Random processes with smooth paths

Question

Is there any prototypical example of a Random process with smooth paths?

I imagine one can simple integrate each path of a Brownian motion and get a $C^{\frac32-\epsilon}$ path.

It's easy to describe paths of Brownian bridges over a circle using Fourier expansion and the normal distribution:

$$ B(w, x) = \sum_{n \in \mathbb{Z}} C_n(w) e^{inx} $$

• $C_n \sim \mathcal{N}(0,1)$ we obtain White Noise.
• $C_n \sim \mathcal{N}(0,\left< n \right>^{-2})$ we obtain a Brownian Bridge, paths are a.s. in $H^{\frac12-\epsilon}$
• $C_n \sim \mathcal{N}(0,\left< n \right>^{-4})$ paths are a.s. in $H^{\frac32-\epsilon}$, so we now have at least one derivative.
• $C_n \sim \mathcal{N}(0,\left< n \right>^{-2k})$ paths are a.s. in $H^{\frac{2k-1}{2}-\epsilon}$, so we now have at least $k$ derivatives.
• For a $C^\infty$ maybe with $C_n \sim \mathcal{N}(0,e^{-\left< n \right>})$ paths would be a.s. in $C^\infty$?.

I can't find any literature about non-trivial random processes with smooth paths.

Do you know of any literature related with it?

I'm trying to understand something on set-valued functions, and the way an integral for such functions is defined seems related to random processes (http://econweb.ucsd.edu/~rstarr/201/Aumann%20Journal%20of%20Mathematical%20Analysis%20and%20Applications%201965.pdf).

Thanks.

Answer 1

A good place to read about this is Adler and Taylor's book Random Fields and Geometry. Regular random processes or functions are used more frequently in integral geometry.

Consider the more general Gaussian process on $S^1$ $\newcommand{\bZ}{\mathbb{Z}}$ $\newcommand{\ii}{\boldsymbol{i}}$

$$ F(\theta)=\sum_{n\in\bZ} C_n e^{n\ii \theta}, $$

where $C_n$ are independent centered Gaussian random variables. Then $F$ is a.s. smooth if $\newcommand{\vfi}{\varphi}$ $\newcommand{\bE}{\mathbb{E}}$ $\newcommand{\var}{\boldsymbol{var}}$ the covariance kernel

$$K(\theta,\vfi)=\bE\bigl[ F(\theta)\overline{F(\vfi)}\bigr]= \sum_{n\in\bZ}\var[C_n] e^{\ii n(\theta-\vfi)} $$

is smooth. This happens if

$$\sum_{n\in\bZ} n^{2s}\var[C_n]<\infty,\;\;\forall s>0. $$

For stationary processes like this one this condition is also necessary.

Answer 2

One way to get a $C^{\infty}$ path from a Brownian motion, without changing the original path too much, is to mollify it.