Special Functions, Probability Semigroups, and Hamiltonian Flows
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696 Philip J. Feinsilver
Special Functions, Probability Semigroups, and Hamiltonian Flows
Springer-Verlag Berlin Heidelberg New York 1978
Author Philip J. Feinsilver Department of Mathematics Southern Illinois University Carbondale, II 62901/USA
AMS Subject Classifications (1970): 33A30, 33A65, 39A15, 42A52, 44A45, 47 010, 60H05, 60J35, 81 A20 ISBN ISBN
3-540-09100-9 0-387-09100-9
Springer-Verlag Berlin Heidelberg New York Springer-Verlag New York Heidelberg Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin Heidelberg 1978 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Summary of Contents.
Chapter I. We introduce the generator corresponding to a process with independent increments.
Assuming we have a convolution semigroup of measures satisfying
fezxpt(dx) = etL(z) where L(z) is analytic in a neighborhood of DeC, only special L's arise.
Conversely, for any L(z) analytic near DeC, L(O) = 0,
define a general translation-invariant process to have corresponding densities
pt(x)
~fe-i~xetL(i~)d~· 2TI
'
pt(x) may not be positive measures.
Our basic tools to analyze the corresponding processes w(t) such that 1 zw(t) .
tlog( e
) = L (z) , 0
denobng expected value are:
(1) Hamiltonian flows, e.g. choosing
H = L(z),
z =momentum.
(2) Iterated stochastic integrals.
Chapter II. We study the basic theory of operators analytic in
d
(x,D), D
dx
functions f(x), f(D), feS (Schwartz space) or feS * .
and of
Chapter III. We study the generalized powers x(t)n where x(t) = etHxe-tH, H a Hamiltonian such that Hl = 0.
We introduce the operator
A = x(t)z(t), a
generalization of xo. Chapter IV. We study orthogonal polynomials corresponding to a certain class of generators, which we call Bernoulli generators.
The main feature is that
the orthogonal systems are actually generalized powers and so the orthogonal series' are isomorphic to Taylor series'. Chapter V. We study in detail the five standard Bernoulli-type processes: Symmetric Bernoulli, Exponential, Poisson, Brownian Motion. familiar special functions appear.
Bernoulli,
The most
In general we see that the relevant
functions are confluent hypergeometric functions.
IV
Chapter VI. We discuss the relationships among the five standard processes.
We have
~Exponential, e(t) ~ Bernoulli Symmetric;: d(t) x(t) ~ Bernoulh
~
Poisson, N(t)
~
~
Brownian Motion b(t)
The arrows indicate limits taken by the vanishing of the indicated parameter (these parameters determine the various generators L). Chapter VII. We discuss the theory of discrete it
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