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By Born M.

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Rλ f (x) = IEx e−λt f (Xt )dt , x ∈ Rn , 0 for sufficiently integrable f on Rn , where IEx denotes the conditional expectation given that {X0 = x}. It satisfies the resolvent equation Rλ − Rµ = (µ − λ)Rλ Rµ , λ, µ > 0. We refer to [Kal02] for the following result. 2. Let (Tt )t∈R+ be a Feller semigroup on C0 (Rn ) with resolvent Rλ , λ > 0. Then there exists an operator A with domain D ⊂ C0 (Rn ) such that λ > 0. 3) Rλ−1 = λI − A, The operator A is called the generator of (Tt )t∈R+ and it characterizes (Tt )t∈R+ .

Next is a constructive approach to the definition of Brownian motion, using the decomposition ∞ 1[0,t] = t en n=0 en (s)ds. 0 34 N. 8 1 Fig. 8 Sample paths of one-dimensional Brownian motion. 5. For all t ∈ R+ , let ∞ Bt (ω) := J1 (1[0,t] ) = t ξn (ω) n=0 en (s)ds. 4) cf. Figure 8. 2) shows that if u1 , . . , un are orthogonal in L2 (R+ ) then J1 (u1 ), . . 1 of Jacod and Protter [JP00], we get the following. 6. Let u1 , . . e. ui , uj L2 (R+ ) = 0, 1 ≤ i = j ≤ n. Then (J1 (u1 ), . . , J1 (un )) is a vector of independent Gaussian centered random variables with respective variances u1 2L2 (R+ ) , .

E. µs,t depends only on the difference t−s, and we will denote it by µt−s . In this case the family (T0,t )t∈R+ is denoted by (Tt )t∈R+ . It defines a transition semigroup associated to (Xt )t∈R+ , with Tt f (x) = IE[f (Xt ) | X0 = x] = Rn f (y)µt (x, dy), x ∈ Rn , and satisfies the semigroup property Tt Ts f (x) = IE[Ts f (Xt ) | X0 = x] = IE[IE[f (Xt+s ) | Xs ] | X0 = x] = IE[IE[f (Xt+s ) | Fs ] | X0 = x] = IE[f (Xt+s ) | X0 = x] = Tt+s f (x), which can be formulated as the Chapman-Kolmogorov equation µs+t (x, A) = µs ∗ µt (x, A) = Rn µs (x, dy)µt (y, A).

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