Let $X$ be an infinite dimensional separable Hilbert Space with norm $||\cdot||$ and let $\mu$ be a Gaussian measure on $X$ such that $\mu(X) = 1$. What do we know about $\mu(B(0,1))$, where $B(0,1)$ is the unit ball w.r.t the norm?


This seems to me like a fundamental question but I cannot seem to find anything. Any information/references would be most appreciated.


EDIT: A related question which is of interest to me: Do there exist asymptotically tight bounds to 
$\int\_{||u||> K}||u||^2 \mu(du)$?



What is known about the Gaussian measure of the unit ball in a Hilbert Space?
You can't talk about "the" Gaussian measure on an infinite-dimensional Hilbert space, for the same reason that you can't talk about a uniform probability distribution over all integers. It doesn't exist; see Richard's answer. However, there are a lot of non-uniform Gaussian measures on infinite dimensional Hilbert spaces. 


Consider the measure on $\mathbb{R}^\infty$ where the $j$th coordinate is a Gaussian with mean 0 and variance $\sigma\_j^2$, where $\sum\_{j=1}^{\infty} \sigma\_j^2 < \infty$ (and different coordinates are independent). This is almost surely bounded in the $\ell\_2$ metric, and any projection onto a finite-dimensional space has a Gaussian distribution. The squared length of a vector drawn from this measure is a sum of squares of Gaussians, and so follows some kind of generalized $\chi$-square distribution. If I knew more about generalized $\chi$-square distributions, I might be able to tell you what the measure of the unit ball was.


This kind of Gaussian distribution is very important in quantum optics. In fact, in quantum optics, a thermal state is Gaussian, so "the" Gaussian measure actually makes some sense.


