Motivated by real world applications, three topics - deterministic quantities, uncertainty quantification and data assimilation, are considered for non-Gaussian stochastic dynamics. More... Show moreMotivated by real world applications, three topics - deterministic quantities, uncertainty quantification and data assimilation, are considered for non-Gaussian stochastic dynamics. More specifically, three problems are formulated to investigate non-Gaussian dynamics: (i) exit problem and time-dependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and time-dependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non- Gaussian -stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on non-Gaussian models is studied. For continuous-discrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, time-dependent transition probability between metastable states are investigated. xiMotivated by real world applications, three topics - deterministic quantities, uncertainty quantification and data assimilation, are considered for non-Gaussian stochastic dynamics. More specifically, three problems are formulated to investigate non-Gaussian dynamics: (i) exit problem and time-dependent probability density; (ii) parameter and function estimation for stochastic differential equations driven by L´evy motion; and (iii) nonlinear data assimilation to infer transition phenomena. First, numerical algorithms are developed to study important metrics: mean exit time, escape probability and time-dependent probability density, which can be utilized to quantify dynamical behaviors of stochastic differential equations with non- Gaussian -stable L´evy motion. Moreover, detailed numerical analysis work is done to ensure the algorithms accurate, fast and stable considering the singular nature of the L´evy jump measure. Second, new approaches on parameter and function estimation in stochastic dynamical systems are devised. Taking advantage of observations on mean exit time, escape probability or probability density, model uncertainty can be quantified by some optimization methods. These methods are beneficial to systems for which mean exit time, escape probability or probability density are feasible to observe. Finally, nonlinear data assimilation on non-Gaussian models is studied. For continuous-discrete filtering, a recursive Bayesian approach is used, and for continuous filtering, Zakai equation is solved to provide the system state estimation. In both cases, time-dependent transition probability between metastable states are investigated. Show less