Estimating transitions using the Potential-Theoretic approach and a data-based method
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A large variety of random dynamical systems describing real world phenomena show metastable behaviours, namely rapid transitions to an alternate stable equilibrium occurring after a random time. Is it possible to estimate quantities related to transitions only using a relatively small amount of data from a time series of the system state? Can one do it without explicitly knowing the underlying equations for the system? We address the problem for a specific class of stochastic processes, namely continuous-time Markov processes which represent solutions to Stochastic Differential Equations. We propose a method based on the Potential-Theoretic approach to metastability developed by Bovier and Den Hollander. The Potential Theory provides the theoretical framework of the work and, importantly, it defines the Partial Differential Equation with Dirichlet boundary conditions associated with the stochastic process that we aim to solve. We combine the theory with numerical methods in order to estimate the unknown stochastic generator of the process, by means of a series of approximations. The recent Extended Dynamic Mode Decomposition method inspires the data-based approach. As a result, a Dirichlet problem is successfully solved for the stochastic diffusion in a double-well potential in one dimension.