Parabolic Harnack Inequality Research Articles

Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms

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(dx,dy), where E(c) is a strongly local symmetric bilinear form and J(dx,dy) 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.

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

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 the 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 of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

On the length of chains in a metric space

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.

Parabolic Harnack inequality for time-dependent non-symmetric Dirichlet forms

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.

On the parabolic Harnack inequality for non-local diffusion equations

We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $$d\ge \beta $$ , where $$\beta \in (0,2]$$ is the order of the equation with respect to the spatial variable. The equation can be non-local both in time and in space but for the counter-example it is important that the equation has a fractional time derivative. In this case, the fundamental solution is singular at the origin for all times $$t>0$$ in dimensions $$d\ge \beta $$ . This underlines the markedly different behavior of time-fractional diffusion compared to the purely space-fractional case, where a local Harnack inequality is known. The key observation is that the memory strongly affects the estimates. In particular, if the initial data $$u_0 \in L^q_{loc}$$ for q larger than the critical value $$\tfrac{d}{\beta }$$ of the elliptic operator $$(-\Delta )^{\beta /2}$$ , a non-local version of the Harnack inequality is still valid as we show. We also observe the critical dimension phenomenon already known from other contexts: the diffusion behavior is substantially different in higher dimensions than $$d=1$$ provided $$\beta >1$$ , since we prove that the local Harnack inequality holds if $$d<\beta $$ .

Equivalence between radial solutions of different parabolic gradient-diffusion equations and applications

We consider a general form of a parabolic equation that generalizes both the standard parabolic $p$-Laplace equation and the normalized version that has been proposed in stochastic game theory. We establish an equivalence between this equation and the standard $p$-parabolic equation posed in a fictitious space dimension, valid for radially symmetric solutions. This allows us to find suitable explicit solutions for example of Barenblatt type, and as a consequence we settle the exact asymptotic behaviour of the Cauchy problem even for nonradial data. We also establish the asymptotic behaviour in a bounded domain. Moreover, we use the explicit solutions to establish the parabolic Harnack's inequality.

Brownian motion on some spaces with varying dimension

In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.

Random walks among time increasing conductances: heat kernel estimates

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.

Parabolic Harnack inequality on fractal-type metric measure Dirichlet spaces

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.

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

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.

Random walks on infinite percolation clusters in models with long-range correlations

For a general class of percolation models with long-range correlations on $\mathbb{Z}^{d}$, $d\geq2$, introduced in [J. Math. Phys. 55 (2014) 083307], we establish regularity conditions of Barlow [Ann. Probab. 32 (2004) 3024–3084] that mesoscopic subballs of all large enough balls in the unique infinite percolation cluster have regular volume growth and satisfy a weak Poincare 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 [Probab. Theory Related Fields 166 (2016) 619–657] allow to extend various other known results about Bernoulli percolation by mimicking their proofs, for instance, the local central limit theorem of [Electron. J. Probab. 14 (209) 1–27] or the result of [Ann. Probab. 43 (2015) 2332–2373] that the dimension of at most linear harmonic functions on the infinite cluster is $d+1$. In terms of specific models, all these results are new for random interlacements at every level in any dimension $d\geq3$, as well as for the vacant set of random interlacements [Ann. of Math. (2) 171 (2010) 2039–2087; Comm. Pure Appl. Math. 62 (2009) 831–858] and the level sets of the Gaussian free field [Comm. Math. Phys. 320 (2013) 571–601] in the regime of the so-called local uniqueness (which is believed to coincide with the whole supercritical regime for these models).

Differential Harnack inequalities and Perelman type entropy formulae for subelliptic operators

In this paper, under the generalized curvature-dimension inequality recently introduced by F. Baudoin and N. Garofalo, we obtain differential Harnack inequalities (Theorem 2.1) for the positive solutions to the Schrödinger equation associated to subelliptic operator with potential. As applications of the differential Harnack inequality, we derive the corresponding parabolic Harnack inequality (Theorem 4.1). Also we define the Perelman type entropy associated to subelliptic operators and derive its monotonicity (Theorem 5.3).

Supercritical loop percolation on [formula omitted] for [formula omitted

We consider supercritical percolation on Zd (d≥3) induced by random walk loop soup. Two vertices are in the same cluster if they are connected through a sequence of intersecting loops. We obtain quenched parabolic Harnack inequalities, Gaussian heat kernel bounds, the invariance principle and the local central limit theorem for the simple random walks on the unique infinite cluster. We also show that the diameter of finite clusters have exponential tails like in Bernoulli bond percolation. Our results hold for all d≥3 and all supercritical intensities despite polynomial decay of correlations.

The Li–Yau inequality and applications under a curvature-dimension condition

We prove a global Li-Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li-Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequents bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.

The Effect of Climate Shift on a Species Submitted to Dispersion, Evolution, Growth, and Nonlocal Competition

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 {\it 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 {\it confined case} we determine a critical climate change speed for the extinction or survival of the population, the latter case taking place by \lq\lq strictly following the climate shift''. In the so-called {\it environmental gradient case}, or {\it unconfined case}, we additionally determine the propagation speed of the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider {\it 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.

Moser–Nash kernel estimates for degenerate parabolic equations

In this paper we deal with the Cauchy problem associated to a class of nonlinear degenerate parabolic equations, whose prototype is the parabolic p-Laplacian (2<p<∞). In his seminal paper, after stating the Harnack estimates, Moser proved almost optimal estimates for the parabolic kernel by using the so called ‘Harnack chain’ method. In the linear case sharp estimates come by using Nash's approach. Fabes and Stroock proved that Gaussian estimates are equivalent to a parabolic Harnack inequality. In this paper, by using the DiBenedetto–DeGiorgi approach we prove optimal kernel estimates for degenerate quasilinear parabolic equations. To obtain this result we need to prove the finite speed of propagation of the support and to establish optimal estimates. Lastly we use these results to prove existence and sharp pointwise estimates from above and from below for the fundamental solutions.

Gaussian bounds and parabolic Harnack inequality on locally irregular graphs

A well known theorem of Delmotte is that Gaussian bounds, parabolic Harnack inequality, and the combination of volume doubling and Poincare inequality are equivalent for graphs. In this paper we consider graphs for which these conditions hold, but only for sufficiently large balls, and prove a similar equivalence.

Heat Kernel Bounds on Metric Measure Spaces and Some Applications

Let $(X,d,\mu)$ be a $RCD^\ast(K, N)$ space with $K\in \mathbb{R}$ and $N\in [1,\infty]$. For $N\in [1,\infty)$, we derive the upper and lower bounds of the heat kernel on $(X,d,\mu)$ by applying the parabolic Harnack inequality and the comparison principle, and then sharp bounds for its gradient, which are also sharp in time. When $N=\infty$, we also establish a sharp upper bound of the heat kernel by using the dimension free Harnack inequality. For applications, we study the large time behavior of the heat kernel, the stability of solutions to the heat equation, and show the $L^p$ boundedness of (local) Riesz transforms.

Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model

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.

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack inequalities of solutions to quasilinear degenerate parabolic equations. Lastly we focus our attention on Harnack inequalities of solutions to quasilinear singular parabolic equations where the theory did important steps forward in the last few years but still there are some points to be fully understood.