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 nondissipative 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.