Abstract
‘Numbering (in italics). (1) means formula 1 in the present section; (2.1) means formula (1) of Section 15.2, etc.’
Fulbright grantee 1957–1958 during which time the major part of this material was obtained; the support of the Office of Naval Research, U.S. Govt. during the summer of 1961 is gratefully acknowledged also.
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Notes
- 1.
B(R n) is the usual topological Borel field of the n-dimensional euclidean space R n.
- 2.
C d(R 1) is the space of bounded continuous functions \(f: R^{1} \rightarrow R^{1}\) with d bounded continuous derivatives.
- 3.
\(C^{2}[0, +\infty )\) is the space of functions \(u \in C[0, +\infty )\) with \(D^{2}u \in C(0, +\infty )\) and \((D^{2}u)(0) \equiv (D^{2}u)(0+)\) existing. \(u^{+}(0) =\lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}[u(\varepsilon ) - u(0)]\).
- 4.
\(a \wedge b\) is the smaller of a and b. \(\int _{0+}\) means \(\int _{0<l<+\infty }\).
- 5.
\(\mathfrak{p}\mathfrak{p}^{-1}\mathfrak{t}^{+}\) means \(\mathfrak{p}(\mathfrak{p}^{-1}(\mathfrak{t}^{+}))\).
- 6.
\(\mathbf{B}[\mathfrak{q}(t): a \leqq t < b]\) means the smallest Borel subfield of \(\mathbf{B}\) measuring the motion indicated inside the brackets.
- 7.
\((\mathfrak{m} < t)\) is short for \((w: \mathfrak{m} < t)\).
- 8.
\(\infty \) is an extra state \(\not\in R^{1}\).
- 9.
\(\mathfrak{m}_{0+}^{\bullet }\) is identical in law to the standard Brownian passage time \(\mathfrak{m}_{0} =\min (t: \mathfrak{x}(t) = 0)\), and hence \(E_{l}^{\bullet }(\exp (-\alpha \mathfrak{m}_{0+}^{\bullet })) =\exp (-(2\alpha )^{1/2}l)\) (see (6)).
- 10.
\(\mathfrak{G} = D^{2}/2\).
- 11.
\((\alpha -\mathfrak{G}^{\bullet })G_{\alpha }^{\bullet } = 1\).
- 12.
\(\int _{0}f(l \wedge 1)D^{-1}p_{\varepsilon }(dl)\) converges as \(\varepsilon \downarrow 0\) to \(\int fp_{{\ast}}(dl)\) extended over \([0, +\infty ]\) for each \(f \in C[0, +\infty ]\).
- 13.
\(P_{\infty }[\mathfrak{x}^{\bullet } \equiv \infty ] = 1\) as usual.
- 14.
\(\mathfrak{f}^{-1}\) is the inverse function of \(\mathfrak{f}\).
- 15.
\(\mathfrak{t}^{+}(dt) = 0\) off \(\mathfrak{Z}^{+} = (t: \mathfrak{x}^{+}(t) = 0)\).
- 16.
17
- 17.
18
- 18.
19
- 19.
See also K. Itô [8].
- 20.
\(\mathfrak{p}\mathfrak{p}^{-1}\mathfrak{t}^{+}(t)\) is short for \(\mathfrak{p}(\mathfrak{p}^{-1}(\mathfrak{t}^{+}(t)))\).
- 21.
\(a \vee b\) is the larger of a and b.
- 22.
\(\partial \mathfrak{Q}^{+}\) denotes the boundary of \(\mathfrak{Q}^{+}\).
- 23.
\(\#(l_{n}: \text{etc.})\) denotes the number of jumps l n with the properties described inside.
- 24.
\(\mathfrak{x}^{\bullet } = \mathfrak{x}^{+}\) and \(\mathfrak{t}^{\bullet } = 0\) up to time \(\mathfrak{m}_{0} =\min (t: \mathfrak{x}^{+} = 0)\), and \(\mathfrak{x}^{\bullet }\) starts afresh at that moment.
- 25.
p 3∕(p 1 +α p 3) = 0 if p 3 = 0.
- 26.
G α + is the reflecting Brownian Green operator.
- 27.
\(u^{-}(+1) =\lim _{\varepsilon \downarrow 0}\varepsilon ^{-1}[u(1) - u(1-\varepsilon )]\).
- 28.
P. , E. , \(\mathfrak{x}\), \(\mathfrak{m}\) are the standard Brownian probabilities, expectations, sample paths, and passage times.
- 29.
\(C^{\bullet 2}(R^{1}) = C^{2}(-\infty, 0] \cap C^{2}[0, +\infty ) \cap (u: u''(0-) = u''(0+))\).
- 30.
\(\vert \mathfrak{Z}\vert = 0\).
- 31.
\(\vert [0, +\infty ) - \mathfrak{Q}^{+}\vert = 0\) because \(\mathfrak{p}(t)\) has no linear part p 2 t.
- 32.
Z 1 is the integers.
- 33.
\(C^{\bullet 2}(S^{1}) = C(S^{1}) \cap C^{2}(S^{1} - 0) \cap (u: u''(0-) = u''(0+))\).
- 34.
See J. L. Doob [1].
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Itô, K., McKean, H.P. (2015). Brownian Motions on a Half Line. In: Grünbaum, F., van Moerbeke, P., Moll, V. (eds) Henry P. McKean Jr. Selecta. Contemporary Mathematicians. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22237-0_15
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