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Retrieving the Green’s function of the diffusion equation from the response to a random forcing

It is known that the Green’s function for non-dissipative acoustic or elastic
wave propagation can be extracted by correlating noise recorded at different
receivers. This property is often related to the invariance for time-reversal of
the acoustic or elastic wave equations. The diffusion equation is not invariant
for time-reversal. It is shown in this work that the Green’s function of the
diffusion equation can also be retrieved by correlating solutions of the diffusion
equation that are excited randomly and are recorded at different locations. This
property can be used to retrieve the Green’s function for diffusive systems from
ambient fluctuations. Potential applications include the fluid pressure in porous
media, electromagnetic fields in conducting media, the diffusive transport of
contaminants, and the intensity of multiply scattered waves.