Constructing Nonhomeomorphic Stochastic Flows by R. W. R. Darling

By R. W. R. Darling

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6. 4) is also an FRPM on B(H. Proof. 1. First, Q * R is a Radon probability measure by definition. 10. 3, (iii). 2(b), this proves that Q * R has almost no fixed points of discontinuity. For (iii), let E be a Borel set in M with |i(E) = 0, and let x be a point in M. 1) and the fact that R ({g : g(x) e T}) = 1 . 7. Suppose Qv ... ,Qn are FRPM's on B ( r ) . 6, we may define the convolution Qj * ... *Qn = (Ql*-*Qn-l)*Qif (8-8> Moreover (i) Every such composition convolution is an FRPM. , fn) = fx o ...

Then the map c : T2 —> T, c(f,g) = f og (composition), is Q ® R - measurable (Radon product) into B 0 (r). 3 Suppose Q and R are FRPM'son BQT). Then (i) Hxand Cx belong to 3(T2)Jorall x in T, (ii) Q ® R(HX) = I for all x in T, and (iii) Q ® R(CX) = 1 for all [\i] - almost all x in T. Proof, (i) Define a mapping 0x : T2 —> M x T by 0x(f,g) = (g(x),f). It is easy to verify that 0x is a continuous map. 1(i). Therefore Hxc e 3(T2). 4) and Fx is in B(H; thus Cx is in B(r 2 ). 2)). 1) implies that R(Ax v) = 1, where y A x = { g e r : g ( x ) E J).

Zk) and Zj < z 2 < ... ^Z l (z k )) = l. 1 . 1, with the same finite-dimensional distributions. 1; in particular, (x,co) -» Xst(x,co) is m <8> P - measurable (m = Lebesgue measure ). Proof . 2. 2). Fix x in IR *, t > 0, and e > 0. Suppose x -1/n < zx< z2 < ... < z{ = x < ... < z^ < x + 1/n. 2) < z \ n Zt(zk) - Zt(x)] + E[ Zt(x) - Zt(Zl)]] <2/(en). 2. 2 . • Part V Covariance functions and the corresponding sets of finite-dimensional motions. In section 10 we gave assumptions on a set of finite-dimensional motions that allow the construction of a convolution semigroup of probability measures.

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