Unendlich-dimensionale Wienerprozesse mit Wechselwirkung

Abstract

Suppose Ф is a superstable pair potential on ℝϒ with finite range and three times continuously differentiable and Ω i(t) (i=1,2,⋯; t≧0) are independent ϒ-dimensional standard-Wiener-processes related to a probability space (Ω, $$\mathfrak{F}$$ ), Ω: = (Ω i)i=1,2,⋯ The atoms a iεℝϒ (i = 1,2,⋯) of an element aεℳ, the space of the Radon counting measures on ℝv, may move according to the following equations (G) $$x_i (t;a,\omega ) = a_i + \int\limits_o^t {ds( - \tfrac{1}{2}\sum\limits_{j \pm i} {grad} } \Phi (x_i (s;a,\omega ) - x_j (s;a,\omega ))) + \omega _i (t)$$ $$(\omega \in \Omega ; i = 1,2,...; t\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \geqslant } 0)$$ (ΩεΩ; i = 1,2,⋯; t≧0). Let ∇ the σ-algebra generated by the vague topology on ℳ, Μ a probability measure on (ℳ, ∇). Call a ℳ-valued stochastic process x(t) = x i(t): i = 1,2,⋯ (x i(t)εℝv, t≧0,iεIN) on (ℳ × Ω, ∇, Μ ⊗ Q) a Μ-solution of (G), if x(t) satisfies some measurability conditions and the x i(t; a, Ω) (i= 1, 2,⋯; t ≧0) satisfy the equations (G) for Μ ⊗ Q-almost every (a, Ω). Call a Μ-solution x(t) Μ ⊗ Q-invariant, if Μ ⊗ Q x(t)εB=Μ(B) for all t≧0 and Bε∇Be93. Then for all tempered Gibbs-measures Μ associated to Ф holds: A Μ-solution of (G) exists. There is a Μ ⊗ Q-a.e. unique Μ ⊗ Q-invariant solution of (G). The Μ ⊗ Q-invariant solution x(t) is a reversible markov process. This result is the starting point for more research in a forthcoming paper [4].