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

By R. W. R. Darling

Publication through Darling, R. W. R.

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2)). 1) implies that R(Ax v) = 1, where y A x = { g e r : g ( x ) E J). 2)); Q R(Hx) = J r Q(Fg(x)) R(dg) = J Ax Q(Fg(x)) R(dg) = 1. 4). 2. 5) for some x in M and some open subset of G of M. ) is open in T, so the union over y is open in T . Hence W is a Borel subset of T2. Moreover c"1 (A) is contained in W, for if (f,g) belongs to c"1 (A), then f(y) e G and d(g(x), y) = 0 when y = g(x). To show that c"1 (A) is Q ® R - measurable, it suffices to prove that W - c"1 (A) is a Q a R - nullset.

1. We shall now show that these special assumptions are not needed. l). For each t > 0, and each z in IR1, E IZt(z)l < <~ and E [Zt(z)] = z. , 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 .

Continuity on D|< follows from this. Darling 54 12. ,zk) in ([Rd)k. 4, C(k) = (cr^), 1 < m, n < kd, denotes the non-negative definite square root of B (k). W^t), t > 0) is a kd-dimensional Wiener process. Let CQ((RC1)1<) denote the C00 functions from (Rd)k to (R with compact support. Let Qkd denote the space of continous mappings co : [0,©°) —>(IRd)k with the topology of uniform convergence on compact sets. Putting Zt(co) = co(t) = cot, let Gtk =Q\ be the a-field generated by {Zs, 0 < s < t), giving aright-continuousfiltration of G = Vt Gt.