Metastable Markov chains: from the convergence of the trace to the convergence of the finite-dimensional distributions

We consider continuous-time Markov chains which display a family of wells at the same depth. We provide sufficient conditions which entail the convergence of the finite-dimensional distributions of the order parameter to the ones of a finite state Markov chain. We also show that the state of the process can be represented as a time-dependent convex combination of metastable states, each of which is supported on one well

Metastability in a condensing zero-range process in the thermodynamic limit

Zero-range processes with decreasing jump rates are known to exhibit condensation, where a finite fraction of all particles concentrates on a single lattice site when the total density exceeds a critical value. We study such a process on a one-dimensional lattice with periodic boundary conditions in the thermodynamic limit with fixed, super-critical particle density. We show that the process exhibits metastability with respect to the condensate location, i.e. the suitably accelerated process of the rescaled location converges to a limiting Markov process on the unit torus. This process has stationary, independent increments and the rates are characterized by the scaling limit of capacities of a single random walker on the lattice. Our result extends previous work for fixed lattices and diverging density in [J. Beltran, C. Landim, Probab. Theory Related Fields, 152(3-4):781-807, 2012], and we follow the martingale approach developed there and in subsequent publications. Besides additional technical difficulties in estimating error bounds for transition rates, the thermodynamic limit requires new estimates for equilibration towards a suitably defined distribution in metastable wells, corresponding to a typical set of configurations with a particular condensate location. The total exit rates from individual wells turn out to diverge in the limit, which requires an intermediate regularization step using the symmetries of the process and the regularity of the limit generator. Another important novel contribution is a coupling construction to provide a uniform bound on the exit rates from metastable wells, which is of a general nature and can be adapted to other models.Comment: 55 pages, 1 figur

On the optimally controlled stochastic shallow lake

We consider the stochastic control problem of the shallow lake and continue the work of G. T. Kossioris, Loulakis, and Souganidis (2019) in three directions. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tail asymptotics for the invariant distribution and to extend results of Grass, Kiseleva, and Wagener (2015) beyond the small noise limit

Managing geotechnical risk on US design-build transport projects

Awarding design-build (DB) contracts before a complete subsurface investigation is completed, makes mitigating the risk of differing site conditions difficult, if not impossible. The purpose of the study was to identify effective practices for managing geotechnical risk in DB projects, and it reports the results of a survey that included responses from 42 of 50 US state departments of transportation and a content analysis of DB requests for proposals from 26 states to gauge the client’s perspective, as well as 11 structured interviews with DB contractors to obtain the perspective from the other side of the DB contract. A suite of DB geotechnical risk manage tools is presented based on the results of the analysis. Effective practices were found in three areas: enhancing communications on geotechnical issues before final proposals are submitted; the use of project-specific differing site conditions clauses; and expediting geotechnical design reviews after award. The major finding is that contract verbiage alone is not sufficient to transfer the risk of changed site conditions. The agency must actively communicate all the geotechnical information on hand at the time of the DB procurement and develop a contract strategy that reduces/retires the risk of geotechnical uncertainty as expeditiously as possible after award