<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>
</mml:math>
</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,∞).

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.

For each n let Y-t((n)) be a continuous time symmetric Markov chain with state space n(-1)Z(d). Conditions in terms of the conductances are given for the convergence of the Y-t((n)) to a symmetric Markov process Y-t on R-d. We have weak convergence of {Y-t((n)) : t &lt;= t(0)} for every t(0) and every starting point. The limit process Y has a continuous part and may also have jumps.

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 consider the following type of non-local (pseudo-differential) operators L on R(d):
Lu(x) = 1/2 d Sigma i,j=1 partial derivative/partial derivative x(i) (aij(x)partial derivative u(x)/partial derivative x(j))
+lim3 down arrow 0 integral{is an element of R(d):/y-x vertical bar &gt;}
where A(x) = (aij(x))1 &lt;= i j &lt;= d is a measurable dxd matrix-valued function on R(d) that is uniformly elliptic and bounded and J is a symmetric measurable non-trivial non-negative kernel on d x Rd satisfying certain conditions. Corresponding to is a symmetric strong Markov process X on i d that has both the diffusion component and pure jump component. We establish a priori Holder estimate for bounded parabolic functions of L and parabolic Harnack principle for positive parabolic functions of C. Moreover, two-sided sharp heat kernel estimates are derived for such operator and jump-diffusion X. In particular, our results apply to the mixture of symmetric diffusion of uniformly elliptic divergence form operator and mixed stable-like processes on Rd. To establish these results, we employ methods from both probability theory and analysis.

We prove that, up to scalar multiples, there exists only one local regular Dirichlet form on a generalized Sierpinski carpet that is invariant with respect to the local symmetries of the carpet. Consequently, for each such fractal the law of Brownian motion is uniquely determined and the Laplacian is well defined.

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.

It is well-known that there is a deep interplay between analysis and probability theory. For example, for a Markovian infinitesimal generator L, the transition density function p(t, x, y) of the Markov process associated with L (if it exists) is the fundamental solution (or heat kernel) of L. A fundamental problem in analysis and in probability theory is to obtain sharp estimates of p(t, x, y). In this paper, we consider a class of non-local (integro-differential) operators L on R(d) of the form
Lu(x) = lim(epsilon down arrow 0) integral({y is an element of Rd :vertical bar y-x vertical bar &gt;epsilon}) (u(y) - u(x)) J(x ,y)dy,
where J (x, y) = c(x, y)/vertical bar x-y vertical bar(d+alpha) 1({vertical bar x-y vertical bar &lt;= k}) for constant k &gt; 0 and a measurable symmetric function c(x, y) that is bounded between two positive constants. Associated with such a non-local operator L is an R(d)-valued symmetric jump process of finite range with jumping kernel J (x, y). We establish sharp two-sided heat kernel estimate and derive parabolic Harnack principle for them. Along the way, some new heat kernel estimates are obtained for more general finite range jump processes that were studied in (Barlow et al. in Trans Am Math Soc, 2008). One of our key tools is a new form of weighted Poincare inequality of fractional order, which corresponds to the one established by Jerison in (Duke Math J 53( 2): 503-523, 1986) for differential operators. Using Meyer&apos;s construction of adding new jumps, we also obtain various a priori estimates such as Holder continuity estimates for parabolic functions of jump processes (not necessarily of finite range) where only a very mild integrability condition is assumed for large jumps. To establish these results, we employ methods from both probability theory and analysis extensively.

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 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.

We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated non-trivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.

We conducted research on Markov processes using stochastic analysis methods for cases of various state spaces, such as Euclid space, Riemannian manifold, and infinite dimensional space such as Wiener space and path space. In the case of the one-dimensional diffusion process, the spectrum of the Kolmogorov diffusion process was determined in the framework of supersymmetry. Also, in the case of Kummer process, spectra were determined in Zygmundt space or Orlicz space.
Furthermore, we characterized the ultracontractivity using the asymmetric Dirichlet form and applied to the asymptotic behavior of the fundamental solution in the case of compact Riemannian manifolds. We also constructed a non-symmetric diffusion process on the Wiener space as a typical example of an infinite dimensional space.

We gave basic progresses of the Dirichlet form theory, which is complementary to the theory of stochastic differential equations in the study of stochastic analysis. Primary results are based on a theme about the relation between the probability theory and the spatial structures on the basis of local structures determined by Dirichlet forms. Theoretical development was carried out by taking into consideration that it is useful for the situation in which the concept of usual differentiation is not defined like fractal sets, as well as for smooth spaces.

We studied the stationary measure of the multi-component coupled KPZ(Kardar-Parisi-Zhang) equation and showed its global well-posedness by applying the paracontrolled calculus. Moreover, we derived such singular stochastic partial differential equation from a particle system with several conservation laws. We also studied the sharp interface limit for mass conserving Allen-Cahn equation with stochastic fluctuation term, the derivation of motion by mean curvature from particle systems, the motion by mean curvature perturbed by a direction-dependent noise, the conservation law with a multiplicative noise term that is a first order stochastic partial differential equation, to establish an affirmative mathematical base to the adaptive dynamics employed in the mathematical genetics, stochastic dynamics related to the random matrices, Markov chains in random environments, and others.

