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 the wave equation with a white noise source.Communications in Mathematical Physics, 332(3):933–953, 2014
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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.