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June, 1978 A Signed Measure on Path Space Related to Wiener Measure
Kenneth J. Hochberg
Ann. Probab. 6(3): 433-458 (June, 1978). DOI: 10.1214/aop/1176995529


The connection between the heat equation and Brownian motion is generalized to a process related to the equation $\partial u/\partial t = (-1)^{n + 1} \partial^{2n}u/\partial x^{2n}, n \geqq 2$. The associated measure is of unbounded variation and signed; the process cannot be realized in the space of continuous functions. Stochastic integrals $\int^t_0 \varphi(x(s))(dx)^j(s), j = 1, 2, \cdots, 2n$, are defined, and an analogue of Ito's lemma for the Brownian integral is proven. Specifically, one gets $2n$ independent differentials $(dx)^j$, with $(dx)^{2n} = (-1)^{n + 1}(2n)! dt$. Applications include the derivation of the analogue of the Brownian exponential martingale $\exp\{\alpha x - \alpha^2t/2\}$ and a class of orthogonal functions which generalize the Hermite polynomials. These are followed by the Feynmann-Kac formula, distribution of the maximum function, arc-sine law, and distribution of eigenvalues. Finally, central limit theorems are proven for convergence of sums of independent random variables identically distributed by a signed measure, normalized to have first $2n - 1$ moments equal to zero and $2n$th moment equal to $(-1)^{n + 1}(2n)!$.


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Kenneth J. Hochberg. "A Signed Measure on Path Space Related to Wiener Measure." Ann. Probab. 6 (3) 433 - 458, June, 1978.


Published: June, 1978
First available in Project Euclid: 19 April 2007

zbMATH: 0378.60030
MathSciNet: MR490812
Digital Object Identifier: 10.1214/aop/1176995529

Primary: 60G20
Secondary: 35K25 , 60F05 , 60H05

Keywords: arc-sine law , Brownian motion (Wiener) process , central limit theorem , Feynmann-Kac formula , Hermite polynomials , Ito lemma , stochastic integral

Rights: Copyright © 1978 Institute of Mathematical Statistics

Vol.6 • No. 3 • June, 1978
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