An ∞-dimensional inhomogeneous Langevin's equation

Stochastic Evolution Equations with Values on the Dual of a Countably Hilbert Nuclear Space.

“…In Section 3 we extend the previous results to stochastic evolution equations with a nuclear space valued martingale as a driving term. Section 4 contains special cases and examples recently considered by Christensen and Kallianpur (1985), Hitsuda and Mitoma (1985) and Mitoma (1985). It is important to observe that the last two examples of Section 4 are instances where the two parameter evolution semigroup T(s,t), its generator A and the perturbator P t t can all be defined directly on a countably Hilbertian nuclear space D so as to satisfy the above assumptions AI-A4.…”

“…(Adapted from Mitoma (1985)). This example is an instance wLere T(s,t), A t and Pt can all be defined directly on a countably Hilbert nuclear space D. It was recently considered by Mitoma (1985) in the case when D is obtained by modifying the space S. For the purpose of illustration we here consider S for which some simplifications are possible. Recall that the topology of S is given by the Hilbertian norms n 2 2n (k 2 (4.3)…”

“…Then if (P ) -2 is any perturbation operator on S that satisfies condi- (Hitsuda-Mitoma (1985), Mitoma (1985)). This example has been considered by Hitsuda and Mitoma (1985) and Mitoma (1985) in connection with central limit theorems for propagation of chaos (see McKean (1967)).…”

“…This example has been considered by Hitsuda and Mitoma (1985) and Mitoma (1985) in connection with central limit theorems for propagation of chaos (see McKean (1967)). dX(t) -ct(X(t),t)d~3t + (X(t),t)dt, X0=a zt where B is a one dimensional Brownian motion and a is a real valued r.v.…”

“…This example has been considered by Mitoma (1985) and Hitsuda and Mitoma (1985) in connection with the following central limit theorem: Consider the n-th interacting particle diffusion process y (t) = (Y(1)(t) .. yGn)(n)) 1 n given by the SDE x where p(t) is the probability distribution of the solution of (4.31). Let S n(t) = wInIu(n)(t)-P(t) ).…”