We construct a compatible sequence of resistance forms on the graphs approximating the infinite Sierpinski gasket and show that the resistance form given by the limit of the compatible sequence induces the Dirichlet form associated with the Brownian motion on the infinite Sierpinski gasket. Moreover, using this characterization of the Dirichlet form, we obtain an explicit expression of the jump kernel associated with the trace process of the Brownian motion onto the half-line [0, ∞).

We derive a heat kernel lower bound estimate for symmetric pure jump processes on general volume doubling metric measure spaces with possible degenerate and/or singular jump kernels using averaged jump kernels. As an application, the main result of this paper is applied to derive a lower bound estimate for the transition density function of the trace of Brownian motions on Sierpinski gaskets on the bottom of the Sierpinski gasket.

We provide a rich family of self-similar sets, called locally symmetric polygon-based self-similar sets, as examples of metric spaces having conductive homogeneity, which was introduced as a sufficient condition for the construction of counterparts of "Sobolev spaces" on compact metric spaces. In particular, our results imply the existence of "Brownian motions" on our family of self-similar sets at the same time. Unlike the known examples like the Sierpinski carpet by Barlow-Bass, unconstrained carpet by Cao and Qiu and the Octa-carpet by Andrews, our examples may have no global symmetries, i.e. the group of isometries is trivial.

The Sierpinski gasket K has three line segments constituting a regular triangle as its border. This paper studies what will happen if one of them, which is called the bottom line and is denoted by I , is removed from K . At a glance, “the Sierpinski gasket minus the bottom line” K\ I has a structure of a tree of Sierpinski gaskets. This observation leads us to the results showing that the boundary of K\ I is not the line segment I but a Cantor set from viewpoints of geometry and analysis. As a by-product, we have an explicit expression of the jump kernel of the trace of the Brownian motion of K on the bottom line I .

In the ordinary theory of Sobolev spaces on domains of $R^n$, the $p$-energy is defined as the integral of $|\nabla{f}|^p$. In this paper, we try to construct $p$-energy on compact metric spaces as a scaling limit of discrete $p$-energies on a series of graphs approximating the original space. In conclusion, we propose a notion called conductive homogeneity under which one can construct a reasonable $p$-energy if $p$ is greater than the Ahlfors regular conformal dimension of the space. In particular, if $p = 2$, then we construct a local regular Dirichlet form and show that the heat kernel associated with the Dirichlet form satisfies upper and lower sub-Gaussian type heat kernel estimates. As examples of conductively homogeneous spaces, we present a new class of square-based self-similar sets and rationally ramified Sierpinski cross, where no diffusion was constructed before.

Successive divisions of a space have played important roles in many areas of mathematics. One of the simplest examples is the binary division of the unit interval [0, 1] shown in Fig. 1.1. Let K ϕ = [0, 1] and divide K ϕ in half as $$K_0 = [0, \frac 12]$$ and $$K_1 = [\frac 12, 1]$$ . Next, K 0 and K 1 are divided in half again and yield K ij for each (i, j) ∈{0, 1}2. Repeating this procedure, we obtain $$\{K_{{i}_1\ldots {i}_{m}}\}_{i_1, \ldots , i_m \in \{0, 1\}}$$ satisfying $$\displaystyle K_{{i}_1\ldots {i}_{m}} = K_{{i}_1\ldots {i}_{m}0} \cup K_{{i}_1\ldots {i}_{m}1}$$ for any m ≥ 0 and i 1…i m ∈{0, 1}m. In this example, there are two notable properties.

In this section, we present a sufficient condition for the existence of an adapted metric to a given weight function. The sufficient condition obtained in this section will be used to construct an Ahlfors regular metric later.

This book presents a systematic framework on geometry and analysis on metric spaces. The central notion is a partition (an iterated decomposition) of a compact metric space. Via a partition, a compact metric space is associated with an infinite graph whose boundary is the original space.

In this section, we define the notion of bi-Lipschitz equivalence of weight functions. Originally the definition, Definition 3.1.1, only concerns the tree structure $$(T, \mathcal {A}, \phi )$$ and has nothing to do with a partition of a space.

In this section, we review basic notions and notations on a tree with a reference point.

In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including n n -dimensional cube [ 0 , 1 ] n [0, 1]^n are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on [ 0 , 1 ] n [0, 1]^n , density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on [ 0 , 1 ] 2 [0, 1]^2 and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to 0 0 . In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance”associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.

The stretched Sierpinski gasket, SSG for short, is the space obtained by replacing every branching point of the Sierpinski gasket by an interval. It has also been called "deformed Sierpinski gasket" or "Hanoi attractor". As a result, it is the closure of a countable union of intervals and one might expect that a diffusion on SSG is essentially a kind of gluing of the Brownian motions on the intervals. In fact, there have been several works in this direction. There still remains, however, "reminiscence" of the Sierpinski gasket in the geometric structure of SSG and the same should therefore be expected for diffusions. This paper shows that this is the case. In this work, we identify all the completely symmetric resistance forms on SSG. A completely symmetric resistance form is a resistance form whose restriction to every contractive copy of SSG in itself is invariant under all geometrical symmetries of the copy, which constitute the symmetry group of the triangle. We prove that completely symmetric resistance forms on SSG can be sums of the Dirichlet integrals on the intervals with some particular weights, or a linear combination of a resistance form of the former kind and the standard resistance form on the Sierpinski gasket.

