Efficient Simulation of Metastable Transitions

Definitions and mathematical statement

Let $N$ be a positive integer and let $\beta>0$, using the implicit dimension and inverse-temperature conventions of the supplied formulation. Let $V:{\mathbb{R}}^{3N}\to{\mathbb{R}}$ be a potential and let $(W_t)_{t\geq0}$ be standard Brownian motion in ${\mathbb{R}}^{3N}$. The dynamics under consideration are \[ {\,\mathrm{d}} X_t=-\nabla V(X_t)\,{\,\mathrm{d}} t+\sqrt{2\beta^{-1}}\,{\,\mathrm{d}} W_t. \] The source describes the invariant density as proportional to $e^{-\beta V}$. More explicitly, if \[ 0<Z_\beta:=\int_{{\mathbb{R}}^{3N}}e^{-\beta V(x)}\,{\,\mathrm{d}} x<\infty, \] its proposed equilibrium probability measure is \[ \pi_\beta({\,\mathrm{d}} x)=Z_\beta^{-1}e^{-\beta V(x)}\,{\,\mathrm{d}} x. \] The record does not specify the regularity, growth, or other hypotheses on $V$ needed for a well-defined nonexplosive diffusion and an invariant probability measure. These requirements must therefore remain explicit qualifications. When such a stationary diffusion exists, an equilibrium trajectory means a trajectory of this dynamics with $X_0$ distributed according to $\pi_\beta$.

Research problem. For a metastable potential $V$, develop efficient methods for generating equilibrium trajectories that leave one potential well and reach another. The supplied expanded statement also asks how available rare-event methods should be classified and improved.

This is a research-program formulation rather than a fully specified conjecture or algorithmic performance assertion. The source does not define the wells, metastability, the observation interval, the precise distribution of transition trajectories to be sampled, or a criterion of efficiency. In particular, sampling stationary trajectories until a transition occurs and sampling trajectories conditioned on a transition specify different tasks; the record does not select a precise target path distribution. No uniform quantifier over potentials, temperature regime, approximation tolerance, or computational complexity bound is supplied.

Short English statement

The problem asks for efficient simulation of transitions between wells of an overdamped Langevin diffusion in equilibrium, together with classification and improvement of rare-event methods. The admissible potentials, transition-path distribution, and meaning of efficiency are unspecified, so this remains a qualified research question requiring human mathematical review.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30001622$, titled Efficient Simulation of Metastable Transitions, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. No individual proposer is supplied.
• Supplied source citation: Large Scale Stochastic Dynamics (2011), Oberwolfach Reports, pp. 2877--2975, DOI 10.4171/owr/2010/50. Source URL and supplied literature-source link, labelled "original OWR report": https://doi.org/10.4171/owr/2010/50. The record gives proposed year $2010$; the relation between this year and the citation's $2011$ label is not resolved here.
• Literature assessment dated exactly 2026-08-21: supplied dated curation, not independently verified. It labels the problem open, reports that the OWR report describes efficient generation of equilibrium trajectories crossing metastable wells as mostly open, and states that modern transition-path and rare-event methods address special models without resolving the general problem. The supplied background explicitly calls for independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and supplies a verification note asserting that the official report defines the Langevin dynamics and potential and describes efficient generation of inter-well equilibrium trajectories as a mostly open research direction. These are supplied provenance assertions; no independent source verification is claimed in this draft.