This article gives an overview of some recent progress in the study of sharp two-sided estimates for the transition density of a large class of Markov processes having both diffusive and jumping components in metric measure spaces. We summarize some of the main results obtained recently in [11, 18] and provide several examples. We also discuss new ideas of the proof for the off-diagonal upper bounds of transition densities which are based on a generalized Davies’ method developed in [10].

In the context of countable groups of polynomial volume growth, we consider a large class of random walks that are allowed to take long jumps along multiple subgroups according to power law distributions. For such a random walk, we study the large time behavior of its probability of return at time n in terms of the key parameters describing the driving measure and the structure of the underlying group. We obtain assorted estimates including near-diagonal two-sided estimates and the Hölder continuity of the solutions of the associated discrete parabolic difference equation. In each case, these estimates involve the construction of a geometry adapted to the walk.

We establish via a probabilistic approach the quenched invariance principle for a class of long range random walks in independent (but not necessarily identically distributed) balanced random environments, with the transition probability from x to y on average being comparable to |x − y|−(d+α) with α ∈ (0, 2]. We use the martingale property to estimate exit time from balls and establish tightness of the scaled processes, and apply the uniqueness of the martingale problem to identify the limiting process. When α ∈ (0, 1), our approach works even for non-balanced cases. When α = 2, under a diffusive with the logarithmic perturbation scaling, we show that the limit of scaled processes is a Brownian motion.

This article investigates the heat kernel of the two-dimensional uniform spanning tree. We improve previous work by demonstrating the occurrence of log-logarithmic fluctuations around the leading order polynomial behaviour for the on-diagonal part of the quenched heat kernel. In addition we give two-sided estimates for the averaged heat kernel, and we show that the exponents that appear in the off-diagonal parts of the quenched and averaged versions of the heat kernel differ. Finally, we derive various scaling limits for the heat kernel, the implications of which include enabling us to sharpen the known asymptotics regarding the on-diagonal part of the averaged heat kernel and the expected distance travelled by the associated simple random walk.

We study random walks on Zd (with d≥ 2) among stationary ergodic random conductances { Cx,y: x, y∈ Zd} that permit jumps of arbitrary length. Our focus is on the quenched invariance principle (QIP) which we establish by a combination of corrector methods, functional inequalities and heat-kernel technology assuming that the p-th moment of ∑x∈ZdC0,x|x|2 and q-th moment of 1 / C,x for x neighboring the origin are finite for some p, q≥ 1 with p- 1+ q- 1< 2 / d. In particular, a QIP thus holds for random walks on long-range percolation graphs with connectivity exponents larger than 2d in all d≥ 2 , provided all the nearest-neighbor edges are present. Although still limited by moment conditions, our method of proof is novel in that it avoids proving everywhere-sublinearity of the corrector. This is relevant because we show that, for long-range percolation with exponents between d+ 2 and 2d, the corrector exists but fails to be sublinear everywhere. Similar examples are constructed also for nearest-neighbor, ergodic conductances in d≥ 3 under the conditions complementary to those of the recent work of Bella and Schäffner (Ann Probab 48(1):296–316, 2020). These examples elucidate the limitations of elliptic-regularity techniques that underlie much of the recent progress on these problems.

We study the quenched invariance principle for random conductance models with long range jumps on Zd, where the transition probability from x to y is, on average, comparable to |x − y|−(d+α) with α ∈ (0, 2) but is allowed to be degenerate. Under some moment conditions on the conductance, we prove that the scaling limit of the Markov process is a symmetric α-stable Lévy process on Rd. The well-known corrector method in homogenization theory does not seem to work in this setting. Instead, we utilize probabilistic potential theory for the corresponding jump processes. Two essential ingredients of our proof are the tightness estimate and the Hölder regularity of caloric functions for nonelliptic α-stable-like processes on graphs. Our method is robust enough to apply not only for Zd but also for more general graphs whose scaling limits are nice metric measure spaces.

