Covariance kernel ($c_X(s,t)$)

When a covariance operator on a function space is an integral operator, its covariance kernel satisfies $(C_{X}f)(s)=\int c_X(s,t)f(t)dt$. For a centered process, $c_X(s,t)=\mathbb{E}[X(s)X(t)]$ whenever point evaluation is meaningful.

The covariance kernel of standard Brownian motion on $[0,1]$ is $c(s,t)=\min (s,t)$. Its integral operator has eigenfunctions $\sqrt{2}\sin ((k-\frac{1}{2})\pi t)$ and eigenvalues $((k-\frac{1}{2})\pi {)}^{-2}$.

For a complete orthonormal basis of $L^2({\mathbb{R}}_{+})$, the Parseval identity for a Hilbertian basis identifies the sum of the feature products with $⟨1_{[0,s]},1_{[0,t]}⟩=\min (s,t)$. The Cauchy-Schwarz inequality gives absolute convergence.

The covariance kernel of a standard Brownian bridge on $[0,1]$ is $c(s,t)=\min (s,t)-st$. Its integral operator has eigenfunctions $\sqrt{2}\sin (k\pi t)$ and eigenvalues $(k\pi {)}^{-2}$ for $k\ge 1$.