A diffusion process in a singular mean-drift-field
A diffusion process in a singular mean-drift-field
复制标题
奇异平均漂移场中的扩散过程
DOI：
10.1007/bf00532640
复制
发表时间：
1985
期刊：
影响因子：
--
通讯作者：
Hiroshi Tanaka
中科院分区：
文献类型：
--
作者：
M. Nagasawa;Hiroshi Tanaka
在第一作者提出的(相互作用的)粒子系统的统计模型中，以下两个事实结合在一起：一是当粒子的数量足够大时，系统确定一个(非线性)扩散过程(例如，Vlasov-McKean(或平均场)极限，见[7]；在[8]中，它被称为系统的“中间描述”)。二是一类扩散过程与Schr6dinger型特征值问题之间存在一一对应关系。更准确地说，存在具有不变密度的扩散过程1012，其中0是本征函数(参见。F~nyes[3]，Nelson[13]，长泽[8]；在[8]中这被称为“宏观描述”)。如果我们假设平均场极限的过程将变得渐近稳定并且具有1~12个不变密度，那么我们就得到了一个连接三个对象的统计模型：相互作用粒子系统、扩散过程和本征值问题。该模型将在w和附录中作简要说明(模型的一些应用见[8，9，12])。在这一背景下，有几个有趣的数学问题：(I)给定一个系统的平衡分布密度，找到系统中粒子之间的一对相互作用。(Ii)给定一个(对)相互作用，证明在平均场极限下扩散过程的存在性。(Iii)证明大数定律(混沌的传播)。(Iv)研究系统t--+o的渐近行为。McKean[7]在相互作用的Lipschitz连续性的假设下讨论了问题(II)和(III)(有关极限定理见Tanaka[16]和Sznitman[14]，以及Dawson[1])。然而，这个假设对于我们的目的来说太严格了，因为产生激发态平衡分布的相互作用不是Lipschitz连续的，并且具有由本征函数的零点引起的奇异性。
In a statistical model of systems of (interacting) particles proposed by the first author (cf.[8, 12]) the following two facts are combined together: One is that a system determines a (non-linear) diffusion process when the number of particles is large enough (eg Vlasov-McKean (or mean-field) limit, see [7]; in [8] it is called" intermediate description" of a system). Another one is that there is one to one correspondence between a class of diffusion processes and an eigenvalue problem of Schr6dinger type. More precisely, there is a diffusion process which has an invariant density 1012 where 0 is an eigenfunction (cf. F~ nyes [3], Nelson [13], Nagasawa [8]; in [8] this is called" macroscopic description"). If we assume that the process of the mean-field limit will become asymptotically stationary and have an invariant density 1~ 12, then we obtain a statistical model which connects three objects; systems of interacting particles, diffusion processes and eigenvalue problems. The model will be explained briefly in w and Appendix (see [8, 9, 12] for some applications of the model). There are several interesting mathematical problems in this context:(i) Given an equilibrium distribution density of a system, find a pair-interaction between particles of the system.(ii) Given a (pair) interaction, prove the existence of the diffusion process in the mean-field limit.(iii) Prove the law of large numbers (propagation of chaos).(iv) Investigate the asymptotic behaviour of the system as t--+ oo. The problem (ii) and (iii) were discussed by McKean [7] under the assumption of the Lipschitz continuity of the interaction (for related limit theorems see Tanaka [16] and Sznitman [14], and also Dawson [1]). However, this assumption is too strict for our purpose, because the interactions which produce equilibrium distributions of excited states are not Lipschitz continuous and have singularities that are caused by zeros of the eigenfunctions.