Background: the prototypical example of---and way to generate---smooth noise is by smoothing a one-dimensional white noise process with a Gaussian kernel.
My question: beyond smoothness, does one-dimensional white noise convolved with a Gaussian kernel have sample paths that are analytic functions with probability 1?
Definitions:
- White noise: White noise $\omega_t$ is a mean-zero stationary Gaussian process on $t\in \mathbb R$ with autocovariance function given by: $\mathbb E[\omega_t \omega_s]= \delta(t-s)$, where $\delta$ is the Dirac delta.
- Gaussian kernel: is the function $\Phi_t = e^{-\beta t^2}$ for all $t \in \mathbb R$ and some parameter $\beta >0$.
- White noise convolved with a Gaussian kernel: We consider the process $w_t=\int_{\mathbb R} \omega_s \Phi_{t-s} d s$
Work so far:
- Properties of $ w_t$: this is a mean-zero Gaussian process with autocovariance given by
$$\mathbb E[w_t w_s] =e^{-\frac\beta 2(t-s)^2}\sqrt{\frac{\pi}{2 \beta}}$$
Because the process is wide-sense stationary and Gaussian it is also stationary. It also has smooth sample paths:
$$\frac{d^k}{dt^k}w_t=\int_{\mathbb R} \omega_s \frac{d^k}{dt^k} \Phi_{t-s} d s$$
- For any $k \in\mathbb N$, the derivative process $\frac{d^k}{dt^k}w_t$ is a mean zero stationary Gaussian process with autocovariance
$$\mathbb E[\frac{d^k}{dt^k}w_t \frac{d^k}{ds^k}w_s]= (-1)^k\frac{d^{2k}}{dt^{2k}} \mathbb E[w_t w_s].$$
- From equivalent characterizations of analyticity: since $ w_t$ is smooth, it is analytic on an open set $D \subset \mathbb R$ if and only if, for each compact $K \subset D$, there exists $ C >0$ such that for all $t \in K$ and $ k \in \mathbb N$:
$$\mid \frac{d^k}{dt^k}w_t \mid \leq C^{k+1} k!$$
- Since the sample paths of $w_t$ are smooth, then for each compact set $K \subset \mathbb R$ and each $k \in \mathbb N$, the path $$\frac{d^k}{dt^k}w_t, t\in K$$ is bounded. The issue I have is in showing whether or not the bounding constants for these paths do not grow too fast as a function of $k$. Any help, pointers or references are appreciated.