共查询到20条相似文献,搜索用时 15 毫秒
1.
František Žák 《随机分析与应用》2016,34(6):1057-1082
We develop and implement new probabilistic strategy for proving basic results about long-time behavior for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process as a solution to suitable infinite-dimensional martingale problem. However, the techniques allow us to consider cases where the generator of the particle is degenerate elliptic operator. As a model example, we present the situation where the operator arises from Heisenberg group. In the last section, we provide further examples that can be handled using our methods. 相似文献
2.
Summary We study the problem of relating the long time behavior of finite and infinite systems of locally interacting components.
We consider in detail a class of lincarly interacting diffusionsx(t)={x
i
(t),i ∈ ℤ
d
} in the regime where there is a one-parameter family of nontrivial invariant measures. For these systems there are naturally
defined corresponding finite systems,
, with
. Our main result gives a comparison between the laws ofx(t
N
) andx
N
(t
N
) for timest
N
→∞ asN→∞. The comparison involves certain mixtures of the invariant measures for the infinite system.
Partly supported by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University, by the
National Science Foundation, and by the National Security Agency
Research supported in part by the DFG
Partly supported by S.R.63540155 of Japan Ministry of Education 相似文献
3.
We survey recent effort in establishing the hydrodynamic limits and the fluctuation limits for a class of interacting diffusions in domains. These systems are introduced to model the transport of positive and negative charges in solar cells. They are general microscopic models that can be used to describe macroscopic phenomena with coupled boundary conditions, such as the popula- tion dynamics of two segregated species under competition. Proving these two types of limits represents establishing the functional law of large numbers and the functional central limit theorem, respectively, for the empirical measures of the spatial positions of the particles. We show that the hydrodynamic limit is a pair of deterministic measures whose densities solve a coupled nonlinear heat equations, while the fluctuation limit can be described by a Gaussian Markov process that solves a stochastic partial differential equation. 相似文献
4.
J. Theodore Cox Klaus Fleischmann Andreas Greven 《Probability Theory and Related Fields》1996,105(4):513-528
Summary A general comparison argument for expectations of certain multitime functionals of infinite systems of linearly interacting diffusions differing in the diffusion coefficient is derived. As an application we prove clustering occurs in the case when the symmetrized interaction kernel is recurrent, and the components take values in an interval bounded on one side. The technique also gives an alternative proof of clustering in the case of compact intervals. 相似文献
5.
6.
We study the limiting behavior of the empirical measure of a system of diffusions interacting through their ranks when the number of diffusions tends to infinity. We prove that under certain assumptions the limiting dynamics is given by a McKean–Vlasov evolution equation. Moreover, we show that the evolution of the cumulative distribution function under the limiting dynamics is governed by the generalized porous medium equation with convection. The implications of the results for rank-based models of capital distributions in financial markets are also explained. 相似文献
7.
A. F. Ramírez 《Probability Theory and Related Fields》1998,110(3):369-395
Summary. Let η be a diffusion process taking values on the infinite dimensional space T
Z
, where T is the circle, and with components satisfying the equations dη
i
=σ
i
(η) dW
i
+b
i
(η) dt for some coefficients σ
i
and b
i
, i∈Z. Suppose we have an initial distribution μ and a sequence of times t
n
→∞ such that lim
n
→∞μS
tn
=ν exists, where S
t
is the semi-group of the process. We prove that if σ
i
and b
i
are bounded, of finite range, have uniformly bounded second order partial derivatives, and inf
i
,ησ
i
(η)>0, then ν is invariant.
Received: 12 September 1996 / In revised form: 10 November 1997 相似文献
8.
Kôhei Uchiyama 《Probability Theory and Related Fields》1994,99(1):97-110
Summary It is remarked that for Brownian particles interacting with a smooth repulsive pair potential the nonlinear diffusion equation which S. Varadhan has derived under an entropy bound for initial densities is valied whatever initial distribution they start with.Research partially supported by Japan Society for the Promotion of Science 相似文献
9.