<p>In this paper, we consider symmetric jump processes of mixed-type on metric measure spaces under general volume doubling condition, and establish stability of two-sided heat kernel estimates and heat kernel upper bounds. We obtain their stable equivalent characterizations in terms of the jumping kernels, variants of cut-off Sobolev inequalities, and the Faber-Krahn inequalities. In particular, we establish stability of heat kernel estimates for <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha">
<mml:semantics>
<mml:mi>α<!-- α --></mml:mi>
<mml:annotation encoding="application/x-tex">\alpha</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula>-stable-like processes even with <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than-or-equal-to 2">
<mml:semantics>
<mml:mrow>
<mml:mi>α<!-- α --></mml:mi>
<mml:mo>≥<!-- ≥ --></mml:mo>
<mml:mn>2</mml:mn>
</mml:mrow>
<mml:annotation encoding="application/x-tex">\alpha \ge 2</mml:annotation>
</mml:semantics>
</mml:math>
</inline-formula> when the underlying spaces have walk dimensions larger than <inline-formula content-type="math/mathml">
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2">
<mml:semantics>
<mml:mn>2</mml:mn>
<mml:annotation encoding="application/x-tex">2</mml:annotation>
</mml:semantics>
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</inline-formula>, which has been one of the major open problems in this area.</p>

In this paper we study homogenization problem for strong Markov processes on ℝd having infinitesimal gener (formula presented) in periodic media, where Π is a nonnegative measure on d that does not charge the origin 0, satisfies (formula presented) and can be singular with respect to the Lebesgue measure on ℝd. Under a proper scaling we show the scaled processes converge weakly to Lévy processes on ℝd. The results are a counterpart of the celebrated work (Asymptotic Analysis for Periodic Structures (1978) North-Holland; Ann. Probab. 13 (1985) 385–396) in the jump-diffusion setting. In particular, we completely characterize the homogenized limiting processes when b(x) is a bounded continuous multivariate 1-periodic ℝd -valued function, k(x,z) is a nonnegative bounded continuous function that is multivariate 1-periodic in both x and z variables and, in spherical coordinate (formula presented) (formula presented) with (formula presented) and e0 being any finite measure on the unit sphere (formula presented) in Rd. Different phenomena occur depending on the values of α; there are five cases: α ∈(0, 1), α = 1, α ∈ (1, 2), α = 2 and α ∈ (2,∞).

Consider random d-regular graphs, i.e., random graphs such that there are exactly d edges from each vertex for some d ≥ 3. We study both the configuration model version of this graph, which has occasional multi-edges and self-loops, as well as the simple version of it, which is a d-regular graph chosen uniformly at random from the collection of all d-regular graphs. In this paper, we discuss mixing times of Glauber dynamics for the Ising model with an external magnetic field on a random d-regular graph, both in the quenched as well as the annealed settings. Let ß be the inverse temperature, ßc be the critical temperature and B be the external magnetic field. Concerning the annealed measure, we show that for ß > ßc there exists Bc(ß) ϵ (0, ∞) such that the model is metastable (i.e., the mixing time is exponential in the graph size n) when ß > ßc and 0 ≤ B < Bc(ß), whereas it exhibits the cut-off phenomenon at c*n logn with a window of order n when ß < ßc or ß > ßc and B > Bc(ß). Interestingly, Bc(ß) coincides with the critical external field of the Ising model on the d-ary tree (namely, above which the model has a unique Gibbs measure). Concerning the quenched measure, we show that there exists Bc(ß) with Bc(ß) ≤ Bc(ß) such that for ß > ßc, the mixing time is at least exponential along some subsequence (nk)k≥1 when 0 ≤ B < Bc(ß), whereas it is less than or equal to Cnlogn when B > Bc(ß). The quenched results also hold for the model conditioned on simplicity, for the annealed results this is unclear.

This paper studies homogenization of symmetric nonlocal Dirichlet forms with stable-like jumping kernels in a one-parameter stationary ergodic environment. Under suitable conditions, we establish results of homogenization and identify the limiting effective Dirichlet forms explicitly. The coefficients in the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded, and the coefficients in the effective Dirichlet forms can also be degenerate.

This paper surveys some recent progress in.

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.

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.

In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel q(t,x,y), which we call the fundamental solution for the time fractional Poisson equation, if the semigroup for L has an integral kernel. We further show that q(t,x,y) can be expressed as a time fractional derivative of the fundamental solution for the homogeneous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution q(t,x,y) through explicit estimates of transition density functions of subordinators.

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 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 consider on-diagonal heat kernel estimates and the laws of the iterated logarithm for a switch-walk-switch random walk on a lamplighter graph under the condition that the random walk on the underlying graph enjoys sub-Gaussian heat kernel estimates.

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.

Based on two-sided heat kernel estimates for a class of symmetric jump processes on metric measure spaces, the laws of the iterated logarithm (LILs) for sample paths, local times and ranges are established. In particular, the LILs are obtained for beta-stable-like processes on alpha-sets with beta &gt; 0.

