^{+\infty}_{-\infty}\frac{f^{(2)}_{1,e}(\omega ,\tau ^+)}{s+\xi
+i\omega }g(\omega )d\omega \ .$$ To find $R_e^{(2)*}(t)$ we take the inverse Laplace transform, $$R_e^{(2)*}(t)=\frac{1}{2\pi i}\int ^{+i\infty +\eta
}_{-i\infty+\eta }e^{st}\hat R_{e*}^{(2)}(s)ds \ , \eta >0 \ .$$ For the purposes of evaluating the integral (31), we recall that $D(s)=0$ has roots whose real parts correspond to the exponential decay rate of a response to an initial stimulus toward the $R=0$ state. Thus, as before in our discussion of the linear response (see Eq. (16)), any poles at the roots of $D(s)=0$ give contributions that we assume decay substantially faster with increasing $t>\tau $ than the diffusion induced exponential decay rate $\xi $. Since we are interested in echoes that we will find occur for $t=2\tau ,3\tau ,\ldots $, we neglect contributions to Eq. (31) from such poles. Thus it suffices to consider only the contribution to Eq. (31) from the pole at $s+\xi +i\omega =0$. Hence Eqs. (30) and (31) yield $$R_e^{(2)*}(t)\cong \int ^{+\infty}_{-\infty}e^{-(i\omega +\xi
)(t-\tau )}\frac{f^{(2)}_{1,e}(\omega ,\tau ^+)}{D[-(i\omega +\xi
)]} g(\omega )d\omega \ .$$

D. Echoes
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In order to see how Eq. (32) results in echoes, we recall our previous results, Eqs. (25), (19) and (21) for $f_{1,e}^{(2)}(\omega ,\tau ^+)$, and combine them to obtain 