We prove a large deviation theorem from the hydrodynamical limit for the empirical measure of Ginzburg-Landau and zero range processes in infinite volume starting from deterministic initial configurations. In the Ginzburg-Landau case the main tool is the study of the evolution of the H?1 norm and in the zero range case the attractiveness which allows couplings. 相似文献
10.
Eva Löcherbach 《Journal of Functional Analysis》2004,215(1):130-177
We consider the invariant measure for finite systems of interacting branching diffusions with immigrations. We use Malliavin calculus in order to show that the intensity measure of the invariant measure admits a density which is continuous, one times partially differentiable and bounded provided the immigration measure is absolute continuous. 相似文献
11.
Philip McDunnough 《Annals of the Institute of Statistical Mathematics》1979,31(1):435-443
This article deals with the estimation of a parameter in the stochastic motion affecting an infinite number of particles.
An estimator, based on a nonstationary time-series is considered and shown to be consistent. A comparison with more well-known
estimates, via asymptotic variances, is also carried out. 相似文献
12.
Clustering of linearly interacting diffusions and universality of their long-time limit distribution
J. M. Swart 《Probability Theory and Related Fields》2000,118(4):574-594
Let K⊂ℝ
d
(d≥ 1) be a compact convex set and Λ a countable Abelian group. We study a stochastic process X in K
Λ, equipped with the product topology, where each coordinate solves a SDE of the form dX
i
(t) = ∑
j
a(j−i) (X
j
(t) −X
i
(t))dt + σ (X
i
(t))dB
i
(t). Here a(·) is the kernel of a continuous-time random walk on Λ and σ is a continuous root of a diffusion matrix w on K. If X(t) converges in distribution to a limit X(∞) and the symmetrized random walk with kernel a
S
(i) = a(i) + a(−i) is recurrent, then each component X
i
(∞) is concentrated on {x∈K : σ(x) = 0 and the coordinates agree, i.e., the system clusters. Both these statements fail if a
S
is transient. Under the assumption that the class of harmonic functions of the diffusion matrix w is preserved under linear transformations of K, we show that the system clusters for all spatially ergodic initial conditions and we determine the limit distribution of
the components. This distribution turns out to be universal in all recurrent kernels a
S
on Abelian groups Λ.
Received: 10 May 1999 / Revised version: 18 April 2000 / Published online: 22 November 2000 相似文献
13.
R. Bantsuri 《Journal of Mathematical Sciences》2013,193(3):374-377
A system of infinite algebraic equations is solved explicitly using a Carleman-type problem. 相似文献
14.
15.
16.
P. Révész 《Journal of Theoretical Probability》1993,6(2):353-383
Let be a Poisson process on d of intensity and letW
1(t),W
2
(t),..., be a sequence of independent Wiener processes. LetW
i
(t)=X
i
+W
i
(t) whereX
1,X
2,..., are the points of . Consider the processess(t)=#{i:X
i
(t)1}. These and related processes are studied. 相似文献
17.
Minoru W. Yoshida 《Probability Theory and Related Fields》1996,106(2):265-297
Summary. By the theory of quasi-regular Dirichletforms and the associated special standard processes, the existence of symmetric diffusion
processes taking values in the space of non-negative integer valued Radon measures on and having Gibbs invariant measures associated with some given pair potentials is considered. The existence of such diffusions
can be shown for a wide class of potentials involving some singular ones. Also, as a consequence of an application of stochastic
calculus, a representation for the diffusion by means of a stochastic differential equation is derived.
Received: 5 September 1995 / In revised form: 14 March 1996 相似文献
18.
《Mathematical and Computer Modelling》2007,45(9-10):1265-1271
Using the classical Schauder fixed point principle we prove an existence result concerning an infinite system of singular integral equations. The obtained result is applied to establish the solvability of an infinite system of differential equations of fractional order. 相似文献
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