Abstract
Duality and time reversal of time-homogeneous diffusion processes will be discussed in the first and second sections. Theorems in these sections will be applied to time-inhomogeneous diffusion processes in the third and fourth sections. Moreover, two different representations of a diffusion process will be established. They will play a crucial role in connection with quantum mechanics in Chapter 4.
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Notes
Cf. Schrödinger(1931)
Cf. S. Ito (1957)
Cf., e.g. Tanaka (1979), Lions-Sznitman (1984), Ikeda-Watanabe (1981, 89)
Cf. Nagasawa (1961)
See examples in Section 7.9, and also in appendix of Nagasawa-Tanaka (1985)
Cf. Faris-Simon (1975), see also Ezawa-Klauder-Shepp (1975), Glimm-Jaffe (1987)
We have shown the claim assuming transition densities, but it is not necessary, cf. Nagasawa (1964, 1970/71)
m is excessive, if ∫ mP t f ≤ ∫ mf, for any non-negative bounded measurable f
C K denotes the space of continuous function of compact support
P x = P μ with μ =δ x
From now on and in Chapter 4 the diffusion coefficient will be assumed to be time-independent. This is a technical assumption. It may be time-dependent. This case can be handled parallel assuming a gauge condition div b — (1/√σ2)(∂√σ2/∂t) = 0
In applications b is the vector potential of the electromagnetic field and denoted usually with A, cf. e.g., Lorentz (1915)
For simplicity we write dx instead of √σ2(x)dx
This is for simplicity. We can reverse the process from the last exit time, if it is natural to do so
See (3.52) and what follows
Compare with (3.6)
We assume the existence of the (weak) fundamental solution in this section. But the existence will be shown in Chapters 5 and 6
If \( \hat \varphi = \varphi \), the requirement reduces to “φ ∈ L2”
Further discussion on the construction of the diffusion process with q(s, x; t, y) will be given in Chapters 5 and 6
Our p (resp. q)-representation has nothing to do with Dirac’s “q-number”
I have called a diffusion process with the representation (3.49) the “Schrödinger process” in Nagasawa (1989,a), since it was first considered by Schrödinger, but I abandon this terminology to avoid confusion, because it is not a diffusion process of a special kind but a special representation of an arbitrary diffusion process
This is technically necessary but harmless
Cf. Lemma 3.2 of Nagasawa (1989, a)
The Schrödinger’s representation was generalized by Kuznetzov (1973) to processes with random birth and death motivated by Hunt (1960). For further discussions see Getoor-Glover (1984), Getoor-Sharpe (1984)
Motivated by Schrödinger (1931), Bernstein (1932) introduced an interesting class of conditional processes. His formulation plays, however, no role in this monograph. For an approach based on Bernstein (1932) cf. Jamison (1974,a; 1975). Influenced by Schrödinger (1931), Fényes (1952) discussed diffusion theory of quantum mechanics (cf. Appendix, and also Jammer (1974) for further literature)
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© 1993 Springer Basel AG
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Nagasawa, M. (1993). Duality and Time Reversal of Diffusion Processes. In: Schrödinger Equations and Diffusion Theory. Monographs in Mathematics, vol 86. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8568-3_3
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DOI: https://doi.org/10.1007/978-3-0348-8568-3_3
Publisher Name: Birkhäuser, Basel
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