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1.
Hideki Tanemura 《Probability Theory and Related Fields》1997,109(2):275-299
Summary. Dirichlet forms associated with systems of infinitely many Brownian balls in ℝ
d
are studied. Introducing a linear operator L
0 defined on a space of smooth local functions, we show the uniqueness of Dirichlet forms associated with self adjoint Markovian
extensions of L
0. We also discuss the ergodicity of the reversible process associated with the Dirichlet form.
Received: 18 July 1996/In revised form: 13 February 1997 相似文献
2.
It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V
(0), which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V
(0). In this paper, I prove that, provided an eigenform exists, even if the form on V
(0) is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket. 相似文献
3.
We define a Dirichlet form ɛ describing diffusion in ℝ
d
and jumps in a fractal Γ ⊂ ℝ
d
. The jump measure J is defined as an image of a jump measure j of a process in a non-Archimedean metric space. As the result the jump intensity depends on the hierarchical structure of
Γ rather than the geometric distance in ℝ
d
. For a class of fractals in ℝ2 we find a condition on the measure j so that the Dirichlet form ɛ is regular. The condition is given in terms of Hausdorff dimension of Γ. 相似文献
4.
We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form
operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these
two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO
Dirichlet problem implies L
p
solvability for all p>p
0. 相似文献
5.
Laurent Denis 《Probability Theory and Related Fields》2000,118(3):406-426
We construct a Dirichlet structure related to a Poisson measure on ℝ+×M, where M is a general measured space, with compensator dt⊗dv. We obtain a criterion of density for variables in the domain of the Dirichlet form and we apply it to S.D.E. driven by this
Poisson measure.
Received: 15 May 1999 / Revised version: 23 February 2000 / Published online: 12 October 2000 相似文献
6.
The concept of complex Dirichlet forms
c
resp. operators L
c
in complex weighted L
2-spaces is introduced. Perturbations of classical Dirichlet forms by forms associated with complex first-order differential operators provide examples of complex Dirichlet forms.Complex Dirichlet operators L
c
are unitarily equivalent with (a family of) Schrödinger operators with electromagnetic potentials.To
c
there is associated a pair of real-valued non symmetric Dirichlet forms on the corresponding real weighted L
2-spaces, which in turn are associated with (non-symmetric) diffusion processes.Results by Stannat on non symmetric Dirichlet forms and their perturbations can be used for discussing the essential self-adjointness of L
c
.New closability criteria for (perturbation of) non symmetric Dirichlet forms are obtained. 相似文献
7.
A Dirichlet series with multiplicative coefficients has an Euler product representation. In this paper we consider the special
case where these coefficients are derived from the numbers of representations of an integer by an integral quadratic form.
At first we suppose this quadratic form to be positive definite. In general the representation numbers are not multiplicative.
Instead we consider the average number of representations over all classes in the genus of the quadratic form. And we consider
only representations of integers of the form tk
2 with t square-free. If we divide the average representation number for these integers by a suitable factor, we do get a multiplicative
function. Using results from Siegel (Ann. Math. 36:527–606, 1935), we derive a uniform expression for the Euler product expansion of the corresponding Dirichlet series. As a special case,
we consider the standard quadratic form in n variables corresponding to the identity matrix. Here we use results from Shimura (Am. J. Math. 124:1059–1081, 2002). For 2≤n≤8, the genus of this particular quadratic form contains only one class, and this leads to a rather simple expression for
the Dirichlet series, where the coefficients are just the number of representations of a square as the sum of n squares. Finally we consider the indefinite case, where we can get results similar to the definite case. 相似文献
8.
We study the Dirichlet problem for fully nonlinear, degenerate elliptic equations of the form F (Hess u) = 0 on a smoothly bounded domain Ω ? ?n. In our approach the equation is replaced by a subset F ? Sym2(?n) of the symmetric n × n matrices with ?F ? { F = 0}. We establish the existence and uniqueness of continuous solutions under an explicit geometric “F‐convexity” assumption on the boundary ?Ω. We also study the topological structure of F‐convex domains and prove a theorem of Andreotti‐Frankel type. Two key ingredients in the analysis are the use of “subaffine functions” and “Dirichlet duality.” Associated to F is a Dirichlet dual set F? that gives a dual Dirichlet problem. This pairing is a true duality in that the dual of F? is F, and in the analysis the roles of F and F? are interchangeable. The duality also clarifies many features of the problem including the appropriate conditions on the boundary. Many interesting examples are covered by these results including: all branches of the homogeneous Monge‐Ampère equation over ?, ?, and ?; equations appearing naturally in calibrated geometry, Lagrangian geometry, and p‐convex Riemannian geometry; and all branches of the special Lagrangian potential equation. © 2008 Wiley Periodicals, Inc. 相似文献
9.
We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W1,p estimates.© 2016 Wiley Periodicals, Inc. 相似文献
10.
For a family of second‐order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic expansions of Poisson kernels and the Dirichlet‐to‐Neumann maps as well as optimal convergence rates in Lp and W1,p for solutions with Dirichlet or Neumann boundary conditions. © 2014 Wiley Periodicals, Inc. 相似文献
11.
Hiroshi Kaneko 《Potential Analysis》2006,24(1):87-103
The space ℱr,p, which was designed so as to play similar roles to the ordinary Sobolev space Wr,p(ℝn), introduced as a cornerstone for analyzing nonlinear potential theoretic features of the state space with a measure-symmetric
transition probability semi-group. The aim of this article is revealing a sufficient condition for the counterpart of the
Sobolev space to coincide with domain of some Dirichlet form on a local field and discussing some other features of those
counterparts on the non-Archimedean metric space. For example, we will see a sufficient condition for the space ℱr,2 to be viewed as domain of some Dirichlet form and microscopic property such as polarity of singleton will be investigated.