干渉ブラウン運動のtagged粒子の不変原理に関して、従来より適切かつ精密な定式化を行い、Kipnis-Varadhanの不変原理の対応物が成立することを証明した。
従来は、Palm測度に対して、無限個の環境粒子系にたいして、tagged粒子のブラウン運動への収束が「測度収束」の意味で成立するという形で、主張が定式化されてきた。これは、80年代前半に行われたGuo-Papanicolauの研究以来の伝統的な定式化ではあるが、tagged粒子から見た他の無限個の粒子家の挙動を記述する確率微分方程式に対する解析となるため、すぐには元来の問題との関係が分かりにくい。
それに対して、今回は、粒子のラベルを考慮し構造に入れることにより、元々の平行移動不変な平衡分布に関して、可逆な確率力学を記述する確率微分方程式を考え、その個々の粒子が、ブラウン運動に初期条件に関して測度収束するという、より自然なわかりやすい定式化となった。
また、応用として、1次元の無限粒子系が互いに衝突しない（順序を入れ替えない）という性質をもつとき、極限が常に退化するという結果を得た。この事実自体は、当然の結果だが、ポイントはこれが常に成立するということを、幾何的な結果から平易に示したという一般性を備えている点である。
ランダム行列に関係する干渉ブラウン運動は、対数関数（2次元クーロンポテンシャル）で相互作用する確率力学である。また、干渉ブラウン運動を含む広い範囲の無限次元確率微分方程式に対して一般論を構築し、更に発展させている最中だが、これに関する最近の研究結果をまとめてreview論文として報告した。

We have studied the relation between geometric and algebraic structures and analysis on fractals such as self-similar sets and invariant sets of various dynamical systems. For example, we have considered time changes of the Brownian motions on generalized Sierpinski carpets with respect to singular measures. We have established the notion of weakly exponential weakly exponential measures. If a measure is weakly exponential, then we have shown tcertain type of Poincare inequality, the existence of jointly continuous heat kernel and an asymptotic heat kernel estimate using the protodistance associated with the weakly exponential measure.

Kardar-Parisi-Zhang (KPZ) equation is a nonlinear stochastic partial differential equation which describes an evolution of growing interfaces with fluctuation. Mathematically, this equation involves a divergent term so that it is ill-posed, but Hairer, a Fields medalist, introduced a method of renormalization which removes the divergent term and gave a mathematical meaning to it. In this research project, we have specified the stationary measures of KPZ equation and multicomponent coupled KPZ equation, and shown the global solvability in time. Moreover, we have found new determinantal structures in related interacting infinite particle systems.

We studied dynamics on random media and their scaling limits in a systematical way. Our major achievements are as follows: i) We proved convergence of Markov chains on random conductances with boundaries under a wide framework. The convergence is with probability one with respect to the randomness of the media. ii) We proved sub-sequential convergence of the random walk on 2-dimensional uniform spanning tree, which is a random media whose scaling limit is conformal invariant, and gave detailed estimates of the heat kernel for the limiting process. iii) We proved stability of heat kernel estimates for symmetric jump processes on metric measure spaces. This was one of the major open problems in the area for more than 10 years.

Relationships among harmonic functions, solutions to the heat equation, the Green and heat kernels and their defining domains, the influence of the space complexity to the boundary behavior were studied. They were applied to various fields such as non-smooth Euclidean domains, manifolds, varifolds, networks and fractals. In particular, new results were obtained in Harnack principle with exceptional sets, estimates of the principal frequency in terms of capacitary width, sufficient conditions for the global boundary Harnack principle based on the capacitary width of sublevel sets of the Green function, conditions for the parabolic boundary Harnack principle (Intrinsic Ultracontractivity), the critical exponent of a graph domain enjoying the global boundary Harnack principle, and the 0-1 law of the capacity density at infinity and so on.

We study mathematical framework for Quantum spin system. Physics on topological Insulator and topologically protected surface/edge state is formulated in K-theory. By using the non-commutative geometry, we generalized it to disordered systems. We also develop discrete surface theory to study the relation of microscopic structural data and macroscopic properties and apply it to carbon networks.

To study analysis on metric-measure spaces, we have investigated important properties of metrics and measures on topological space such as bi-Lipshitz equivalence, volume doubling property, quasisymmetry and Ahlfors regularity form the view point of partitions and associated gauge functions. In particular, we have shown that bi-Lipshitz property between a measure and a metric as gauge functions is equivalent to the Ahlfors regularity between a measure and a metric.

We establish a general theory for solving infinite-dimensional stochastic differential equations (ISDE) with symmetry typically appearing in statistical mechanics. In particular, we prove the pathwise uniqueness and the existence of the strong solution under a very general framework. This method is novel, and regards the tail sigma field of the configuration space as a boundary of the ISDE. Furthermore, if the tail sigma field is trivial, then a strong solution exists. If the set of probability-one events is unique, then the pathwise uniqueness of solution holds.
The method is effective for the ISDE with logarithmic interaction potentials, which appear in random matrix theory.

We studied invariant measures of KPZ equation which describes a growth of interfaces with fluctuations. This stochastic partial differential equation involves a diverging term which makes difficult to give a mathematical meaning to it. We discussed the non-equilibrium fluctuation problem for the dynamics of two-dimensional Young diagrams and derived a stochastic partial differential equation under a scaling limit. The method of the hydrodynamic limit is applied to a system of creatures with an effect of self-organized aggregation and established a link between macroscopic and microscopic descriptions. We proved unique existence of strong solutions of infinite dimensional stochastic differential equations and rigidity for Airy point process and Ginibre point process, which appear in the theory of dynamic random matrices. Furthermore, we studied percolations, nonlinear diffusion equations, stochastic partial differential equations with stable noises and others.

