Isonormal Gaussian process

An isonormal Gaussian process over a real Hilbert space $H$ is a centered Gaussian process $(W(h){)}_{h\in H}$ satisfyingIt turns the geometry of $H$ into the covariance structure of a Gaussian family.

For any real separable Hilbert space $H$, any orthonormal basis $(e_j)$ and independent standard normal random variables $(Z_j)$, the displayed series converges almost surely for each fixed $h\in H$ and in L2 space. Indeed, the Parseval identity for a Hilbertian basis gives ${\sum }_j⟨h,e_j{⟩}^2=\Vert h{\Vert }^2$, so the partial sums form an L2 space-bounded martingale. The martingale convergence theorem supplies both limits. Every finite linear combination has the normal distribution with variance $\Vert {\sum }_j{u}_j{h}_j{\Vert }^2$. Thus its finite-dimensional characteristic function is $\exp (-\Vert {\sum }_j{u}_j{h}_j{\Vert }^2/2)$, independent of the chosen orthonormal basis. This proves equality of the finite-dimensional distributions of the resulting isonormal Gaussian processes; pathwise equality needs a coupling.

For an isonormal Gaussian process $X$ on a real Hilbert space $H$ and $k\in H$, the positive density $\exp (X(k)-\Vert k{\Vert }^2/2)$ has expectation one and defines an equivalent probability measure. Under this measure, $X(h)$ has mean $⟨h,k⟩$ and the same covariance as before. The shifted family $X(h)-⟨h,k⟩$ is again an isonormal Gaussian process. This follows by evaluating the joint Gaussian exponential formula for any finite collection of arguments. It is the Hilbert-space version of exponential tilting.