## The Annals of Probability

- Ann. Probab.
- Volume 6, Number 3 (1978), 433-458.

### A Signed Measure on Path Space Related to Wiener Measure

#### Abstract

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 $(-1)^{n + 1}(2n)!$.

#### Article information

**Source**

Ann. Probab., Volume 6, Number 3 (1978), 433-458.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995529

**Digital Object Identifier**

doi:10.1214/aop/1176995529

**Mathematical Reviews number (MathSciNet)**

MR490812

**Zentralblatt MATH identifier**

0378.60030

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60G20: Generalized stochastic processes

Secondary: 35K25: Higher-order parabolic equations 60H05: Stochastic integrals 60F05: Central limit and other weak theorems

**Keywords**

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

#### Citation

Hochberg, Kenneth J. A Signed Measure on Path Space Related to Wiener Measure. Ann. Probab. 6 (1978), no. 3, 433--458. doi:10.1214/aop/1176995529. https://projecteuclid.org/euclid.aop/1176995529