By D.K.C. MacDonald

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Baxendale trajectories. This point is a stochastic bifurcation point for the two-point motion {(ξ t (x), ξ t (y) : t ≥ 0}, and hence for the stochastic ﬂow {ξ t : t ≥ 0}, even though the law of the one-point motion has not changed. 9 References [1] L. Arnold. Random dynamical systems. Springer, Berlin Heidelberg New York, 1988. [2] L. Arnold, N. Sri Namachchivaya and K. Schenk-Hopp´e. Toward an understanding of stochastic Hopf bifurcation. Internat. J. Bifur. Chaos Appl. Sci. Engrg 6:1947–1975, 1996.

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.

8 gives µα0 = (µα0 )+ = (µα0 )− . Hence µα0 is F + – and F − –measurable. 8 gives the assertion. 11 yields the following characterization of pitchfork and transcritical bifurcations. 4. Let (ϕα )α∈R be the family of RDS induced by (17) and suppose (E) and (IC) are fulﬁlled for all α ∈ R. (ϕα )α∈R undergoes a stochastic pitchfork bifurcation at α = 0, if and only if (i) sign(bα (0)) = sign(α) for all α ∈ R, (ii) mα (I) = ∞ = mα (I) for I = I + , I − for α ≤ 0, (iii) mα (I) < ∞, mα (I \ ] − c, c[) = ∞, c ∈ I for I = I + , I − for α > 0, (iv) να := mα mα (I) −→ δ0 weakly as α ↓ 0, for I = I + , I − , where mα , mα are the speed measures of ϕα respectively ψα for α ∈ R and the constant c ∈ I in (iii) is independent of α.