We investigated recently developed methods on the de Giorgi-Nash-Moser theory and a priori estimates of caloric functions for non-local operators and jump-type stochastic processes. We gave some sufficient conditions for the boundary Harnack inequalities to hold for jump-type processes (non-local operators) on general metric measure spaces, and applied the results to various concrete examples. We also worked on heat kernel estimates for fractional time derivative heat equations.

In this project, we have studied internal structure of fractal sets from the broad viewpoint of analysis, geometry and algebra. In particular, we have considered stochastic processes and potential theory on fractals, which are thought of as models of objects in the nature. For instance, we have obtained conditions to ensure the existence of scaling relation of space-time on the heat conductance on fractals. Moreover, we have studied equilibrium state of heat on spaces with infinitely many holes and obtained a condition for the equilibrium state to be smooth.

We studied Markov processes by using the stochastic analysis. We are mainly interested in the behavior of the semigroup associated with a Markov process. First we considered diffusion processes on a Riemannian manifold. We give a generator as a sum of the Laplace-Beltrami operator and a vector field. We discussed the uniqueness of the semigroup associated with the generator. The condition was given in terms of the vector field.
Next we consider the conditions for which the semigroup reserves a convex set. We consider this problem in the framework of general Banach space. We give some necessary and sufficient conditions in terms of generator. In Hilbert space setting, we formulate this problem by using Dirichlet forms.
In addition, we considered the rate of convergence of the semigroup under the condition of logarithmic Sobolev inequality. We also discussed the dual ultracontractive property of the semigroup.

We analysed Markov chains on disordered media and their scaling limits in a unified manner by using probabilistic and analytic methods. From the viewpoint of constructing general theory, we established the following results; i) heat kernel estimates for non-symmetric Markov chains satisfying some cycle condition and analyzing their scaling limits, ii) equivalence of the sub-Gaussian heat kernel estimates and generalized Parabolic Harnack inequalities on symmetric diffusions for general metric measure spaces, iii) convergence of jump-type processes for general metric measure spaces. From the viewpoint of concrete examples, we established the following results; i) asymptotic behavior and scaling limits of biased random walks on critical percolation clusters (conditioned to survive forever) on trees, ii) convergence of scaled mixing times on Markov chains on random finite graphs.

On non-compact Riemannian manifolds and infinite graphs, therelationship among global geometric structure, geometric inequalities and long time heatkernel behavior is well investigated by many researchers. In this research, we considerconnected sums and we obtain a precise relationship between a structure of bottlenecknessand isoperimetric inequality, Poincare inequality and long time heat kernel behavior.

Problems on important functions such as (super, sub)harmonic functions appearing in analysis, geometry and probability are referred to as potential problems. We have investigated potential problems from various view points and unveiled deep properties of important functions in connections with nonsmooth domains, fractals, manifolds and functions spaces.

Aim of this research proposal is to develop new methods to study geometric objects with singularities, or discrete spaces, which are not accessible by traditional differential geometrical technics. Our idea is to apply probability theory to those geometric objects. Some results are obtained and published from international journals.

本年度行った研究により得られた成果は、以下の通りである。
1. 熊谷は、Croydon氏(Warwick大)、Hambly氏(Oxford大)と共同で、与えられた有限グラフの列に対して、その上の対称マルコフ連鎖の混合時間が収束するための十分条件を与えた。この十分条件は、グラフのグロモフ-ハウスドルフ収束に、熱核の収束の概念を加えた新たな収束概念によって表現することができる。これにより、例えばErdos-Renyiのランダムグラフの臨界確率近傍での最大連結成分上のランダムウォークの混合時間に関する収束定理を証明することができる。この結果は3人の共著論文にまとめ、現在雑誌に投稿中である。
2. 熊谷は、昨年度から継続しているChen氏(Washington大)、Kim氏(Seoul大)との共同研究を論文にまとめ、雑誌に投稿した。その内容は、D次元正方格子上の対称マルコフ連鎖が飛躍型確率過程に収束するための十分条件を、ディリクレ形式の手法を用いて導出するものである。特に、ランダムコンダクタンスモデルへの応用として、2点間のコンダクタンスに長距離相関がある場合に、対応するランダムなマルコフ連鎖がD次元安定過程に収束するための十分条件を与える部分について、昨年度やや曖昧であった条件を明確にした。現在、査読結果を元に改訂版を作成中である。
3. 長田は白井朋之氏(九州大)と共に、Ginibre random point fieldのPalm測度が、もとの測度に対して特異になること、および(任意の相異なる)2点で条件づけたPalm測度は互いに絶対連続になることを示した。

We have studied various aspects of mathematical foundation of fractals. In particular, we have constructed intrinsic geometrical structure associated with analytical objects like stochastic processes on fractals. Furthermore, based on these geometric structure, we have investigated the asymptotic properties of stochastic processes on fractals and/or boundary behaviors of functions on domains with fractal boundary.

To account of not only the members but also of the researchers on probability theory in Japan, we held several conferences every year, including two international ones, and made connections on study and joint research. Moreover we invited several foreign researchers and the members of this grant attended conferences outside Japan and visited foreign researchers. Through these activities we contributed some progress on theory of stochastic analysis and on applications to study on questions with origins from statistical physics, study of differential equations, spectra of manifolds, mathematical finance and so on.

