stochastic calculus - Relation between regularities of the trajectory of a mean zero gaussian process and its covariance operator

Suppose we already know that the process is continuous with probability one, so that the process takes values in the Banach space $X = C([0,1])$ with distribution $mathbb P$. The covariance operator $C : X^* to X$ is then a map from the dual space $X^*$ to $X$. The support of the Gaussian measure $mathbb P$ is then the closure of the image of $X^*$ under $C$:

$$operatorname{supp} mathbb P = overline{ CX^* }.$$ (This is the main theorem of [Vakhania 1975])

Now let's construct the Cameron-Martin space. The operator $C$ defines an inner product on the dual space $X^*$ by

$$langle f, g rangle = f(Cg)$$ for $f, g in X^*$. The space $X^*$ isn't necessarily closed under the topology induced by the inner product, so let $H$ be the Hilbert space completion, and let $iota : X^* hookrightarrow H$ be the inclusion map. Define a map $iota^* : H hookrightarrow X$ first on the dense subspace $iota X^* subseteq H$ by $$iota^*(iota f) = Cf,$$ and extend continuously to all of $H$. Thus the covariance operator factors as $C = iota^* circ iota$.

The Hilbert space $iota^* H$ is a subspace of $X$, and is called the Cameron-Martin space of the process. Interpreting this in the context of the support of the Gaussian measure $mathbb P$, we have

$$operatorname{supp} mathbb P = overline{iota^* H},$$ so that the closure of the Cameron-Martin space (with respect to the original norm of $X$) is exactly the support of $mathbb P$.

I go into these ideas in more detail in Section 2 of my preprint [LaGatta 2010].

Suppose your covariance operator is an integral operator with kernel $c(s,t)$, called the covariance function of the process. That is, if $mu$ is a Radon measure on $[0,1]$, then

$$(Cmu)(s) = int_0^1 c(s,t) , dmu(t).$$

If we write $c_s(t) = c(s,t)$, then the support of $mathbb P$ is the closure of the span of the functions $c_s$ in $C([0,1])$. So to answer your question: if you know that the closure $overline{operatorname{span}{c_s}}$ is a space with regularity property $P$, then the process $xi_t$ satisfies property $P$ with probability one.

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