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