We have studied the interaction between geometric structures and invariants of 3-manifolds in order to get deeper understanding for their topology, and obtained several new results on global behavior of geometric invariants. In particular, the reporter attained a few progress on comparison of dynamical invariants of mapping classes of a surface and simplicial volumes of their mapping tori. Also, we have organized an international conference to review our activities in the last year, and pointed out many issues that suggest further direction.

We have studied large scale interacting systems related to the interface model and the theory of random matrices, based on the stochastic analysis and the theory of nonlinear partial differential equations. In particular, concerning the interface model on a wall or with a pinning effect, we have made a precise analysis and determined the scaling limit when the corresponding large deviation rate functional admits plural minimizers, and moreover, established the hydrodynamic limit for an evolutional model of two dimensional Young diagrams. As a dynamic model in the theory of random matrices, we have investigated the system of Ginibre interacting Brownian particles and found an outstanding property, that is, the sub-diffusive behavior of a tagged particle in this system.

本年度行った研究により得られた成果は以下の通りである。二年間という短期間で多くのモデルの解析を行い、当該研究を大きく推進できたと考える。
1.Diamond latticeと呼ばれるグラフ上の臨界確率におけるパーコレーションクラスターを考え、そのスケール極限が連結なクラスターを持つという条件のもとで、クラスター上の拡散過程を構成し、その熱核に関する詳しい評価を得た。特に、スペクトル次元に関する有名なAlexander-Orbach予想はこのモデルでは成り立たないことを証明した。(なお、このモデルでは臨界確率において連結なクラスターが存在する確率が正になる。)このモデルは2次元モデルの簡易化としていくつもの文献に取り扱われているが、臨界確率におけるダイナミックスの解析は本研究が初めてであると考える。現在この結果を、Oxford大学のHambly氏との共著論文として執筆中である。
2.臨界確率における分枝過程の家系図に対応する樹木を考え、そのincipient infinite cluster上のランダムウオークの熱核を研究した。昨年までのこの研究を発展させ、本年は子孫分布が有限な二次モーメントを持たない場合にこの問題を取り扱い、二次モーメントを持つ場合との間で熱核挙動に顕著な違いがあることを解明した。具体的には、体積増大度が大きく異なるため、熱核の対角評価のオーダーが二次モーメントを持つ場合と異なり、特に上述したAlexander-Orbach予想が成り立たないことが示された。(なお、二次モーメントを持つ場合はこの予想が正しいことを昨年我々が証明している。)

We have obtained the following two results on relations between analysis and geometry on fractals. The first result concerns a heat kernel associated with a time change of a process associated with a self-similar Dirichlet form on a self-similar set. We show an sufficient and necessary condition on the existence of a distance under which the heat kernel satisfy the on-diagonal Li-Yau type sub-Gaussian estimate. The second result is on the measurable Riemannian Geometry on the Sierpinski gasket. We show that the geodesic distance coincides with the shortest path metric on the harrmonic Sierpinski gasket and establilsh Li-Yau type Gaussian estimate of the heat kernel under the geodesic metric.

木上は、Sierpinski gasket上のstandard Dirichlet formに関するKusuokaの結果をもとに、Sierpinski gasket上のMeasurable Riemannian stturcutreについて研究を行った。その過程で、Kusuokaによるgradient operatorとKigamiによるhamonic Sierpinski gasket上のgradient operatorが本質的に同じものであること、Kusuoka measureがユークリッドの距離に関してvolume doublingであることを示した。
また、harmonic Sierpinski gasket上のshortest path metricがmeasurable Riemannian structureから決まる測地線距離で有ることを示した。さらにこれらの結果を組み合わせて、Kusuoka measureに対応する熱核がLi-Yau型のGaussian estimateを満たすことを証明した。
日野は、(無限分岐的なものも含む)自己相似集合上の自己相似的なDirichlet formのエネルギー測度について研究した。
エネルギー測度が自己相似測度に対して特異的になるための条件を示し、いくつかの興味ある場合に適用した。

本年度行った研究により得られた成果は以下の通りである。
1.D次元正方格子上の対称マルコフ連鎖で、2点間のコンダクタンスが、2点間の距離が離れても0にならないようなものを考える。コンダクタンスに関する一様な二乗可積分性条件の下、熱核の評価、exit timeの評価を行った。さらに、コンダクタンスに関する仮定を加えることにより、一様なハルナック不等式が成り立つことを証明し、また、スケール変換したマルコフ過程にある種の条件を課することにより、このマルコフ過程がユークリッド空間上のdivergence formで決まる拡散過程に収束することを証明した。この結果は、Bass氏との共著論文にまとめ、現在雑誌に投稿中である。
2.D次元正方格子上の飛躍型確率過程が放物型ハルナック不等式を満たすための必要十分条件、ある種の多項式的な減衰をする熱核評価を持つための必要十分条件について、Barlow氏、Bass氏と共同で研究を行った。これらの必要十分条件について部分的な結果を得ることができ、今後も継続して共同研究を行うこととなった。
3.フラクタルを典型例とする自己相似な空間上に、自己相似な局所正則ディリクレ形式が与えられ、その定義域となる関数空間がベソフ空間であるとする。このとき、このディリクレ形式の自己相似部分集合へのトレースをとったとき、対応する定義域を決定するという問題を取り扱った。これは、関数空間論的にはベソフ空間のトレース理論の一般化に相当する問題である。もとのディリクレ形式がハルナック不等式などいくつかの条件を満たすとき、トレースで決まる関数空間が再びベソフ空間となることを示し、その特徴づけを行った。この結果は、日野正訓氏との共著論文にまとめ、現在雑誌に投稿中である。

