It is well known that the tensor product of two centered Gaussian measures is a centered Gaussian measure over $X\times X$ where $X$ is a Banach space. But I was wondering whether we have the inverse; i.e. if we have a Gaussian measure $\mu$ can we decomposed it into a tensor product of two Gaussian measures?
2026-05-06 03:14:25.1778037265
Reflection about Gaussian measures
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If an arbitrary centered Gaussian measure is given on a product of a Banach space X with itself, then it is not guaranteed to be the tensor product of two centered Gaussian measures on X. As an example, consider the one-dimensional real Banach space R. On the product space R x R (the Euclidean vector plane) consider the measure with density (up to some normalisation factor)
exp((x-y)^2 + 2(x+y)^2)
This measure diagonalises along the lines x=y and x=-y with different variances. Any product of two measures on R would have to diagonalise along the X- and Y-axis.