Given a sequence of resistance forms that converges with respect to the Gromov-Hausdorff-vague topology and satisfies a uniform volume doubling condition, we show the convergence of corresponding Brownian motions and local times. As a corollary of this, we obtain the convergence of time-changed processes. Examples of our main results include scaling limits of Liouville Brownian motion, the Bouchaud trap model and the random conductance model on trees and self-similar fractals. For the latter two models, we show that under some assumptions the limiting process is a FIN diffusion on the relevant space.

The first main result of this paper is that the law of the (rescaled) two-dimensional uniform spanning tree is tight in a space whose elements are measured, rooted real trees continuously embedded into Euclidean space. Various properties of the intrinsic metrics, measures and embeddings of the subsequential limits in this space are obtained, with it being proved in particular that the Hausdorff dimension of any limit in its intrinsic metric is almost surely equal to 8/5. In addition, the tightness result is applied to deduce that the annealed law of the simple random walk on the two-dimensional uniform spanning tree is tight under a suitable rescaling. For the limiting processes, which are diffusions on random real trees embedded into Euclidean space, detailed transition density estimates are derived.

We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on Z among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. exp(1), and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.

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.

Via a Dirichlet form extension theorem and making full use of two-sided heat kernel estimates, we establish quenched invariance principles for random walks in random environments with a boundary. In particular, we prove that the random walk on a supercritical percolation cluster or among random conductances bounded uniformly from below in a half-space, quarter-space, etc., converges when rescaled diffusively to a reflecting Brownian motion, which has been one of the important open problems in this area. We establish a similar result for the random conductance model in a box, which allows us to improve existing asymptotic estimates for the relevant mixing time. Furthermore, in the uniformly elliptic case, we present quenched invariance principles for domains with more general boundaries.

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e. g., to many subordinate Brownian motions, Levy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schrodinger, drift and jump perturbations of such processes.

We consider the biased random walk on a critical Galton-Watson tree conditioned to survive, and confirm that this model with trapping belongs to the same universality class as certain one-dimensional trapping models with slowly-varying tails. Indeed, in each of these two settings, we establish closely-related functional limit theorems involving an extremal process and also demonstrate extremal aging occurs.

We provide conditions that ensure that the maximum of the Gaussian free field on a sequence of graphs fluctuates at the same order as the field at the point of maximal standard deviation; under these conditions, the expectation of the maximum is of the same order as the maximal standard deviation. In particular, on a sequence of such graphs the recentered maximum is not tight, similarly to the situation in Z but in contrast with the situation in Z(2). We show that our conditions cover a large class of "fractal" graphs.

We consider a certain class of nonsymmetric Markov chains and obtain heat kernel bounds and parabolic Harnack inequalities. Using the heat kernel estimates, we establish a sufficient condition for the family of Markov chains to converge to nonsymmetric diffusions. As an application, we approximate nonsymmetric diffusions in divergence form with bounded coefficients by nonsymmetric Markov chains. This extends the results by Stroock and Zheng to the nonsymmetric divergence forms. © 2011 Wiley Periodicals, Inc.

In this paper we give general criteria on tightness and weak convergence of discrete Markov chains to symmetric jump processes on metric measure spaces under mild conditions. As an application, we investigate discrete approximation for a large class of symmetric jump processes. We also discuss some application of our results to the scaling limit of random walk in random conductance.

We prove the equivalence of parabolic Harnack inequalities and sub-Gaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form.

We establish conditions on sequences of graphs which ensure that the mixing times of the random walks on the graphs in the sequence converge. The main assumption is that the graphs, associated measures and heat kernels converge in a suitable Gromov-Hausdorff sense. With this result we are able to establish the convergence of the mixing times on the largest component of the Erdros-Renyi random graph in the critical window, sharpening previous results for this random graph model. Our results also enable us to establish convergence in a number of other examples, such as finitely ramified fractal graphs, Galton-Watson trees and the range of a high-dimensional random walk.

In this paper, we study sharp heat kernel estimates for a large class of symmetric jump-type processes in R(d) for all t &gt; 0. A prototype of the processes under consideration are symmetric jump processes on R(d) with jumping intensity
1/Phi(vertical bar x - y vertical bar)integral([alpha 1, alpha 2])c(alpha, x, y)/vertical bar x - y vertical bar(d+alpha) nu(d alpha),
where nu is a probability measure on [alpha(1), alpha(2)] subset of (0, 2), Phi is an increasing function on [0, infinity) with c(1)e(c2r beta) &lt;= Phi(r) &lt;= c(3)e(c4r beta) with beta is an element of (0, infinity), and c(alpha, x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in (x, y). They include, in particular, mixed relativistic symmetric stable processes on R(d) with different masses. We also establish the parabolic Harnack principle.