(1)As for stochastic oscillatory integrals (SOI in short), which are Fourie-Laplace transforms on the path space, studies on exact expression, asymptotic behavior, and application to nonlinear partial differential equation were made. In the case of phase function of quadratic Wiener functional, a Levy-Ito type exponential expression was established with the associated Hilbert-Schmidt operator. Moreover, in the case of phase function of stochastic line integral with polynomial coefficients, a concrete asymptotic behavior was shown. A door to the stationary phase method on the path space was opened by showing the concentration around stationary points of the asymptotic behavior in the case of Gaussian amplitude function. Finally, we found out a bijective relation between the tau function of the KdV hierarchy and SOI' s associated with Ornstein-Uhlenbeck processes, which is a completely new application of stochastic analysis to nonlinear PDE theory. (2)We made studies on concrete functionals on path spaces. We computed the distribution of the exponential functional obtained as time-integral of geometric Brownian motion, which plays a key role in studies of mathematical finance, diffusions in random environments, and Brownian motion on hyperbolic space, and established the recursive formula for its density function. Moreover, the absolute continuity of the distribution of the Wishart process was computed, and a probabilistic proof of Selberg trace formulas for Laplacian on forms on hyperbolic spaces was given. Furthermore, the trace formula on p-adic upper half plane was studied via semi-stable processes. (3)An easily-applicable criterion for heat kernel on general space to have a sub-Gaussian estimate was achieved. Diffusion processes on complexity were constructed, and a detailed estimation of associated heat kernel was established.

The main research area of the head investigator is diffusions in infinite dimension. At the same time, a research of diffusions on a Riemannian manifold is accomplished because geometric point of view is important in our research. We gave a probabilistic proof of the Littlewood-Paley inequality, the L^P norm equivalence between the gradient and the square of the generator, essential self-adjointness of a Schrodinger operator, and the spectral gap. The essential matter we used is the intertwining property between gradient and the generator. The use of Functional Analysis is crucial because it is irrelevant of the geometry of the space. In our general framework, we assume the logarithmic Sobolev inequality and the exponential integrability of the remaining term of the intertwining property. We also discussed the similar problem in a setting of Riemannian manifold with convex boundary.
Further we considered a Schrodinger operator on the Wiener space of the form L+V,L beging the Ornstein-Uhlenbeck operator. We gave a characterization of the generator domain and proved the essential self-adjointness and the spectral gap of the Schrodinger operator under a suitable condition of the potiential V.
We also showed the Littlewood-Paley inequality for the the Schrodinger operator. Since we have a potential term, we need a modification of the standard proof. This method works for a Hodge-Kodaira operator on a Riemannian manifold with a potential.
In this project, we have held several symposiums and gave financial support for participants. One of them is "Stochastic Analysis and related fields" that was a research project of the Research Institute of Mathematical Sciences in 2002. We invited Professors McKean, Ustunel, Rockner, etc., from abroad. The others are "Probability Summer School" in 2003,2004. We gave introductory lectures of frontier of recent research for graduated students. We could accomplish stimulating discussions. We also held every year "Stochastic Processes and related fields", which gave fruitful communication between researchers in Japan. As a sum, we held 24 symposiums during three years and invited 14 foreign researchers and produced many results.

The purpose of this project is to study fractal from various mathematical viewpoints, for example, analysis, probability, ergode theory, dynamical systems and applied mathematics. We had two conferences in accordance with the purpose of this project. The first one held in the first year of the project. We discussed what was the main issues and how we should approach them. The second one held in in the last year of the project was to get together all the results we obtained in this project. The followings are the selection of results from this project. Kigami has shown that under the volume doubling condition, the upper Li-Yau type estimate of heat kernels is equivalent to the local Nash inequality and the escape time estimate. Kumagai along with Barlow and Bass has shown that the Li-Yau type heat kernel estimate is stable under a perturbation. Ito has studied beta-transform and the associated tiling of the Euclidean space. Kameya has made clear the relation between Julia sets and the self-similar sets. Hino has shown that the energy measure associated with the self-similar Dirichlet form on the Sierpinski gasket is mutually singular with any self-similar measure. Finally Kigami and Kameyama have obtained a relation between the topological property of a self-similar set and the asymptotic behavior of a diffusion process on it.

1.We have obtained new results for the problem to construct diffusion processes penetrating disordered media and to study properties of the processes when there are countable number of disordered media on a space. When domains of Dirichlet forms on each medium are Besov spaces, we apply the theory of Besov spaces and construct the penetrating diffusion process. We also obtain a short time asymptotic behavior of the heat kernel for the process and express the "most probable path" by means of the solution of variational formula of energy functions. This is a joint work with B.M.Hambly and appears in Probab. Theory Relat. Fields.
2.We have constructed jump-type processes on d-sets (fractal sets) and obtained detailed heat kernel estimates of the processes. The corresponding domains of Dirichlet forms are again Besov spaces and the trace formula of Besov spaces are very useful. Probabilistic techniques are used to obtain the off-diagonal estimates. Using the heat kernel estimates, we show transience/recurrence of the process and computed the Hausdorff dimension of the range of the process. This is a joint work with Z.Q.Chen and appears in Stoch. Proc. Their Appl.
3.We have studied the asymptotic behavior of the heat kernels for diffusion processes on self-similar sets. We prove an equivalence between the volume doubling property of the measure and some on-diagonal heat kernel estimates. Furthermore, we prove an equivalence between some estimates of escaping times plus local Nash inequalities and some type of heat kernel estimates from above.

