Since writing about the simulation of fractional Brownian motion, I’ll spend an hour to write RKHS.
Definition: A Hilbert space is a RKHS if
Theorem: If is a RKHS, then for each there exists a function (called the representer of ) with the reproducing property
Definition: is a reproducing kernel if it’s symmetric and positive definite.
Theorem: A RKHS defines a corresponding reproducing kernel. Conversely, a reproducing kernel defines a unique RKHS.
Once we have the kernels, if ,
When it comes to the fractional Brownian motion
Theorem: For fBm RKHS is symmetric and positive definite, , there exists s.t
Take Wiener integral as an example:
For example, when , , we can check
Claim: By previous theorem, the reproducing kernel uniquely defines a RKHS with the inner product