inequality and Hölder regularity hold for parabolic functions of such processes, and derive sharp two-sided Green function estimates. [ABSTRACT FROM AUTHOR]

We prove a quenched local central limit theorem for continuous-time random walks in Z d , d ≥ 2 , in a uniformly-elliptic time-dependent balanced random environment which is ergodic under space-time shifts. We also obtain Gaussian upper and lower bounds for quenched and (positive and negative) moment estimates of the transition probabilities and asymptotics of the discrete Green's function. [ABSTRACT FROM AUTHOR]

Let Sn be a symmetric simple random walk on the integer lattice Zd. For a Bernstein function ϕ we consider a random walk Snϕ which is subordinated to Sn. Under a certain assumption on the behaviour of ϕ at zero we establish global estimates for the transition probabilities of the random walk Snϕ. The main tools that we apply are a parabolic Harnack inequality and appropriate bounds for the transition kernel of the corresponding continuous time random walk. [ABSTRACT FROM AUTHOR]

In this paper, we establish stability of parabolic Harnack inequalities for symmetric nonlocal Dirichlet forms on metric measure spaces under a general volume doubling condition. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cutoff Sobolev inequalities, and Poincar'e inequalities. In particular, we establish the connection between parabolic Harnack inequalities and two-sided heat kernel estimates, as well as with the H¨older regularity of parabolic functions for symmetric non-local Dirichlet forms. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider the non-local Fisher-KPP equation modeling a population with individuals competing with each other for resources with a strength related to their distance, and obtain the asymptotics for the position of the invasion front starting from a localized population. Depending on the behavior of the competition kernel at infinity, the location of the front is either 2 t − (3 / 2) log ⁡ t + O (1) , as in the local case, or 2 t − O (t β) for some explicit β ∈ (0 , 1). Our main tools here are a local-in-time Harnack inequality and an analysis of the linearized problem with a suitable moving Dirichlet boundary condition. Our analysis also yields, for any β ∈ (0 , 1) , examples of Fisher-KPP type non-linearities f β such that the front for the local Fisher-KPP equation with reaction term f β is at 2 t − O (t β). [ABSTRACT FROM AUTHOR]

For any graph having a suitable uniform Poincaré 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth, and nonlocal competition. We introduce the climate shift due to global warming and discuss the dynamics of the population by studying the long time behavior of the solution of the Cauchy problem. We consider three sets of assumptions on the growth function. In the so-called confined case we determine a critical climate change speed for the extinction or survival of the population, the latter case taking place by "strictly following the climate shift." In the so-called environmental gradient case, or unconfined case, we additionally determine the propagation speed of the population when it survives: thanks to a combination of migration and evolution, it can be different here than the speed of the climate shift. Finally, we consider mixed scenarios, that are complex situations, where the growth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left. The main difficulty comes from the nonlocal competition term that prevents the use of classical methods based on comparison arguments. This difficulty is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, sharp two-sided estimates for the transition densities of relativistic α-stable processes with mass in exterior open sets are established for all time . These transition densities are also the Dirichlet heat kernels of with in exterior open sets. The estimates are uniform in m in the sense that the constants are independent of . As a corollary of our main result, we establish sharp two-sided Green function estimates for relativistic α-stable processes with mass in exterior open sets. [Copyright &y& Elsevier]

In this paper, we derive global sharp heat kernel estimates for symmetric α-stable processes (or equivalently, for the fractional Laplacian with zero exterior condition) in two classes of unbounded C open sets in $${\mathbb R^d}$$: half-space-like open sets and exterior open sets. These open sets can be disconnected. We focus in particular on explicit estimates for p( t, x, y) for all t > 0 and $${x, y\,{\in}\,D}$$. Our approach is based on the idea that for x and y in D far from the boundary and t sufficiently large, we can compare p( t, x, y) to the heat kernel in a well understood open set: either a half-space or $${\mathbb R^d}$$; while for the general case we can reduce them to the above case by pushing x and y inside away from the boundary. As a consequence, sharp Green functions estimates are obtained for the Dirichlet fractional Laplacian in these two types of open sets. Global sharp heat kernel estimates and Green function estimates are also obtained for censored stable processes (or equivalently, for regional fractional Laplacian) in exterior open sets. [ABSTRACT FROM AUTHOR]

We consider the fractional Laplacian --(--Δ)α/2 on an open subset in ℝd with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such a Dirichlet fractional Laplacian in C1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric α-stable process killed upon leaving a C1,1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets. [ABSTRACT FROM AUTHOR]

In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality. [ABSTRACT FROM AUTHOR]

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to establish a gradient estimate and a parabolic Harnack inequality for the non-symmetric transition semigroup with respect to the Gibbs measure on a path space. This semigroup is related to a diffusion process which is represented by the solution of a certain parabolic stochastic partial differential equation (SPDE, in abbreviation) containing rotation. We also discuss the relationship between the Gibbs measure and stationary measures of our dynamics. For the proof of our functional inequalities, we formulate a suitable domain of the infinitesimal generator for the semigroup. As an application of our results, we study a certain lower estimate on the transition probability for our dynamics. [Copyright &y& Elsevier]

The notion of d-set arises in the theory of function spaces and in fractal geometry. Geometrically self-similar sets are typical examples of d-sets. In this paper stable-like processes on d-sets are investigated, which include reflected stable processes in Euclidean domains as a special case. More precisely, we establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such stable-like processes. Results on the exact Hausdorff dimensions for the range of stable-like processes are also obtained. [Copyright &y& Elsevier]

In this paper, we consider the following symmetric Dirichlet forms on a metric measure space (M , d , μ) : E (f , g) = E (c) (f , g) + ∫ M × M (f (x) − f (y)) (g (x) − g (y)) J (d x , d y) , where E (c) is a strongly local symmetric bilinear form and J (d x , d y) is a symmetric Radon measure on M × 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 inequalities, and the Faber-Krahn inequalities (resp. the Poincaré inequalities). We also obtain characterizations of parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2. [ABSTRACT FROM AUTHOR]

We obtain an upper bound on the minimal number of points in an ε -chain joining two points in a metric space. This generalizes a bound due to Hambly and Kumagai (1999) for the case of resistance metric on certain self-similar fractals. As an application, we deduce a condition on ε -chains introduced by Grigor'yan and Telcs (2012). This allows us to obtain sharp bounds on the heat kernel for spaces satisfying the parabolic Harnack inequality without assuming further conditions on the metric. A snowflake transform on the Euclidean space shows that our bound is sharp. [ABSTRACT FROM AUTHOR]