Abstract
Given a nice topological space Q (for example, a several-dimensional differentiable manifold), let ∞ be an extra point which is either isolated or the one-point compactification of Q according as Q is compact or not, let C∞(Q) be the space of bounded continuous functions f: Q ⋃ ∞ → R1 with f(∞) ≡ 0 , introduce the (continuous) sample paths w:t → x(t) ∈ Q ⋃ ∞ with x(t)∈Q (t + ∞) and x(+∞) ≡ ∞, define Markov times m, shifted paths w +m , and Borel algebras B and Bm+ as usual, take probabilities Pa(B) (a ∈Q⋃∞, B ∈ B) with the usual properties including P∞ [x(t) ≡ ∞ (t z 0)] = 1, and call the associated motion a diffusion if it starts afresh at each Markov time, i.e., if
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1)
Pα(wm+∈B∣Bm+)=Pb(B) a∈Q,B∈B,b=x(m),m +∞ for each Markov time m and if, in addition, it is smooth, i.e., if the Green operators\({G_\alpha }:f \to E.\left[ {\int\limits_0^{ + \infty } {{e^{ - \alpha t}}fdt} } \right]\)map C ∞ (Q) into itself:
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2)
GαC∞⊂C∞ α 0
1) and 2) are not Unrelated.
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© 1996 Springer-Verlag Berlin Heidelberg
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Itô, K., McKean, H.P. (1996). Brownian motion in several dimensions. In: Diffusion Processes and their Sample Paths. Classics in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-62025-6_8
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DOI: https://doi.org/10.1007/978-3-642-62025-6_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-60629-1
Online ISBN: 978-3-642-62025-6
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