The main result of this thesis is the two-sided heat kernel estimates for both Dirichlet and Neumann problem in a inner uniform domain of Rn, and many other spaces with Gaussian-type heat kernel estimates. We assume that the heat equation is associated with a local divergence form differential operator, or more generally with a strictly local Dirichlet form on a complete locally compact metric space. Other results include the (parabolic) Harnack inequality and the boundary Harnack principle.

We introduce conditions on the symmetric and skew-symmetric parts of timedependent, local, regular forms that imply a parabolic Harnack inequality for appropriate weak solutions of the associated heat equation, under natural assumptions on the underlying space. In particular, these local weak solutions are locally bounded and Holder continuous. Precise two-sided heat kernel estimates are deo rived from this parabolic Harnack inequality. For Dirichlet forms satisfying our conditions we prove a scale-invariant boundary Harnack principle in inner uniform domains. Inner uniformity is a condition on the boundary of the domain that is described solely in terms of the intrinsic length metric of the domain. In addition, we show that the Martin boundary of an inner uniform domain is homeomorphic to the boundary of the domain with respect to its completion in the inner distance. The main result of this work are two-sided Gaussian bounds for Dirichlet heat kernels corresponding to (non- )symmetric, local, regular Dirichlet forms. These bounds hold in domains that satisfy the inner uniformity condition. The proof uses the parabolic Harnack inequality and the boundary Harnack principle described above, as well as the Doob h-transform technique. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa, A. Ancona, P. Gyrya, L. Saloff-Coste and K.-T. Sturm.

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This monograph focuses on the heat equation with either the Neumann or the Dirichlet boundary condition in unbounded domains in Euclidean space, Riemannian manifolds, and in the more general context of certain regular local Dirichlet spaces. In works by A. Grigor'yan, L. Saloff-Coste, and K.-T. Sturm, the equivalence between the parabolic Harnack inequality, the two-sided Gaussian heat kernel estimate, the Poincare inequality and the volume doubling property is established in a very general context. The authors use this result to provide precise two-sided heat kernel estimates in a large class of domains described in terms of their inner intrinsic metric and called inner (or intrinsically) uniform domains. Perhaps surprisingly, they treat both the Neumann boundary condition and the Dirichlet boundary condition using essentially the same approach, albeit with the additional help of a Doob's h-transform in the case of Dirichlet boundary condition. The main results are new even when applied to Euclidean domains with smooth boundary where they capture the global effect of the condition of inner uniformity as, for instance, in the case of domains that are the complement of a convex set in Euclidean space.

Abstract: "In this paper, we investigate symmetric jump-type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces. The class of metric measure spaces includes the Alfors d-regular sets, which is a class of fractal sets that contains geometrically self-similar sets. A typical example of our jump-type processes is the symmetric jump process with jumping intensity [formula] where v is a probability measure on [[alpha]1, [alpha]2] [subset of] (0,2), c([alpha], x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between two positive constants, and c0(x, y) is a jointly measurable function that is symmetric in (x, y) and is bounded between [gamma]1 and [gamma]2, where either [gamma]2 [> or =] [gamma]1> 0 or [gamma]1 = [gamma]2 = 0. This example contains mixed symmetric stable processes on R[superscript n] as well as mixed relativistic symmetric stable processes on R[superscript n]. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes."

Focusing on Poincaré, Nash and other Sobolev-type inequalities and their applications to the Laplace and heat diffusion equations on Riemannian manifolds, this text is an advanced graduate book that will also suit researchers.

Assume that there is some analytic structure, a differential equation or a stochastic process for example, on a metric space. To describe asymptotic behaviors of analytic objects, the original metric of the space may not be the best one. Every now and then one can construct a better metric which is somehow ``intrinsic'' with respect to the analytic structure and under which asymptotic behaviors of the analytic objects have nice expressions. The problem is when and how one can find such a metric. In this paper, the author considers the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. The author's main concerns are the following two problems: (I) When and how to find a metric which is suitable for describing asymptotic behaviors of the heat kernels associated with such processes. (II) What kind of requirement for jumps of a process is necessary to ensure good asymptotic behaviors of the heat kernels associated with such processes.