木上は一般の測度-距離空間上のディリクレ形式から定義される熱核に対して、測度のvolume doubling propertyのもとでexit timeの評価とlocal Nash inequalityが熱核のある種の上からの評価と同値であることを示した。その結果を線分上のブラウン運動を自己相似測度に関して時間変更した拡散過程の熱核に応用し、測度がvolume doublingであるときに熱核の詳細な評価を得た。さらに一般のresistance formに対して、境界が有限個の点である場合のgreen関数の定義をあたえ、そのgreen関数がresistance metricに関して一様Lipschitz連続であることを示した。また、green関数を用いて測度に値を持つLaplacianの定義を与えるとともに自己相似集合上のディリクレ形式の定義域、Laplacianの定義域の関数のresistance metricに関するLipschitz連続性を考察した。
熊谷は、異なった次元を持つフラクタルを張り合わせた空間上の拡散過程の研究を行った。まずベゾフ空間のトレースの理論を援用することで2次形式の正則性を示し、熱核の短時間挙動についての詳しい評価をえた。その評価を用いて短い時間スケールでみたとき粒子が最も通りやすい経路のエネルギー関数の変分問題の解としての特徴付けを得た。さらにベゾフ空間の理論の応用として、(フラクタル集合を自然に含む)d-setと呼ばれるクラスの集合の上に飛躍型対象マルコフ過程と対応するディリクレ形式を構成した。また対応する熱核の詳細な評価を得て、過程の再帰性や粒子の軌跡のハウスドルフ次元に関する結果を得た。
宍倉は自己相似集合上のLaplacianに付随する高次元の複素力学系に関係して、高次元複素力学系の不変集合(ジュリア集合、ファツウ集合を含む)の構造、繰り込みおよび力学系に付随するポテンシャル(グリーン関数)に関する研究を行った。

Large deviation principle is one of the most basic laws in probability theory as well as the law of large numbers and the central limit theorem. We have studied the various features of the large deviation principle, starting from the elucidation of the structure of a few new classes of stochastic processes.
We(Shirai and Takahashi) introduced classes of random point fields or point processes, parameterized by a real number α, including fermion point processes(α= -1) and boson point processes(α= 1). They are associated with Fredholm determinants (to the power 1/α)of symmetric integral operators. We established the existence theorem of such random point fields for α= -1/n(n = 1,2,....) and for α = 2/m(M = 1, 2,...) by constructing them in a probabilistic manner. We also proposed a-statistics generalizing. Fermi and Bose statistics and a conjecture that such random fields exist for other values of a. Moreover, w answered to the question raised by Spohn, Johanson and others affirmatively byy showing that such random point fields do exist even for nonsymmetric integral operators provided that they are transition operators of one dimensional diffusions or birth and death processes. Based upon these facts we have established the large deviation principle in addition to other basic limit theorems and ergodic properties such as the estimates of metric entropy and Bernoulli and other properties. These results are published in Ann. Probability, J. Functional Analysis and, ASPM Series vol. 39. Besides them Shirai published an interesting application to Glauber dynamics.
Other investigators have obtained their results related to the large deviation on their own fields: Higuchi (with Shirai) on the random walk and the Schrodinger operator on infinite graphs, Kumagai on the diffusions on fractals, Shigekawa and Hino on the Wiener space and Hara on quadratic Wiener functionals.

・境界の容量密度条件の下で,一様領域,内部一様領域,John領域が境界Harnack原理や調和測度の評価で特徴付けられることを示した.また,非可積分な核に対する境界挙動を調べ,Fatou型の定理とLittlewood型の定理を導いた.3G不等式を内部一様領域に対して示すとともに,3次元以上の次元ではMartin境界が位相境界と一致するにもかかわらず,Cranston-McConnellの不等式が成立せず,その結果3G不等式が成立しない領域の例を構成した.p-調和関数に関するCarleson評価を導いた.
・ベソフ空間の理論の応用として、フラクタルを典型例として持つd-setと呼ばれるクラスの上に構成した飛耀型対称確率過程が、Triebel氏を初めとした関数空間の専門家が研究している作用素とどのような関係にあるかを調べ、論文にまとめた。

本研究は、14年度に予定されている、京都大学数理解析研究所プロジェクト研究「確率解析とその周辺」、日本数学会国際研究集会「大規模相互作用系に関する確率解析」という、確率解析に関する2つの大きな国際研究集会・共同研究を有意義かつ円滑に実施するための企画調査を目的として行われたこのために、以下のように全体会議、海外動向調査、研究者招聘によるコミュニケーションの深化を図った。
1.全体会議について:5月19日に大阪大学において初めての全体会合を持ち、各研究集会の基本的な枠組みについて話し合った。その後、代表者・分担者が各方面と連絡を取り合い、具体的な案作りを行った。これを受けて、10月4日に九州大学において2回目の全体会合を持ち、具体案の確認と今後の作業の分担の仕方について話し合った。
2.海外動向調査について:6月上旬に木上は、オーストリアでフラクタルの国際研究集会において視察・情報収集を行った。また、10月に松本は、フランスでの確率解析の国際研究集会を視察し、Elworthy教授、Yor教授らと、次年度にプロジェクト研究の一環として行われる研究集会の打ち合わせを行った。熊谷は、10月中旬よりイギリスを中心に、フランス、ドイツを2ヶ月に渡り歴訪し、イギリスのLyons教授、フランスのYor教授、Emery教授、ドイツのAlbeverio教授、Sturm教授らと事前の意思疎通を図るとともに近隣大学での最新の研究動向を詳細に調査した。これらの渡航の結果、ヨーロッパを中心とした確率解析の研究動向の最新情報を得るとともに、関連分野研究者とのコミュニケーションを深めることが出来た。
3.研究者招聘について:12月にOxford大学のHambly講師を招聘し、当該分野研究動向の報告を受けた。また、次年度に向けての日本側の準備状況を説明し、理解を得た。

