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Convergence of the heavy-tailed underdamped slow-fast multiscale system to its averaged dynamics

Establish strong convergence of the slow component (X_t, V_t) to the averaged underdamped annealed Langevin dynamics as the scale-separation parameter ε → 0 for the multiscale system with a Student’s t heavy-tailed base distribution and a fast modified Itô diffusion dY_t = −((α + d − 1)/ε) ∇U(Y_t) dt + √(2 U(Y_t)/ε) dŴ_t targeting the conditional distribution along the diffusion path, where U is the potential defining the heavy-tailed conditional density. Provide sufficient conditions replacing exponential ergodicity of the frozen process, which fails in the heavy-tailed setting.

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Background

For Gaussian base distributions, the authors prove strong convergence of their proposed slow-fast multiscale systems (overdamped and underdamped) to averaged dynamics using stochastic averaging theory that relies on exponential ergodicity of the frozen fast process. This yields rigorous guarantees for MultALMC and MultCDiff in the log-concave regime.

In the heavy-tailed setting, they propose using a modified Itô diffusion for the fast process to improve mixing but note that exponential ergodicity of the frozen process generally fails. As a result, the main convergence theorem does not apply, and the authors explicitly defer the analysis of convergence of the underdamped slow-fast system with heavy-tailed noising to future work.

References

We leave the analysis of the convergence of eq:slow_fast_underdamped_heavy_tailed_appendix to the averaged dynamics eq:annealed_langevin_sde_underdamped for future work.

eq:slow_fast_underdamped_heavy_tailed_appendix:

{dXt=M−1VtdtdVt={(11−λκt∇log⁡ν(Xt−Yt1−λκt)−ΓM−1Vt)dt+2ΓdBtif    λκt<λ~(−1λκt∇Vπ(Yt1−λκt)−ΓM−1Vt)dt+2ΓdBtif    λκt≥λ~dYt=−1ε(α+d−1)∇Uρ^t,Xt(y)dt+2Uρ^t,XtεdB~t.\begin{cases} d X_t =& M^{-1} V_t d t\\ d V_t =& \begin{cases} \left(\frac{1}{\sqrt{1-\lambda_{\kappa t}}}\nabla\log\nu\left(\frac{X_t-Y_t}{\sqrt{1-\lambda_{\kappa t}}}\right)-\Gamma M^{-1}V_t\right)d t+ \sqrt{2\Gamma} d B_t&\text{if}\;\;\lambda_{\kappa t}<\tilde{\lambda}\\ \left(-\frac{1}{\sqrt{\lambda_{\kappa t}}}\nabla V_\pi\left(\frac{Y_t}{\sqrt{1-\lambda_{\kappa t}}}\right)-\Gamma M^{-1}V_t\right)d t+ \sqrt{2\Gamma} d B_t &\text{if}\;\;\lambda_{\kappa t}\geq\tilde{\lambda} \end{cases}\\ d Y_t =& -\frac{1}{\varepsilon}\left({\alpha +d } -1\right) \nabla U_{\hat{\rho}_{t,X_t}}(y) d t + \sqrt{\frac{2 U_{\hat{\rho}_{t,X_t}}}{\varepsilon}} d \tilde{B}_t. \end{cases}

eq:annealed_langevin_sde_underdamped:

$\begin{cases} d \bar{X}_t &= M^{-1}\bar{V}_td t\\ d \bar{V}_t &= \nabla \log \hat{\mu}_t(\bar{X}_t) d t -\Gamma M^{-1} \bar{V}_td t+\sqrt{2\Gamma}d B_t. \end{cases} \quad\quad\quad\quad\text{for $t\in[0, 1/\kappa]$. } $

— Sampling by averaging: A multiscale approach to score estimation (2508.15069 - Cordero-Encinar et al., 20 Aug 2025) in Appendix, Subsection “Challenges of extension to heavy-tailed diffusions”