By Giuseppe Da Prato

In this revised and prolonged model of his path notes from a 1-year path at Scuola Normale Superiore, Pisa, the writer offers an advent вЂ“ for an viewers figuring out easy practical research and degree concept yet no longer inevitably likelihood conception вЂ“ to research in a separable Hilbert house of endless measurement.

Starting from the definition of Gaussian measures in Hilbert areas, thoughts reminiscent of the Cameron-Martin formulation, Brownian movement and Wiener critical are brought in an easy way.В These suggestions are then used to demonstrate a few simple stochastic dynamical structures (including dissipative nonlinearities) and Markov semi-groups, paying unique cognizance to their long-time habit: ergodicity, invariant degree. right here primary effects just like the theorems ofВ Prokhorov, Von Neumann, Krylov-Bogoliubov and Khas'minski are proved. The final bankruptcy is dedicated to gradient structures and their asymptotic behavior.

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**Example text**

16. We note that if f ∈ C 1 ([0, T ]) it is possible to express the Wiener integral 0T f (s)dB(s) in terms of a Riemann integral as the following integration by parts formula shows. 12 If f ∈ C 1 ([0, T ]) we have T T f (s)dB(s) = f (T )B(T )− 0 f (s)B(s)ds 0 in L2 (Ω, F , P). 19) Proof. Let σ = {0 = t0 < t1 < · · · < tn = T } ∈ Σ(0, T ). Then we have n f (tk−1 )(B(tk ) − B(tk−1 )) Iσ (f ) = k=1 n (f (tk )B(tk ) − f (tk−1 )B(tk−1 )) = k=1 n − (f (tk ) − f (tk−1 ))B(tk ) k=1 n (f (tk ) − f (tk−1 ))B(tk ) = f (T )B(T ) − k=1 n f (αk )B(tk )(tk − tk−1 ), = f (T )B(T ) − k=1 where αk is a suitable number lying in the interval [tk−1 , tk ], k = 1, .

If n > k the identity above is equivalent to R∞ ϕ(x)ν(dx) = R∞ ϕ(x)fn (x)µ(dx). As n → ∞ we obtain ﬁnally R∞ ϕ(x)ν(dx) = R∞ ϕ(x)f (x)µ(dx). This proves that ν µ in view of the arbitrariness of k and ϕ. Similarly we have µ ν. 4. 3 The Cameron–Martin formula Here we consider two Gaussian measures µ = NQ and ν = Na,Q on (H, B(H)), where a ∈ H and Q ∈ L+ 1 (H). 8 (i) If a ∈ / Q1/2 (H) then µ and ν are singular. 1/2 (ii) If a ∈ Q (H) then µ and ν are equivalent. dν is given by (iii) If µ and ν are equivalent the density dµ dν 1 (x) = exp − |Q−1/2 a|2 + WQ−1/2 a (x) , dµ 2 x ∈ H.

Bn (t)), is a Brownian motion in Rn . 18 Let B be a Brownian motion in Rn . Then the following properties are easy to check. (i) For all t > s > 0, B(t) − B(s) is a Gaussian random variable with law N(t−s)In , where In is the identity operator in Rn . (ii) For all t, s > 0, E(Bi (t)Bj (s)) = 0 if i = j, i, j = 1, . . , n. (iii) We have E |B(t) − B(s)|2 = n(t − s). 23) Let us check (iii). We have n E |B(t) − B(s)|2 = E |Bk (t) − Bk (s)|2 = n(t − s). 19 Prove that for 0 ≤ s < t we have E |B(t) − B(s)|4 = (2n + n2 )(t − s)2 .