Given a Gaussian process on some topological space $T$, with a continuous covariance kernel $C(.,.): TxT -> R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in  , for instance, R J Adler & J E Taylor: "Random Fields and Geometry", and surely a lot of other places. We can suppose the topological space $T$ is separable.

A very rapid review of the construction is: Define an inner product space $H_0$ as consisting of all real-valued functions on $T$ of the form $f(x) = \sum_{i=1}^n a_i C(x_i,x)$, for real numbers $a_i$ and points in $T$, $x_i$. We can define an inner product on $H_0$ by $< \sum a_i C(x_i,.), \sum b_j C(y_j,.) > = \sum \sum a_i b_j C(x_i,y_j)$.

Then the reproducing kernel Hilbert space associated with our gaussian process is the completion $H$ of $H_0$.

Now, this strongly suggests (to be usefull, and by the Karhunen-Loéve theorem, which is based on this construction) that sample paths of our Gaussian process belongs to H with probability 1. This must be proved somewhere, but where? Anybody knows a reference?

Given a Gaussian process on some topological space $T$, with a continuous covariance kernel $C(\cdot,\cdot)\colon T\times T\to R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in, for instance, R J Adler & J E Taylor: "Random Fields and Geometry", and surely a lot of other places. We can suppose the topological space $T$ is separable.

A very rapid review of the construction is: Define an inner product space $H_0$ as consisting of all real-valued functions on $T$ of the form $f(x) = \sum_{i=1}^n a_i C(x_i,x)$, for real numbers $a_i$ and points in $T$, $x_i$. We can define an inner product on $H_0$ by $\left\langle\sum a_i C(x_i,\cdot), \sum b_j C(y_j,\cdot)\right\rangle = \sum \sum a_i b_j C(x_i,y_j)$.

Then the reproducing kernel Hilbert space associated with our gaussian process is the completion $H$ of $H_0$.

Now, this strongly suggests (to be usefull, and by the Karhunen-Loéve theorem, which is based on this construction) that sample paths of our Gaussian process belongs to H with probability 1. This must be proved somewhere, but where? Anybody knows a reference?

Given a Gaussian process on some topological space $T$, with a continuous covariance kernel $C(.,.): TxT -> R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in , for instance, R J Adler & J E Taylor: "Random Fields and Geometry", and surely a lot of other places.

A very rapid review of the construction is: Define an inner product space $H_0$ as consisting of all real-valued functions on $T$ of the form $f(x) = \sum_{i=1}^n a_i C(x_i,x)$, for real numbers $a_i$ and points in $T$, $x_i$. We can define an inner product on $H_0$ by $< \sum a_i C(x_i,.), \sum b_j C(y_j,.) > = \sum \sum a_i b_j C(x_i,y_j)$.

Then the reproducing kernel Hilbert space associated with our gaussian process is the completion $H$ of $H_0$.

Now, this strongly suggests (to be usefull, and by the Karhunen-Loéve theorem, which is based on this construction) that sample paths of our Gaussian process belongs to H with probability 1. This must be proved somewhere, but where? Anybody knows a reference?

Given a Gaussian process on some topological space $T$, with a continuous covariance kernel $C(.,.): TxT -> R$, we can associate a Hilbert space, which is the reproducing kernel Hilbert space of real-valued functions on $T$, with $C$ as kernel function. This contruction is given in , for instance, R J Adler & J E Taylor: "Random Fields and Geometry", and surely a lot of other places. We can suppose the topological space $T$ is separable.

A very rapid review of the construction is: Define an inner product space $H_0$ as consisting of all real-valued functions on $T$ of the form $f(x) = \sum_{i=1}^n a_i C(x_i,x)$, for real numbers $a_i$ and points in $T$, $x_i$. We can define an inner product on $H_0$ by $< \sum a_i C(x_i,.), \sum b_j C(y_j,.) > = \sum \sum a_i b_j C(x_i,y_j)$.

Then the reproducing kernel Hilbert space associated with our gaussian process is the completion $H$ of $H_0$.

Now, this strongly suggests (to be usefull, and by the Karhunen-Loéve theorem, which is based on this construction) that sample paths of our Gaussian process belongs to H with probability 1. This must be proved somewhere, but where? Anybody knows a reference?