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arXiv math/0311444math.PRmath-phmath.MP
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无限相互作用扩散粒子I：平衡过程及其标度极限

Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

Yuri Kondratiev, Eugene Lytvynov, Michael Röckner

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摘要：

经典连续系统的随机动力学$({\bf X}(t))_{t\ge0}$是一个随机过程，它取$\Bbb R$中所有局部有限子集（配置）的空间$Γ$中的值，并具有Gibbs测度$μ$作为不变测度。我们假设$μ$对应于对称对势$ϕ(x-y)$。一类重要的经典连续系统的随机动力学是由扩散形成的。到目前为止，只有一种类型的这样的动力学-所谓的梯度随机动力学，或相互作用的布朗粒子-已被研究。利用Dirichlet型理论，构造并研究了一类新的随机动力学，我们称之为无限相互作用扩散粒子。在$L^2(Γ;μ)$上引入了Dirichlet形式${\cal E}_μ^Γ$，并在一般条件下证明了它在位势$ϕ$上的闭性.对于一个潜在的$ϕ$有一个“弱”奇点在零，我们也写下了一个明确的形式的生成元的${\cal E}_μ^Γ$上的光滑圆柱函数。最后，我们研究了相互作用扩散的标度极限，并证明了这些标度极限在${\cal E}_μ^Γ$中是紧的，其中${\cal D}'$是${\cal D}{:=}C_0^\infty({\Bbb R})$的对偶空间。

英文摘要：

A stochastic dynamics $({\bf X}(t))_{t\ge0}$ of a classical continuous system is a stochastic process which takes values in the space $Γ$ of all locally finite subsets (configurations) in $\Bbb R$ and which has a Gibbs measure $μ$ as an invariant measure. We assume that $μ$ corresponds to a symmetric pair potential $ϕ(x-y)$. An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics--the so-called gradient stochastic dynamics, or interacting Brownian particles--has been investigated. By using the theory of Dirichlet forms, we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form ${\cal E}_μ^Γ$ on $L^2(Γ;μ)$, and under general conditions on the potential $ϕ$, prove its closability. For a potential $ϕ$ having a ``weak'' singularity at zero, we also write down an explicit form of the generator of ${\cal E}_μ^Γ$ on the set of smooth cylinder functions. We then show that, for any Dirichlet form ${\cal E}_μ^Γ$, there exists a diffusion process that is properly associated with it. Finally, we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in $C([0,\infty),{\cal D}')$, where ${\cal D}'$ is the dual space of ${\cal D}{:=}C_0^\infty({\Bbb R})$.

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