By Smelyakov S.V., Stoyan Y.G.

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5 holds for a semigroup {T t } of measure preserving transformations. 2 Ergodic theory of random processes Recall that a random process is called stationary if its law is invariant under time shifts. l. processes with state space S . The space of sample paths E = D(R; S ) with the Skorohod topology is itself a Polish space. It is well known that any Polish space E can be homeomorphically embedded into a Gδ subset of [0, 1]∞ [32, p. 2]. , E is ¯ Then M := P(E) ¯ is densely continuously embedded in the compact Polish space E.

1) under the same control U ∈ U with initial conditions X0n such that X0n converges in 38 Controlled Diﬀusions H 2 as n → ∞ to some X0 . 1) under U and initial condition X0 , we have Xn − X Proof HT2 −−−−→ 0 , n→∞ ∀T > 0 . Fix n ∈ N, and for i ∈ {n, ∞}, with X ∞ ≡ X, define Xti,R := Xti I |X0n | ∨ |X0∞ | ≤ R /8 , t ≥ 0, 3 τiR := inf t > 0 : Xti,R > R , and Xˆ i,R := X i − X i,R . We write Xn − X∞ HT2 ≤ X n,R − X ∞,R HT2 + Xˆ n,R HT2 + Xˆ ∞,R HT2 . 24) Let AnR := |X0n | > R ∪ |X0∞ | > R . 19) and conditioning, we obtain Xˆ n,R 2 HT2 ≤ E I |X0n | > R E X n ≤ E IAnR E X n 2 ∞,T 2 ∞,T X0n X0n ≤ 2C˜ 12 E 1 + |X0n |2 IAnR = 2C˜ 12 P(AnR ) + X0n IAnR 2 H2 −−−−→ 0 .

Define the operator St by St f (x) := t 1 t f T s (x) ds . 3) and f ∗ = E f | I , where I is the µ-completion of the σ-field {A ∈ E : T t−1 A = A , ∀t ∈ R+ } . In particular, E f ◦ T t | I = E f | I for all t ∈ R+ . Moreover, if f ∈ L p (µ), with p ∈ [1, ∞), then St f − f ∗ −−−→ 0 . Proof Without loss of generality assume that f ≥ 0. Define f1 (x) := for x ∈ E. Then 1 f1 (x) µ(dx) = E E 0 = f (T s (x)) ds µ(dx) 0 1 = f (T s (x)) µ(dx) ds E f (x) µ(dx) . 4) L p (µ) t→∞ 1 0 f (T s (x)) ds 20 Markov Processes and Ergodic Properties Therefore f1 ∈ L1 (µ).