An isonormal process is a fundamental concept in stochastic analysis and Gaussian processes. It provides a way to define a Gaussian process indexed by elements of a Hilbert space. Here’s an overview of the concept:
1. Definition
An isonormal process is a collection of random variables $\{ X(h) : h \in H \}$ defined on a probability space such that:
- $H$ is a real separable Hilbert space.
- $X(h)$ is a centered Gaussian process, meaning $\mathbb{E}[X(h)] = 0$ for all $h \in H$ .
- The covariance structure is given by:
$\mathbb{E}[X(h) X(g)] = \langle h, g \rangle_H,$
where $\langle \cdot, \cdot \rangle_H$ denotes the inner product in $H$ .
2. Properties
- Linearity: The mapping $h \mapsto X(h)$ is linear in distribution, meaning if $a, b \in \mathbb{R}$ and $h_1, h_2 \in H$ , then:
$X(ah_1 + bh_2) \stackrel{d}{=} aX(h_1) + bX(h_2).$
- Centered Gaussian Process: Each $X(h)$ is a Gaussian random variable with mean zero.
- Covariance Structure: The covariance function uniquely characterizes the process and is determined by the inner product of the Hilbert space.
3. Examples and Applications
- Wiener Process (Brownian Motion): The Wiener process $W(t)$ can be viewed as an isonormal process on the Hilbert space $H = L^2([0, T])$ . For any function $h(t) \in L^2([0, T])$ , the corresponding isonormal process $X(h)$ can be represented by the stochastic integral:
$X(h) = \int_0^T h(t) \, dW(t).$
The covariance is $\mathbb{E}[X(h) X(g)] = \int_0^T h(t) g(t) \, dt$ , which matches the $L^2$ inner product.
- White Noise: An isonormal process indexed by the space of square-integrable functions can be interpreted as a white noise process when considering $X(h)$ with rapid oscillations.
4. Mathematical Context
Isonormal processes are used extensively in the study of Gaussian measures on infinite-dimensional spaces and in the Malliavin calculus, which is a stochastic calculus of variations. They are instrumental in defining stochastic integrals with respect to Gaussian processes and in proving central limit theorems for functionals of Gaussian fields.
5. Generalization
Isonormal processes can be generalized beyond basic Gaussian processes to include various fields in probability theory and stochastic analysis:
- Hilbert Space Approach: This framework allows working with stochastic processes in more abstract settings and facilitates the use of functional analysis in stochastic analysis.
- Applications in SPDEs: In the context of stochastic partial differential equations, isonormal processes can serve as noise terms that drive the equations, aiding in the analysis and solution of such equations.
6. Typical Use Case