Constructing Nonhomeomorphic Stochastic Flows (Memoirs of by R. W. R. Darling

By R. W. R. Darling

Ebook by way of Darling, R. W. R.

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Extra resources for Constructing Nonhomeomorphic Stochastic Flows (Memoirs of the American Mathematical Society)

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For the sake of brevity we will restrict ourselves to the case where M is the one-point compactification T u H of an infinite separable metric space (T,d); the case where M is compact, without a distinguished point labelled «>, may be treated in a similar way. 7, the set { ... 2) is closed, and so Fx is an F a 5 set in T, hence Borel. For each f in T, define 0(f) = U m n n {x € T: p(f(y), f(z)) > 1/m for some y,z e B(x,l/n)}. 3) Evidently 0(f) is the set of discontinuities of f in T. The set { ...

T > 0 . 5), we see that Qt(A> = ^n0 Qs * Q*"(A) p i ( N W = n) = Q s * Q t (A). 5). Proof. 1. 5. 5)). ,t), t > 0) is a Poisson process with rate X . As X tends to «>, we would expect the convolution semigroup {Q^t, t > 0} to converge in some sense to {Rt, t > 0}. Part III. Construction of a stochastic flow assuming almost no fixed points of discontinuity.

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. Now W - c"1 (A) = {(f,g): for all k, there exists y such that d(g(x),y) < 1/k, f(y) € G, but f(g(x)) e G c ). c: {(f,g): f is discontinuous at g(x)} = Hxc .

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