Unendlich-dimensionale Wienerprozesse mit Wechselwirkung

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