共查询到20条相似文献,搜索用时 46 毫秒
1.
Brownian motion on the continuum tree 总被引:1,自引:1,他引:0
W. B. Krebs 《Probability Theory and Related Fields》1995,101(3):421-433
Summary We construct Brownian motion on a continuum tree, a structure introduced as an asymptotic limit to certain families of finite trees. We approximate the Dirichlet form of Brownian motion on the continuum tree by adjoining one-dimensional Brownian excursions. We study the local times of the resulting diffusion. Using time-change methods, we find explicit expressions for certain hitting probabilities and the mean occupation density of the process. 相似文献
2.
Naotaka Kajino 《Journal of Functional Analysis》2010,258(4):1310-1360
Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function. 相似文献
3.
Denis Borisov 《Journal of Differential Equations》2009,247(11):3028-1127
We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem. 相似文献
4.
Jun Kigami 《Advances in Mathematics》2010,225(5):2674-2730
Transient random walk on a tree induces a Dirichlet form on its Martin boundary, which is the Cantor set. The procedure of the inducement is analogous to that of the Douglas integral on S1 associated with the Brownian motion on the unit disk. In this paper, those Dirichlet forms on the Cantor set induced by random walks on trees are investigated. Explicit expressions of the hitting distribution (harmonic measure) ν and the induced Dirichlet form on the Cantor set are given in terms of the effective resistances. An intrinsic metric on the Cantor set associated with the random walk is constructed. Under the volume doubling property of ν with respect to the intrinsic metric, asymptotic behaviors of the heat kernel, the jump kernel and moments of displacements of the process associated with the induced Dirichlet form are obtained. Furthermore, relation to the noncommutative Riemannian geometry is discussed. 相似文献
5.
Angelo Alvino 《Journal of Differential Equations》2010,249(12):3279-3290
We prove a comparison principle for second order quasilinear elliptic operators in divergence form when a first order term appears. We deduce uniqueness results for weak solutions to Dirichlet problems when data belong to the natural dual space. 相似文献
6.
Small time asymptotics of diffusion processes 总被引:1,自引:0,他引:1
We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms
under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic
operators. In the degenerate case the asymptotics incorporate possible non-ergodicity. 相似文献
7.
Let M be a complete Riemannian manifold and D⊂M a smoothly bounded domain with compact closure. We use Brownian motion to study the relationship between the Dirichlet spectrum of D and the heat content asymptotics of D. Central to our investigation is a sequence of invariants associated to D defined using exit time moments. We prove that our invariants determine that part of the spectrum corresponding to eigenspaces which are not orthogonal to constant functions, that our invariants determine the heat content asymptotics associated to the manifold, and that when the manifold is a generic domain in Euclidean space, the invariants determine the Dirichlet spectrum. 相似文献
8.
We obtain the equivalence conditions for an on-diagonal upper bound of heat kernels on self-similar measure energy spaces. In particular, this upper bound of the heat kernel is equivalent to the discreteness of the spectrum of the generator of the Dirichlet form, and to the global Poincaré inequality. The key ingredient of the proof is to obtain the Nash inequality from the global Poincaré inequality. We give two examples of families of spaces where the global Poincaré inequality is easily derived. They are the post-critically finite (p.c.f.) self-similar sets with harmonic structure and the products of self-similar measure energy spaces. 相似文献
9.
Summary We study Dirichlet forms associated with random walks on fractal-like finite grahs. We consider related Poincaré constants and resistance, and study their asymptotic behaviour. We construct a Markov semi-group on fractals as a subsequence of random walks, and study its properties. Finally we construct self-similar diffusion processes on fractals which have a certain recurrence property and plenty of symmetries.Partly supported by the JSPS Program 相似文献
10.
B. M. Hambly Michel L. Lapidus 《Transactions of the American Mathematical Society》2006,358(1):285-314
In this paper a string is a sequence of positive non-increasing real numbers which sums to one. For our purposes a fractal string is a string formed from the lengths of removed sub-intervals created by a recursive decomposition of the unit interval. By using the so-called complex dimensions of the string, the poles of an associated zeta function, it is possible to obtain detailed information about the behaviour of the asymptotic properties of the string. We consider random versions of fractal strings. We show that by using a random recursive self-similar construction, it is possible to obtain similar results to those for deterministic self-similar strings. In the case of strings generated by the excursions of stable subordinators, we show that the complex dimensions can only lie on the real line. The results allow us to discuss the geometric and spectral asymptotics of one-dimensional domains with random fractal boundary.
11.
Svante Janson 《Random Structures and Algorithms》2003,22(4):337-358
Asymptotics are obtained for the mean, variance, and higher moments as well as for the distribution of the Wiener index of a random tree from a simply generated family (or, equivalently, a critical Galton–Watson tree). We also establish a joint asymptotic distribution of the Wiener index and the internal path length, as well as asymptotics for the covariance and other mixed moments. The limit laws are described using functionals of a Brownian excursion. The methods include both Aldous' theory of the continuum random tree and analysis of generating functions. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 22: 337–358, 2003 相似文献
12.