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.

We consider symmetric processes of pure jump type. We prove local estimates on the probability of exiting balls, the Holder continuity of harmonic functions and of heat kernels, and convergence of a sequence of such processes.

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.

We investigate the relationships between the parabolic Harnack inequality, heat kernel estimates, some geometric conditions, and some analytic conditions for random walks with long range jumps. Unlike the case of diffusion processes, the parabolic Harnack inequality does not, in general, imply the corresponding heat kernel estimates.

We establish general estimates for simple random walk on an arbitrary infinite random graph, assuming suitable bounds on volume and effective resistance for the graph. These are generalizations of the results in Barlow et al. (Commun. Math. Phys. 278:385-431, 2008, Sects. 1, 2) and in particular imply the spectral dimension of the random graph. We will also give an application of the results to random walk on a long-range percolation cluster.

We establish a variety of properties of the discrete time simple random walk on a Galton-Watson tree conditioned to survive when the offspring distribution, Z say, is in the domain of attraction of a stable law with index alpha is an element of (1,2]. In particular, we are able to prove a quenched version of the result that the spectral dimension of the random walk is 2 alpha/(2 alpha-1). Furthermore, we demonstrate that when alpha is an element of (1,2) there are logarithmic fluctuations in the quenched transition density of the simple random walk, which contrasts with the log-logarithmic fluctuations seen when alpha=2. In the course of our arguments, we obtain tail bounds for the distribution of the nth generation size of a Galton-Watson branching process with offspring distribution Z conditioned to survive, as well as tail bounds for the distribution of the total number of individuals born up to the nth generation, that are uniform in n.

We consider simple random walk on the incipient infinite cluster for the spread-out model of oriented percolation on Z(d) x Z(+). In dimensions d &gt; 6, we obtain bounds on exit times, transition probabilities, and the range of the random walk, which establish that the spectral dimension of the incipient infinite cluster is 4/3, and thereby prove a version of the Alexander-Orbach conjecture in this setting. The proof divides into two parts. One part establishes general estimates for simple random walk on an arbitrary infinite random graph, given suitable bounds on volume and effective resistance for the random graph. A second part then provides these bounds on volume and effective resistance for the incipient infinite cluster in dimensions d &gt; 6, by extending results about critical oriented percolation obtained previously via the lace expansion.

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 e(-c0(x,y)|x-y|) integral(alpha 2)(alpha 1)c(alpha,x,y)/|x-y|(d+alpha) nu(d alpha)
where nu 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 c(0)(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) &gt;= gamma(1) &gt; 0 or gamma(1) = gamma(2) = 0. This example contains mixed symmetric stable processes on R-n as well as mixed relativistic symmetric stable processes on R-n. We establish parabolic Harnack principle and derive sharp two-sided heat kernel estimate for such jump-type processes.

We consider symmetric Markov chains on Z(d) where we do not assume that the conductance between two points must be zero if the points are far apart. Under a uniform second moment condition on the conductances, we obtain upper bounds on the transition probabilities, estimates for exit time probabilities, and certain lower bounds on the transition probabilities. We show that a uniform Harnack inequality holds if an additional assumption is made, but that without this assumption such an inequality need not hold. We establish a central limit theorem giving conditions for a sequence of normalized symmetric Markov chains to converge to a diffusion on R-d corresponding to an elliptic operator in divergence form.

We study the off-diagonal estimates for transition densities of diffusions and jump processes in a setting when they depend essentially only on the time and distance. We state and prove the dichotomy for the tail of the transition density.

We consider a trace theorem for self-similar Dirichlet forms on self-similar sets to self-similar subsets. In particular, we characterize the trace of the domains of Dirichlet forms on Sierpinski gaskets and Sierpinski carpets to their boundaries, where the boundaries are represented by triangles and squares that confine the gaskets and the carpets. As an application, we construct diffusion processes on a collection of fractals called fractal fields. These processes behave as an appropriate fractal diffusion within each fractal component of the field. (C) 2006 Elsevier Inc. All rights reserved.