本研究に際しては,各研究分担者は密接に連絡を取り合いながら各々が独自のテーマを研究するという形態をとった.実施結果は以下の通りである.
若野は,二次元弾性体中の曲線亀裂先端での応力集中現象の数学解析と数値解析について,二次元「全平面」内の亀裂問題に対して得られていた結果が二次元「有界領域」内の亀裂問題についても同様に成り立つことを検証した.
日野は,一般の局所Dirichlet形式に付随するMarkov半群に関して,積分化されたVaradhan型の短時間漸近挙動について研究を行い,極限が内在距離を用いて表現できることを証明した.系として,フラクタル集合上の対称拡散過程についてwalk次元の評価式を得た.(J.A.Ramirezとの共同研究)
重川は,半群の優評価定理と,交換定理(あるいは交差定理)について主に研究してきた.さらにその応用として,Riemann多様体上でのLittlewood-Paleyの不等式の証明や,L^p乗法作用素定理などの証明を与えた.
熊谷は,空間内に複雑な系が可算無限個存在し,それぞれの系については熱伝導に関する情報がある程度分かっているようなモデルについて,各々の系にしみ込む拡散過程を構成し,その熱核の短時間挙動についての詳しい評価を得た.
渡辺は,WalshのBrown運動勲に関連して,Riemann面上のBrown運動の道に沿った有理型微分形式の積分(Abel積分)の漸近挙動について研究した.また,Tsirelsonが確率過程の表現の問題に関連して導入したノイズの概念に関し,Fellerの1次元拡散過程から比較的容易に構成出来るblack noiseの例を与えた.

1. We have obtained an almost sure invariance principle for the range of random walks on 3-dimensional lattice. This result is a refinement of the known results like, central limit theorem, and various limit theorem can be deduced as a corollary to this result. For the range of random walks on 2-dimensional lattice, we have obtained the law of iterated logarithm. Instead of the expected loglog term, logloglog term appears.
This work will appear in Ann. Probab.
2. We have studied a problem that when disordered media are in a Euclidean space, how does the heat transfers from the space to the media. Applying the theory of Besov spaces, we have constructed a penetrating diffusion process which behaves like the diffusion on the media and like Brownian motion on the Euclidean space outside the media. This work appears in J. Funct. Anal. We then continue to work this problem and obtain a short time asymptotic behavior of the heat kernel for the diffusion process. We also obtain a functional type large deviation for the process.
We are now writing a paper on these results.
3. We have obtained asymptotic behavior of the transition probabilities between two sets on infinite dimensional symmetric diffusion processes. We adopted intrinsic distance instead of the usual distance between sets and our result is fairly general. In particular, we have proved the asymptotic behavior for the Orstein-Uhlenbeck processes on loop spaces on Riemannian manifolds, which had been an open problem. Our result is new also for degenerated symmetric diffusions on Euclidean spaces. This is a joint work with J. A. Ramirez and is now submitted for publication.

In this research project, we obtained the following five main results related with analysis on fractals.
(1) Markov property of Dirichlet forms on self-similar sets
We showed the Markov property of Kusuoka-Zhou's Dirichlet forms on self-similar sets.
(2) Self-similarity of volume measures associated with Laplacians on p.c.f. self-similar fractals
We obtained a sufficient condition for the self-similarity of the volume measure, which is defined by using operator theoretic trace. We also showed that the sufficient condition holds in the case of standard Laplacian on the Sierpinski gasket.
(3) Green's function on fractals
We obtained an algorithm to calculate the diagonal of Green's function and used the algorithm to investigate the maximum value of Green's function.
(4) Large deviations for Brownian motion on the Sierpinski gasket
We showed that Varadhan type estimate and Schilder type Large deviation do not hold for the case of the standard Laplacian on the Sierpinski gasket
(5) Multifractal formalisms for the local spectral and walk dimensions
We showed the multifractal nature of the local spectral and walk dimensions associated with the Laplacians on the self-similar sets.

We have established the following results throughout this project.
1. On spectra of self-adjoint operators on fractals
We study the short time asymptotics for heat kernels when the self-similarity of the measure and that of the diffusion process are different. In this case, the asymptotics highly depend on the initial points and there is a multi-fractality. We compute the Hausdorff dimension and the paper will appear. Further, we define a family of quasi-distances and show that the Hausdorff dimension can be expressed in a simple way using some special quasi-distance.
2. On stochastic processes on random fractals
(1) We construct a diffusion process and study its heat kernel properties on a homogeneous random Sierpinski carpet.
(2) We study a short time heat kernel asymptotics for a diffusion process on a random recursive Sierpinski gasket, which does not have spatial symmetries.
These results are in our paper that have already appeared.
3. Stochastic analysis on fractals
(1) We construct a diffusion process on a (Euclidean) space where countable numbers of disordered media (such as fractals) are embedded.
We apply a trace theory of Besov spaces for the proof. The paper has already appeared and we are now working on large deviations for the process.
(2) Concerning the "stochastic analysis for stochastic processes with rough paths" studied by T.Lyons, we have not obtained a new result so far. It is one of the future problem to study the detailed properties of the stochastic differential equations established by him.

