14 problems

Let $\rho_n^m$ be the particle approximation produced at iteration $n$ with $m$ particles, let $\rho^*$ be the target measure, and let $\tilde{\rho}_\tau^*$ be a numerical referenc…

Let $\mathcal{L}_{\mathcal{D}^{\star}}$ be the operator associated with the optimal diffusion coefficient in a one-dimensional setting, and let…

Let $\phi\colon \mathbf R^d\to\mathbf R$ be such that $\operatorname{e}^{-\phi}$ is integrable, and let $\overline X_t$ denote the limiting particle process with law…

Multimodal Langevin-sampling conjecture. The observed separatrices further demonstrate the difficulty of sampling from multimodal distributions with Langevin dynamics.

Let $\eta_{\rm p,OGP}$ denote the OGP barrier value for the spherical spin model, and let Langevin dynamics be run for $N^{O(1)}$ time. Langevin dynamics conjecture. A natural conj…

Small-friction derivative estimate. It is conjectured that

Slow-thermalization conjecture. Starting from the uniform measure, Langevin dynamics takes exponential time to reach equilibrium for every $T<T_{sh}$: there is a constant $c>0$ suc…

Let $T_{BBM}>T_{sh}$ be the temperature associated with metastability of Langevin dynamics for spherical pure $p$-spin glasses, and let the spectral gap refer to the Langevin dynam…

Convergence conjecture. The underdamped Langevin dynamics shows superior convergence over the overdamped Langevin equation.

Higher-order Langevin diffusion conjecture. The exponent on $d$ in the mixing-time complexity can be further reduced, with a trade-off between the dependency on the dimension and t…

Consider the $(\Delta_p,1/\Delta_2)$ phase diagram of the spiked matrix-tensor problem, and let the glassy-state analysis identify a line delimiting the region where stable one-ste…

Let $H_0$ denote the pure-noise Hamiltonian, let $\alpha_c(\infty)=(k-2)/2$, and consider Langevin dynamics with signal-to-noise parameter $\lambda=N^\alpha$, initialized uniformly…

Let $\pi$ be the invariant target measure of the perturbed underdamped Langevin dynamics, and let $\sigma_f^2$ denote the asymptotic variance of an observable $f$. Assume the frict…

The BAOAB scheme is an integrator for Langevin dynamics, and the configurational temperature is an observable used to assess its sampling accuracy. BAOAB accuracy conjecture. For B…