In this article, we consider the problem of estimating the heatkernel on measure-metric spaces equipped with a resistance form.Such spaces admit a corresponding resistance metric that reflectsthe conductivity properties of the set. In this situation, ithas been proved that when there is uniform polynomial volumegrowth with respect to the resistance metric the behaviour ofthe on-diagonal part of the heat kernel is completely determinedby this rate of volume growth. However, recent results haveshown that for certain random fractal sets, there are globaland local (point-wise) fluctuations in the volume as r 0 andso these uniform results do not apply. Motivated by these examples,we present global and local on-diagonal heat kernel estimateswhen the volume growth is not uniform, and demonstrate thatwhen the volume fluctuations are non-trivial, there will benon-trivial fluctuations of the same order (up to exponents)in the short-time heat kernel asymptotics. We also provide boundsfor the off-diagonal part of the heat kernel. These resultsapply to deterministic and random self-similar fractals, andmetric space dendrites (the topological analogues of graph trees). 相似文献
13.
Marco Biroli Silvia Mataloni Michele Matzeu 《NoDEA : Nonlinear Differential Equations and Applications》2005,12(3):295-321
Some stability results for Mountain Pass and Linking type solutions of semilinear problems involving a very general class
of Dirichlet forms are stated. The non linear terms are supposed to have a suitable superlinear growth and the family of Dirichlet
forms is required to be dominated from below and from above by a fixed diffusion type form. Some concrete examples are also
given. 相似文献
14.
Jun Kigami 《Mathematische Annalen》2008,340(4):781-804
We study the standard Dirichlet form and its energy measure,called the Kusuoka measure, on the Sierpinski gasket as aprototype
of “measurable Riemannian geometry”. The shortest pathmetric on the harmonic Sierpinski gasket is shown to be thegeodesic
distance associated with the “measurable Riemannianstructure”. The Kusuoka measure is shown to have the volumedoubling property
with respect to the Euclidean distance and alsoto the geodesic distance. Li–Yau type Gaussian off-diagonal heatkernel estimate
is established for the heat kernel associated withthe Kusuoka measure. 相似文献
15.
Stefan Holst 《Numerische Mathematik》2008,109(1):101-119
We present a local exponential fitting hybridized mixed finite-element method for convection–diffusion problem on a bounded
domain with mixed Dirichlet Neuman boundary conditions. With a new technique that interpretes the algebraic system after static
condensation as a bilinear form acting on certain lifting operators we prove an a priori error estimate on the Lagrange multipliers
that requires minimal regularity. While an extension of more classical arguments provide an estimate for the other solution
components. 相似文献
16.
J. A. Carrillo A. Jüngel P. A. Markowich G. Toscani A. Unterreiter 《Monatshefte für Mathematik》2001,133(1):1-82
We analyse the large-time asymptotics of quasilinear (possibly) degenerate parabolic systems in three cases: 1) scalar problems
with confinement by a uniformly convex potential, 2) unconfined scalar equations and 3) unconfined systems. In particular
we are interested in the rate of decay to equilibrium or self-similar solutions. The main analytical tool is based on the
analysis of the entropy dissipation. In the scalar case this is done by proving decay of the entropy dissipation rate and
bootstrapping back to show convergence of the relative entropy to zero. As by-product, this approach gives generalized Sobolev-inequalities,
which interpolate between the Gross logarithmic Sobolev inequality and the classical Sobolev inequality. The time decay of
the solutions of the degenerate systems is analyzed by means of a generalisation of the Nash inequality. Porous media, fast
diffusion, p-Laplace and energy transport systems are included in the considered class of problems. A generalized Csiszár–Kullback inequality
allows for an estimation of the decay to equilibrium in terms of the relative entropy.
(Received 11 October 2000; in revised form 13 March 2001) 相似文献
17.
Niels Jacob 《Potential Analysis》1992,1(3):221-232
It is shown that a special class of symmetric elliptic pseudo differential operators do generate a Feller semigroup and therefore a non-local Dirichlet form. 相似文献
18.
We consider an inverse boundary value problem for identifying the inclusion inside a known anisotropic conductive medium. We give a reconstruction procedure for identifying the inclusion from the Dirichlet–Neumann map or the Neumann–Dirichlet map associated with the mixed type boundary condition. 相似文献
19.
Minoru Murata 《Journal of Differential Equations》2003,195(1):82-118
We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on Rd. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of Rd with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function. 相似文献
20.
We consider the Dirichlet problem for a class of anisotropic degenerate elliptic equations. 相似文献