Matter, and more specifically molecules, are constantly moving... However, a mathematical model can explain this movement, which is both ordered and chaotic : the Langevin equation. So let Ω ⊂ R^d be a bounded smooth domain and b : Ω→ R^d be a smooth vector field. We focus on the associated overdamped Langevin equation : \partial_t Xt = b(Xt ) + h^1/2Bt in the low temperature regime h→ 0 and in the case where b admits the decomposition b = −∇ f− ℓ with ∇ f· ℓ= 0 on \partial Ω. To study this equation, we analyse the spectrum of the infinitesimal generator of the dynamics: Lh = −∆ + ∇ f · ∇ + ℓ · ∇ with Neumann boundary conditions. In this case, moving particles will remain trapped inside the domain and more precisely the process remains trapped, for some time, in a certain region of the domain before going to another area. These regions are called metastables and correspond to neighborhoods of minima of f. Finally, thanks to spectral theory and more specifically small eigenvalues of Lh , we can describe the return to equilibrium of this metastable dynamic.

State-Space Models (SSMs) are deterministic or stochastic dynamical systems defined by two processes. The state process, which is not observed directly, models the transformation of the states over time. On another hand, the observation process produces the observables on which model fitting and prediction are based. Ecology frequently uses stochastic SSMs to represent the imperfectly observed dynamics of population sizes or animal movement. However, several simulation-based evaluations of model performance suggest broad identifiability issues in ecological SSMs. Formal SSM identifiability is typically investigated using exhaustive summaries, which are simplified representations of the model. The theory on exhaustive summaries is largely based on continuous-time deterministic modelling and those for ecological SSMs have developed by analogy. While the discreteness of time does not constitute a challenge, finding a good exhaustive summary for a stochastic SSM is more difficult. The strategy adopted so far has been to create exhaustive summaries based on a transfer function of the expectations of the stochastic process. However, this evaluation of identifiability does not allow to take into account the possible dependency between the variance parameters and the process parameters. We show that the output spectral density plays a key role in stochastic SSM identifiability assessment. This allows us to define a new suitable exhaustive summary. Using several ecological examples, we show that usual ecological models are often theoretically identifiable, suggesting that most SSM estimation problems are due to practical rather than theoretical identifiability issues.

Abstract: Piecewise Deterministic Markov Processes (PDMPs) describe deterministic dynamical systems whose parameters undergo random jumps, making them versatile tools for modeling complex stochastic phenomena. Yet, simulating their trajectories can be computationally demanding. For a broad class of inference problems, an optimal sampling strategy can be characterized in terms of a generalized "committor function". We introduce a new adaptive importance sampling method designed to efficiently generate rare PDMP trajectories. The approach unfolds in two stages. First, in an offline phase, the PDMP is coarse-grained into a simpler graph-based process, enabling explicit computation of key quantities and yielding a low-cost approximation of the committor function. Then, in an online phase, trajectories are sampled from a distribution guided by this approximation and iteratively improved via cross-entropy minimization. We provide asymptotic guarantees for the method and demonstrate its effectiveness through the estimation of the failure probability of a complex industrial system.

Bayesian Inversion consists of deriving the posterior distribution of unknown parameters or functions from partial and indirect observations of a system. When the dimension of the search space is high or infinite, methods leveraging local information, such as derivatives of different orders, of the target probability measure have the advantages to converge faster than Monte-Carlo sampling techniques. Nevertheless, many applications are characterized by posterior distributions with low regularity or gradients that are intractable to compute. An interesting research direction consists in using interacting particle systems to explore the potential landscape, and Ensemble Kalman Sampler (EKS) is one of those. In this talk, we consider a simplified EKS dynamics, where the gradient of the potential is approximated by finite differences using independent Ornstein-Uhlenbeck processes that explore the neighborhood of the candidate parameter. We will characterize the invariant distribution of this system and compare its dynamics to the overdamped Langevin process.