The full law of the Dirichlet-to-Neumann map determines the law of the random potential, but its expectation or finite moments need not determine even the mean, while the averaged Green's operator determines the pointwise mean and variance.
Inverse problem for a random Schr¨ odinger equation with unknown source and potential.Mathematische Zeitschrift, 304(2):Paper No
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A note on several inverse problems with generally random coefficients
The full law of the Dirichlet-to-Neumann map determines the law of the random potential, but its expectation or finite moments need not determine even the mean, while the averaged Green's operator determines the pointwise mean and variance.