We use an elementary method to obtain Nash-type inequalities for non-local Dirichlet forms on d-sets. We obtain two-sided estimates for the corresponding heat kernels if the walk dimensions of heat kernels are less than two; these estimates are obtained by combining probabilistic and analytic methods. Our arguments partly simplify those used in Chen and Kumagi (Heat kernel estimates for stable-like processes on d-sets.

Let (X, d, mu) 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 beta &gt;= 2 to hold. We show that this parabolic Harnack inequality is stable under rough isometrics. 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.

Sub-Gaussian estimates for random walks are typical of fractal graphs. We characterize them in the strongly recurrent case, in terms of resistance estimates only, without assuming elliptic Harnack inequalities. (c) 2005 Wiley Periodicals, Inc.

We give a sufficient condition to construct non-trivial mu-symmetric diffusion processes on a locally compact separable metric measure space (M, rho, mu). These processes are associated with local regular Dirichlet forms which are obtained as continuous parts of Gamma-limits for approximating non-local Dirichlet forms. For various fractals, we can use existing estimates to verify our assumptions. This shows that our general method of constructing diffusions can be applied to these fractals.

We summarize recent work on heat kernel estimates and parabolic Harnack inequalities for graphs, where the time scale is the beta-th power of the space scale for some,8 greater than or equal to 2. We then discuss self-adjoint operators induced by resistance forms. Using a resistance metric, we give a simple condition for detailed heat kernel estimates and parabolic Harnack inequalities. As an application, we show that on trees a detailed two-sided heat kernel estimate is equivalent to some volume growth condition.

We examine a class of fractal graphs which arise from a subclass of finitely ramified fractals. The two-sided heat kernel estimates for these graphs are obtained in terms of an effective resistance metric and they are best possible up to constants. If the graph has symmetry, these estimates can be expressed as the usual Gaussian or sub-Gaussian estimates. However, without symmetry, the off-diagonal terms show different decay in different directions. We also discuss the stability of the sub-Gaussian heat kernel estimates under rough isometrics.

We summarize recent work on function spaces and stochastic processes on fractals. We discuss relations between various non-local Dirichlet forms on fractals whose domains are Besov spaces. The corresponding stochastic processes are jump-type processes. Results on heat kernel estimates for the processes are introduced. We will also discuss how jump processes and diffusion processes are related by observing their function spaces.

A fractal field is a collection of fractals with, in general, different Hausdorff dimensions, embedded in R-2. We will construct diffusion processes on such fields which behave as Brownian motion in R-2 outside the fractals and as the appropriate fractal diffusion within each fractal component of the field. We will discuss the properties of the diffusion process in the case where the fractal components tile R-2. By working in a suitable shortest path metric we will establish heat kernel bounds and large deviation estimates which determine the trajectories followed by the diffusion over short times.

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. (C) 2003 Elsevier B.V. All rights reserved.

We summarize recent work on non-local Dirichlet forms on fractals whose corresponding processes are stable-like jump processes. Especially, we introduce three natural non-local Dirichlet forms on d-sets and prove that these forms are equivalent.

We discuss the behavior of the dynamic dimension exponents for families of fractals based on the Sierpinski gasket and carpet. As the length scale factor for the family tends to infinity, the lattice approximations to the fractals look more like the tetrahedral or cubic lattice in Euclidean space and the fractal dimension converges to that of the embedding space. However, in the Sierpinski gasket case, the spectral dimension converges to two for all dimensions. In two dimensions, we prove a conjecture made in the physics literature concerning the rate of convergence. On the other hand, for natural families of Sierpinski carpets, the spectral dimension converges to the dimension of the embedding Euclidean space. In general, we demonstrate that for both cases of finitely and infinitely ramified fractals, a variety of asymptotic values for the spectral dimension can be achieved.

Let S-n be a random walk in Z(d) and let R-n be the range of Sn. We prove an almost sure invariance principle for R-n when d = 3 and a law of the iterated logarithm for R-n when d = 2.

We consider a class of random recursive Sierpinski gaskets and examine the short-time asymptotics of the on-diagonal transition density for a natural Brownian motion. In contrast to the case of divergence form operators in R-n or regular fractals we show that there are unbounded fluctuations in the leading order term. Using the resolvent density we are able to explicitly describe the fluctuations in time at typical points in the fractal and, by considering the supremum and infimum of the on-diagonal transition density over all points in the fractal, we can also describe the fluctuations in space. (C) 2001 Elsevier Science B.V. All rights reserved.