Mathematics Subject Classifications (2000) 31C45, 11D88 , 46E35, 60J25. 相似文献
12.
Giles Auchmuty 《Numerical Functional Analysis & Optimization》2018,39(10):999-1010
This paper describes the rate of convergence of solutions of Robin boundary value problems of an elliptic equation to the solution of a Dirichlet problem as a boundary parameter decreases to zero. The results are found using representations for solutions of the equations in terms of Steklov eigenfunctions. Particular interest is in the case where the Dirichlet data is only in L2(?Ω,dσ). Various approximation bounds are obtained and the rate of convergence of the Robin approximations in the H1 and L2 norms are shown to have convergence rates that depend on the regularity of the Dirichlet data. 相似文献
13.
K. Rybnikov 《Discrete and Computational Geometry》1999,21(4):481-517
This paper introduces a general notion of stress on cell-complexes and reports on connections between stresses and liftings
(generalization of C
1
0 -splines) of d -dimensional cell-complexes in R
d
. New sufficient conditions for the existence of a sharp lifting for a ``flat" piecewise-linear realization of a manifold
are given. Our approach also gives some new results on the equivalence between spherical complexes and convex and star polytopes.
As an application, two algorithms are given that determine whether a piecewise-linear realization of a d -manifold in R
d
admits a lifting to R
d+1
which satisfies given constraints. We also demonstrate connections between stresses and Voronoi—Dirichlet diagrams and show
that any weighted Voronoi—Dirichlet diagram without non-compact cells can be represented as a weighted Delaunay decomposition
and vice versa.
Received April 24, 1997, and in revised form July 31, 1998. 相似文献
14.
We introduce here some Itô calculus for non-continuous Dirichlet processes. Such calculus extends what was known for continuous Dirichlet processes or for semimartingales. In particular we prove that non-continuous Dirichlet processes are stable under C
1 transformation. 相似文献
15.
In this paper we develope a perturbation theory for second order parabolic operators in non‐divergence form. In particular we study the solvability of the Dirichlet problem in non cylindrical domains with Lp ‐data on the parabolic boundary (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
16.
Various Poincaré–Sobolev type inequalities are studied for a reaction–diffusion model of particle systems on Polish spaces.
The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the
system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which
we call reaction–diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on
the product spaces En (n≥1) which determine the motion of particles, and the reaction part induced by a Q-process on ℤ+ and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced
particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction
part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently,
for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction–diffusion Dirichlet form satisfies
a super Poincaré inequality (e.g., the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples
are presented to illustrate the main results.
Mathematics Subject Classifications (2000) 4FD0F, 60H10.
Feng-Yu Wang: Supported in part by the DFG through the Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic
Dynamics”, the BiBoS Research Centre, NNSFC(10121101), and RFDP(20040027009). 相似文献
17.
18.
Mu‐Tao Wang 《纯数学与应用数学通讯》2004,57(2):267-281
Let Ω be a bounded C2 domain in ?n and ? ?Ω → ?m be a continuous map. The Dirichlet problem for the minimal surface system asks whether there exists a Lipschitz map f : Ω → ?m with f|?Ω = ? and with the graph of f a minimal submanifold in ?n+m. For m = 1, the Dirichlet problem was solved more than 30 years ago by Jenkins and Serrin [12] for any mean convex domains and the solutions are all smooth. This paper considers the Dirichlet problem for convex domains in arbitrary codimension m. We prove that if ψ : ¯Ω → ?m satisfies 8nδ supΩ |D2ψ| + √2 sup?Ω |Dψ| < 1, then the Dirichlet problem for ψ|?Ω is solvable in smooth maps. Here δ is the diameter of Ω. Such a condition is necessary in view of an example of Lawson and Osserman [13]. In order to prove this result, we study the associated parabolic system and solve the Cauchy‐Dirichlet problem with ψ as initial data. © 2003 Wiley Periodicals, Inc. 相似文献
19.
E. A. Volkov 《Proceedings of the Steklov Institute of Mathematics》2010,269(1):57-64
We study the Dirichlet problem for the Laplace equation in an infinite rectangular cylinder. Under the assumption that the
boundary values are continuous and bounded, we prove the existence and uniqueness of a solution to the Dirichlet problem in
the class of bounded functions that are continuous on the closed infinite cylinder. Under an additional assumption that the
boundary values are twice continuously differentiable on the faces of the infinite cylinder and are periodic in the direction
of its edges, we establish that a periodic solution of the Dirichlet problem has continuous and bounded pure second-order
derivatives on the closed infinite cylinder except its edges. We apply the grid method in order to find an approximate periodic
solution of this Dirichlet problem. Under the same conditions providing a low smoothness of the exact solution, the convergence
rate of the grid solution of the Dirichlet problem in the uniform metric is shown to be on the order of O(h
2 ln h
−1), where h is the step of a cubic grid. 相似文献
20.
Harmonic maps between two Riemannian manifolds M and N are often constructed as energy minimizing maps. This construction is extended for the Dirichlet problem to the case where the Riemannian energy functional on M is replaced by a more general Dirichlet form. We obtain weakly harmonic maps and prove that these maps send the diffusion to N-valued martingales. The basic tools are the reflected Dirichlet space and the stochastic calculus for Dirichlet processes. 相似文献