Successive divisions of compact metric spaces appear in many different areas of mathematics such as the construction of self-similar sets, Markov partitions associated with hyperbolic dynamical systems, dyadic cubes associated with a doubling metric space. The common feature in these is to divide a space into a finite number of subsets, then divide each subset into finite pieces and repeat this process again and again. In this paper we generalize such successive divisions and call them partitions. Given a partition, we consider the notion of a weight function assigning a "size" to each piece of the partition. Intuitively we believe that a partition and a weight function should provide a "geometry" and an "analysis" on the space of our interest. We are going to pursue this idea in three parts. In the first part, the metrizability of a weight function, i.e. the existence of a metric "adapted to" a given weight function, is shown to be equivalent to the Gromov hyperbolicity of the graph associated with the weight function. In the second part, the notions like bi-Lipschitz equivalence, Ahlfors regularity, the volume doubling property and quasisymmetry will be shown to be equivalent to certain properties of weight functions. In particular, we find that quasisymmetry and the volume doubling property are the same notion in the world of weight functions. In the third part, a characterization of the Ahlfors regular conformal dimension of a compact metric space is given as the critical index p of p-energies associated with the partition and the weight function corresponding to the metric.

Adaptive dynamics combines deterministic population dynamics of groups having different trait values and random process describing mutation and tries to predict the course of evolution of a species of interest. One of basic interests is to know which group survives, residents or mutants. By using invasion fitness as the primary tool, “invasion implies substitution” principle, IIS principle for short, has been established under the existence of a generating function in the sense of Brown and Vincent (Theor Popul Biol 31(1):140–166, 1987) and Vincent and Brown (Evolutionary game theory, natural selection, and darwinian dynamics. Cambridge University Press, Cambridge, 2005). This principle essentially says that the local gradient of invasion fitness ultimately determines the outcome of the competition. However, as we will see in this paper, even if a system is within the scope of IIS principle, its neighborhood always contains systems which are beyond this scope. In this paper, in order to overcome such a limitation, we establish a wider class of systems which is still reasonable as a model of evolution of a species. For our wider class, the notion of raw invasion fitness is introduced. In terms of raw invasion fitness, an explicit criterion for the existence of generating function and a counterpart of IIS principle are obtained. This enables us to discuss small perturbations of a system within or without the scope of generating functions/IIS principle. Eventually, we understand why invasion implies substitution, i.e. why the method using invasion fitness works well with the existence of generating function, from our broader point of view.

Using the notions of scales and their gauge functions associated with self-similar sets, we give a necessary and sufficient condition for two metrics on a self-similar set being quasisymmetric to each other. As an application, we construct metrics on the Sierpinski carpet which is quasisymmetric with respect to the Euclidean metrics and obtain an upper estimate of the conformal dimension of the Sierpinski carpet.

First, noncompact Cantor sets along with their defining trees are introduced as a natural generalization of p-adic numbers. Secondly we construct a class of jump processes on a noncompact Cantor set from given pairs. of eigenvalues and measures. At the same time, we have concrete expressions of the associated jump kernels and transition densities. Then we construct intrinsic metrics on noncompact Cantor set to obtain estimates of transition densities and jump kernels under some regularity conditions on eigenvalues and measures. Finally transient random walks on the defining tree are shown to induce a subclass of jump processes discussed in the second part.

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, we consider the above problem in the case of stochastic processes associated with Dirichlet forms derived from resistance forms. Our main concerns are following two problems: (I) When and how can we 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? Note that in general stochastic processes associated with Dirichlet forms have jumps, i. e. paths of such processes are not continuous. The answer to (I) is for measures to have volume doubling property with respect to the resistance metric associated with a resistance form. Under volume doubling property, a new metric which is quasisymmetric with respect to the resistance metric is constructed and the Li-Yau type diagonal sub-Gaussian estimate of the heat kernel associated with the process using the new metric is shown. About the question (II), we will propose a condition called annulus comparable condition, (ACC) for short. This condition is shown to be equivalent to the existence of a good diagonal heat kernel estimate. As an application, asymptotic behaviors of the traces of 1-dimensional α-stable processes are obtained. In the course of discussion, considerable numbers of pages are spent on the theory of resistance forms and quasisymmetric maps. 1991 Mathematics Subject Classification. Primary 30L10, 31E05, 60J35; Secondary 28A80, 43A99, 60G52.

Transient random walk on a tree induces a Dirichlet form on its Martin boundary, which is the Cantor set. The procedure of the inducement is analogous to that of the Douglas integral on S1 associated with the Brownian motion on the unit disk. In this paper, those Dirichlet forms on the Cantor set induced by random walks on trees are investigated. Explicit expressions of the hitting distribution (harmonic measure) ν and the induced Dirichlet form on the Cantor set are given in terms of the effective resistances. An intrinsic metric on the Cantor set associated with the random walk is constructed. Under the volume doubling property of ν with respect to the intrinsic metric, asymptotic behaviors of the heat kernel, the jump kernel and moments of displacements of the process associated with the induced Dirichlet form are obtained. Furthermore, relation to the noncommutative Riemannian geometry is discussed.

This paper studies the following three problems. 1. When does a measure on a self-similar set have the volume doubling property with respect to a given distance? 2. Is there any distance on a self-similar set under which the contraction mappings have the prescribed values of contractions ratios? 3. When does a heat kernel on a self-similar set associated with a self-similar Dirichlet form satisfy the Li-Yau type sub-Gaussian diagonal estimate? Those three problems turns out to be closely related. We introduce a new class of self-similar set, called rationally ramified self-similar sets containing both the Sierpinski gasket and the (higher dimensional) Sierpinski carpet and give complete solutions of the above three problems for this class. In particular, the volume doubling property is shown to be equivalent to the upper Li-Yau type sub-Gaussian diagonal estimate of a heat kernel. Received by the editor May 26, 2005; and in revised form September 18, 2006. 2000 Mathematics Subject Classification. Primary 28A80, 60J35; Secondary 31C25, 60J45.

We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate is established for the heat kernel associated withthe Kusuoka measure.