The purpose of this note is to give a complete and detailed proof of the fundamental Yamada-Watanabe Theorem on infinite dimensional spaces, more precisely in the framework of the variational approach to stochastic partial differential equations. Framework and definitionsLet H be a separable Hilbert space, with inner product ·, · H and norm · H . Let V, E be separable Banach spaces with norms · V and · E , such that V ⊂ H ⊂ E continuously and densely. For a topological space X let B(X) denote its Borel σ-algebra. By Kuratowski's theorem we have that V ∈ B(H), H ∈ B(E) and B(V ) = B(H)∩V , B(H) = B(E)∩H.Setting x V := ∞ if x ∈ H \ V , we extend · V to a function on H. We recall that this extension is B(H)-measurable and lower semicontinuous (cf. e.g. [4, Exercise 4.2.3]). Hence the following path space is well-defined:equipped with the metricObviously, (B, ρ) is a complete separable metric space. Let B t (B) denote the σ-algebra generated by all maps π s : B → H, s ∈ [0, t], where π s (w) := w(s), w ∈ B. For t 0 and w ∈ B define the stopped path w t by w t (s) = w(s ∧ t), s 0.Below we shall use the following elementary, but useful measure theoretic facts.Lemma 1.1. The map w → w t is B t (B)/B(B) measurable.Proof. It suffices to show that π q (w t ) is B t (B)/B(H) measurable for q 0. But π q (w t ) = π q (w) if q t, and π q (w t ) = π t (w) otherwise. In either case, we get a B t (B)/B(H) measurable map.
We study the problem of identification of a proper state-space for the stochastic dynamics of free particles in continuum, with their possible birth and death. In this dynamics, the motion of each separate particle is described by a fixed Markov process M on a Riemannian manifold X. The main problem arising here is a possible collapse of the system, in the sense that, though the initial configuration of particles is locally finite, there could exist a compact set in X such that, with probability one, infinitely many particles will arrive at this set at some time t > 0. We assume that X has infinite volume and, for each α ≥ 1, we consider the set Θ α of all infinite configurations in X for which the number of particles in a compact set is bounded by a constant times the α-th power of the volume of the set. We find quite general conditions on the process M which guarantee that the corresponding infinite particle process can start at each configuration from Θ α , will never leave Θ α , and has cadlag (or, even, continuous) sample paths in the vague topology. We consider the following examples of applications of our results: Brownian motion on the configuration space, free Glauber dynamics on the configuration space (or a birth-and-death process in X), and free Kawasaki dynamics on the configuration space. We also show that if X = R d , then for a wide class of starting distributions, the (non-equilibrium) free Glauber dynamics is a scaling limit of (non-equilibrium) free Kawasaki dynamics. 2000
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