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

Therefore, .

**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 ,

then

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:

where ,

and

For example, when , , we can check

**Claim**: By previous theorem, the reproducing kernel uniquely defines a RKHS with the inner product