By S.K. Srinivasan

ISBN-10: 0444000518

ISBN-13: 9780444000514

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**Additional resources for Stochastic Theory and Cascade Processes**

**Example text**

They obtained a stochastic bifurcation scenario similar to the deterministic one. Arnold and Schmalfuß [3] add a nonlinear smooth perturbation to the drift term in (1) (see Example 2). Then they use a type of random ﬁxed point theorem based on the negativity of Lyapunov exponents to state growth conditions on the perturbation under which the bifurcation pattern is preserved. In [14], Xu considers (1) and (2) with real noise and shows that under certain conditions this leads to bifurcation patterns diﬀering from the deterministic ones.

Thus for (37) the question of stability along trajectories is exactly the same as the question of stability of the ﬁxed point 0. In particular the corresponding ˜ and λ agree whenever λ > 0. Lyapunov exponents λ In this section we study this situation from the viewpoint of stochastic ﬂows of diﬀeomorphisms. e. U0 (x) = a0 + A0 x for some a0 ∈ Rd and A0 ∈ L(Rd ), and the vector ﬁelds U1 , U2 , . . satisfy Uα (x) ⊗ Uα (y) = B(x, y) ∈ Rd ⊗ Rd (41) α≥1 where for all u, x, y ∈ Rd . B(x + u, y + u) = B(x, y) (42) We note that the collection U1 , U2 , .

E. f dL∗ ν = Lf dν. A similar statement holds for (P¯t ) and L. Let c ∈ I and m(dx) = ρ(x) dx on (I, B(I)) with ρ(x) = 2 exp 2 |σ(x)| x x c b(y) dy . σ 2 (y) (11) c Here we use the convention c · = − x · for x < c, valid for Lebesgue integrals. The σ–ﬁnite measure m on (I, B(I)) is called speed measure of ϕ. The speed measure of ψ is given by m(dx) = ρ(x)dx with ρ(x) = 2 exp 2 |σ(x)| x c −b(y) dy . σ 2 (y) (12) The speed measure depends on the real number c ∈ I. But the ﬁniteness of m does not depend on c (see Karatzas and Shreve [11, p.

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