本年度もセミナーや研究会を通じて、スピン系等の数理モデルの性質を中心に最新のプレプリントの紹介、各自の研究の報告・議論を行ったが、スピングラスのモデル自体について大きな進展を得ることは残念ながら出来なかった。以下では、セミナー等で得られた、このテーマに関連した問題に関する実績概要を述べる。
1.吉田は、wetting tansitionの問題に興味を持ち、E.BolthausenやP.Caputoらのプレプリントを読むとともに具体的な計算を行った。Wetting transitionの問題とは、固形物の上に液体がありpinningとentropy repulsionという対立する力がかかるとき、そのinterfaceの局在(dry phase)・非局在(wet phase)がどのようなときに起こるかという問題である。吉田は、非負に条件付けられた一次元のランダムウォークにpinningとしてdiluted local timeを与えたとき、pinningの係数が小さければ非局在、大きければ局在が起こることを示した。
2.熊谷は、フラクタルのような複雑な形の不純物が空間内に存在するときに、空間からこの物質内への熱伝導はどのようになるかという問題を扱い、実解析で用いられるBesov空間の理論を援用することによりある条件のもとでこのような熱伝導を表す拡散過程(物質内ではその物質の拡散をし、外では空間の拡散に従うようなもの)が構成できることを示した。この結果は、最近雑誌に掲載された。
3.村井は篠田氏(奈良女)との議論を通じて、ある範疇の自己相似なグラフ上ではρ-パーコレーションのρ→1における相転移点と普通のパーコレーションの相転移点が一致することを示し、現在論文を執筆中である。

During these three years we achieved the expected results in various areas in probability theory and contributed to promote a few new directions for the future as follows.
In stochastic analysis we have made, among others, two remarkable progresses. One is the study of inequalities and their role in the analysis of function spaces on infinite dimensional, Wiener space led by Aida and Shigekawa. The other is the study of asymptotic behaviors for the Wiener functionals of exponential type by Taniguchi, Matsumoto et al. Besides them, we should mention on the trend to generalize the stochastic analysis based on the Dirichlet space theory shown, for instance, in the work of Osada.
In the theory of hydrodynamic limit Funaki, Uchiyama and thier pupil attacked more realistic models successfully and reached to the promissing problem to compare and synthesize their results with the results obtained in the Dirichlet space theory mentioned above.
In ergodic theory and its around the most remarkable progress was made in the study of the decay of correlation functions by Morita. The summer school in 1997 in this area influenced the information theory people and Han made a large deviation theory for coding.
The renewal of the classical theories and techniques was one of the slogans of the project, which was also successful as is shown by Kumagai on analysis on fractals, Atsuji on Nevalinna theory, Hamana on random walk functionals and Shirai et al. on the spetrum of graph.
Moreover, Nagai et al. applied stochastic control techniques to mathematical finance and Sugita, Takanobu et al. contributed to quasi-random nember theory by using Weyltransformation.
We owed much to PE S. Watanabe to have organized a summer school in 1999 on the noise problem. Also we owed much to the modification on the international exchange under the grant-in-aids by JSPS which has brought more fruitful results.

ホワイトノイズ超関数に作用する作用素の統一理論は、94年に出版された私の著書で一通りの完成を見たが、最近のホワイトノイズ超関数論の発展に合わせて、理論を組み立て直す必要が出てきた。本研究では、まず、増大度の条件を緩めたホワイトノイズ超関数でベクトル値の場合も含むように、従来の理論を発展させた。応用として、量子適合過程・量子的マルチンゲ-ルの一般化と特徴づけが得られた。この理論では、ホワイトノイズの高次の巾乗を取り扱うことができるが、これは確率解析を非線形の方向で一般化するための基礎になるのではないか、という知見を得た。関連する量子的伊藤公式の導出は興味深い課題として残った。別の応用として、ある種の量子確率微分方程式の解の一意存在定理を証明し、ウィック指数関数による解の構成を論じた。また、散逸を含む量子系を記述する量子ランジュヴァン方程式もこの範疇で捉えることができた。解の滑らかさは、極めて興味深い問題である。本研究では、古典的な微分方程式における手法が応用できるか、との関連で研究を開始したが、今のところ手探りの状態である。量子系の中心極限定理も本研究に含まれる課題であったが、群や超群上のランダムウォークのスペクトル解析を背景に、新しい組合せ論的アプローチを開発した。その結果、新しい極限分布の族が構成され、ウィグナーの半円則とその変形が得られた。この流れでダイソンのランダム行列モデルを解析することは今後の課題となった。シュレ-ディンガー作用素の固有値問題・量子スピン系・フラクタル上の確率過程などを量子確率論の観点から見直すことで、これらは今後の研究の展開の上で重要であると認識を新たにした。
本研究遂行にあたり、研究代表者・分担者ともに海外も含め多くの研究集会で口頭発表し、周辺研究者との交流を深めた。また、来日中の外国人研究者や国内研究者多数をセミナーに招待することができ有意義な研究討論を行なった。