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# Brief Introduction to White Noise Analysis

## Contact

**Prof. Kuo works on three areas in stochastic analysis:**:

- White noise analysis,
- Stochastic differential equations, and
- Probability theory on infinite dimensional spaces.

For the third area, he wrote a very popular book in 1975, *Gaussian measures in Banach spaces* (Lecture Notes in Math., volume 463, Springer-Verlag). This book has influenced much of the research in infinite dimensional analysis.

For the second area, he is currently writing a book with M. de Faria and L. Streit, *Stochastic integration with applications*. The book is expected to be published in the year 2000.

For the first area, he has made fundamental contributions. He has coauthored with T. Hida, J. Potthoff, and L. Streit the book, *White Noise: An Infinite Dimensional Calculus* (Kluwer Academic Publishers, 1993). More recently he wrote a book in 1996, *White Noise Distribution Theory* (CRC Press, Boca Raton) to give a friendly presentation of white noise analysis.

Kuo's recent work is concentrated in the theory of white noise. This theory was initiated by T. Hida in 1975. It is nowadays regarded as an infinite dimensional distribution theory and has applications to physics, quantum probability, stochastic integration, biology, control theory, among others.

Below is a brief introduction to white noise theory and some crucial ideas to give its mathematical meaning.

## What is white noise?

White noise is a sound with equal intensity at all frequencies within a broad band. Rock music, the roar of a jet engine, and the noise at a stock market are examples of white noise. We use the word “white” to describe this kind of noise because of its similarity to “white light” which is made up of all different colors (frequencies) of light combined together. In applied science white noise is often taken as a mathematical idealization of phenomena involving sudden and extremely large fluctuations.

## White noise as the derivative of a Brownian motion

White noise can be thought of as the derivative of a Brownian motion. But what is a Brownian motion? As is well-known, Robert Brown made microscopic observations in 1827 that small particles contained in the pollen of plants, when immersed in a liquid, exhibit highly irregular motions. This highly irregular motion is called Brownian motion. Mathematically, a Brownian motion is a continuous stationary stochastic process $B(t)$ having independent increments and for each $t$, $B(t)$ is a Gaussian random variable with mean $0$ and variance $t$. It can be shown that $B(t)$ is nowhere differentiable, a mathematical fact explaining the highly irregular motions that Robert Brown observed. This means that white noise, being thought of as the derivative $dB(t)/dt$ of $B(t)$, does not exist in the ordinary sense.

## White noise as a generalized stochastic process

In order to motivate a mathematical definition of white noise, we need a comparison between functions and stochastic processes. An (ordinary) function is a function $f(t)$ of a real number $t$. A generalized function is a function $f[u]$ of a test function $u$. An (ordinary) stochastic process is a function $X(t)$ of $t$ such that for each $t$, $X(t)$ is a random variable. Thus a generalized stochastic process is a function $X[u]$ of $u$ such that for each $u$, $X[u]$ is a random variable. White noise is defined as a generalized stochastic process $X[u]$ such that for each $u$, the random variable $X[u]$ is Gaussian with mean $0$ and variance the integral of $u$-square.

## White noise as an infinite dimensional generalized function

Defining white noise as a generalized stochastic process is not so satisfactory because its sample path property is lost and nonlinear functionals of white noise cannot be defined in a unified way. To overcome these difficulties, T. Hida introduced in 1975 the theory of white noise. In this theory, for each $t$, the white noise $dB(t)/dt$ is a generalized function on an infinite dimensional space. Not only $dB(t)/dt$, but also all derivatives of Brownian motion are generalized functions on the same space.

## An application of white noise to stochastic integration

Informal integrals involving white noise had already appeared before Ito introduced the stochastic integral (called Ito's integral nowadays) in 1944. Ito combined $dB(t)/dt$ with $dt$ to get $dB(t)$ as an integrator and defined stochastic integrals of nonanticipating stochastic processes with respect to $B(t)$. The celebrated Ito's formula is the ordinary chain rule plus a correction term. It has applications to almost all applied sciences involving sudden and extremely large fluctuations. For example, P. Samuelson in 1970 and R. Merton and M. Scholes in 1997 won Nobel Prizes in economics for their works where Ito's theory plays an essential role. From the white noise point of view, we do not need to combine $dB(t)/dt$ and $dt$ together. In fact, recent applications of white noise to quantum probability show that it is better not to do so. An important fact for white noise integration is that we do not need to assume the nonanticipating property for integrands. For more information, see Kuo's 1996 CRC Press book.