Closed-form time-domain expressions for drift–diffusion moments

Derive closed-form time-domain expressions for the mean position and second moment of a drift–diffusion particle subject to threshold resetting, beyond the available summation and numerical-integral representations.

Background

The paper derives exact Laplace-space expressions for the first two moments of the drift–diffusion process under threshold resetting and provides time-domain representations through inverse Laplace tables. It also develops a Bromwich-integral representation for the mean position involving pole and branch-cut contributions.

The authors explicitly state that they were unable to obtain a closed-form solution, although the resulting summation expressions can be evaluated computationally. Obtaining such a closed form remains an unresolved analytical problem within the treatment of the model.

References

Although we could not find a closed-form solution, those expressions are sometimes relatively easier to compute in Mathematica as we don't need to do numerical integration as in Eq. DD_mean_il.

DD_mean_il:

⟨x(t)⟩=⟨x⟩ss+∑n=1∞Cncos⁡(ωnt+θn)e−γnt−H(t).\langle x(t)\rangle = \langle x\rangle_{\text{ss}} + \sum_{n=1}^\infty \mathcal{C}_n \cos(\omega_n t+\theta_n)e^{-\gamma_n t} - \mathcal{H}(t).

— Non-equilibrium dynamics of drift-diffusion process under threshold resetting  (2609.24438 - Das et al., 21 Sep 2026) in Appendix, Section 'Alternate method using Laplace table'