We study the asymptotics as t→0 of the transition density of a class of μ-symmetric diffusions in the case when the measure μ has a multi-fractal structure. These diffusions include singular time changes of Brownian motion on the unit cube. © 2001 Applied Probability Trust.

We consider homogeneous random Sierpinski carpets, a class of infinitely ramified random fractals which have spatial symmetry but which do not have exact self-similarity. For a fixed environment we construct "natural" diffusion processes on the fractal and obtain upper and lower estimates of the transition density for the process that are up to constants best possible. By considering the random case, when the environment is stationary and ergodic, we deduce estimates of Aronson type.

We study large deviations for Brownian motion on the Sierpinski gasket in the short time limit. Because of the subtle oscillation of hitting times of the process, no large deviation principle can hold. In fact, our result shows that there is an infinity of different large deviation principles for different subsequences, with different (good) rate functions. Thus, instead of taking the time scaling epsilon --&gt; 0, we prove that the large deviations hold for E-n(z) = (2/5)(n)z as n --&gt; infinity, using one parameter family of rate functions I-z (z is an element of [2/5, 1)). As a corollary, we obtain Strassen-type laws of the iterated logarithm. (C) 2000 Elsevier Science B.V. All rights reserved. MSG. 60F10; 60J60; 60J80.

For a closed connected set F in R-n, assume that there is a local regular Dirichlet form (a symmetric diffusion process) on F whose domain is included in a Lipschitz space or a Besov space on F. Under some condition for the order of the space and the Newtonian 1-capacity of F; we prove that there exists a symmetric diffusion process on R-n which moves like the process on F and like Brownian motion on R-n outside F. As an application, we will show that when F is a nested fractal or a Sicrpinski carpet whose Hausdorff dimension is greater than n - 2, we can construct Brownian motion penetrating the fractal. For the proof we apply the technique developed in the theory of Besov spaces. (C) 2000 Academic Press.

We consider the class of diffusions on fractals first constructed in [12] on the Sierpinski and abe gaskets. We give an alternative construction of the diffusion process using Dirichlet forms and extend the class of fractals considered to some nested fractals. We use the Dirichlet form to deduce Nash inequalities which give upper bounds on the short and long time behaviour of the transition density of the diffusion process. For short times, even though the diffusion lives mainly on a lower dimensional subset of the fractal, the heat flows slowly. For the long time scales the diffusion has a homogenization property in that rescalings converge to the Brownian motion on the fractal.

We study so called Varadhan type short time asymptotic estimates of heat kernels and Shilder type large deviation for Brownian motion on some affine nested fractals introduced in [7]. As a corollary to our approach, we obtain sharper estimates of heat kernels for a class of one dimensional diffusion processes studied in [8].

We study tile Bernoulli bond percolation problem on planar pre-Sierpinski carpets, a class af infinitely ramified pre-fractals which has self-similarity but which is not periodic. Let p be the probability for each bond to be open. We show that for large p between 1/2 and 1, the probability that there is an open infinity-cluster from the origin is positive. Under some assumption, we also show that the critical probability is unique.

We study the homogenization problem on nested fractals. Let X(t) be the continuous time Markov chain on the pre-nested fractal given by putting i.i.d. random resistors on each cell, It is proved that under some conditions, alpha(-n)X(tnEt) converges in law to a constant time change of the Brownian motion on the fractal as n --&gt; infinity, where alpha is the contraction rate and t(E) is a time scale constant. As the Brownian motion on fractals is not a semi-martingale, we need a different approach from the well-developed martingale method.

A class of affine nested fractals is introduced which have different scale factors for different similitudes but still have the symmetry assumptions of nested fractals. For these fractals estimates on the transition density for the Brownian motion are obtained using the associated Dirichlet form. An upper bound for the diagonal can be found using a Nash-type inequality, then probabilistic techniques are used to obtain the off-diagonal bound. The approach differs from previous treatments as it uses only the Dirichlet form and no estimates on the resolvent. The bounds obtained are expressed in terms of an intrinsic metric on the fractal.

We obtain upper and lower bounds for the transition densities of Brownian motion on nested fractals. Compared with the estimate on the Sierpinski gasket, the results require the introduction of a new exponent, d(J), related to the ''shortest path metric'' and ''chemical exponent'' on nested fractals. Further, Holder order of the resolvent densities, sample paths and local times are obtained. The results are obtained using the theory of multi-type branching processes.