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Conservation laws are an inherent feature in many systems modeling real world phenomena, in particular, those modeling biological and chemical systems. If the form of the underlying dynamical system is known, linear algebra and algebraic…

Continuous time random walks combining diffusive and ballistic regimes are introduced to describe a class of L\'evy walks on lattices. By including exponentially-distributed waiting times separating the successive jump events of a walker,…

We present a numerical method to learn an accurate predictive model for an unknown stochastic dynamical system from its trajectory data. The method seeks to approximate the unknown flow map of the underlying system. It employs the idea of…

Experimental data is often affected by uncontrolled variables that make analysis and interpretation difficult. For spatiotemporal systems, this problem is further exacerbated by their intricate dynamics. Modern machine learning methods are…

We present an outline of the theory of certain L\'evy-driven, multivariate stochastic processes, where the processes are represented by rational transfer functions (Continuous-time AutoRegressive Moving Average or CARMA models) and their…

We find analytical solution of pair of stochastic equations with arbitrary forces and multiplicative L\'evy noises in a steady-state nonequilibrium case. This solution shows that L\'evy flights suppress always a quasi-periodical motion…

In this work, an adaptive predictive control scheme for linear systems with unknown parameters and bounded additive disturbances is proposed. In contrast to related adaptive control approaches that robustly consider the parametric…

Large classes of multi-dimensional Gaussian processes can be enhanced with stochastic Levy area(s). In a previous paper, we gave sufficient and essentially necessary conditions, only involving variational properties of the covariance.…

The problem of determining dynamical models and trajectories that describe observed time-series data allowing for the understanding, prediction and possibly control of complex systems in nature is of a great interest in a wide variety of…

We investigate the lifetime of dynamical regimes under the impact of noise motivated by low-dimensional models of the atmosphere. One may expect that the inclusion of noise tends to make the system leave prescribed regions of the state…

Predicting the response of nonlinear dynamical systems subject to random, broadband excitation is important across a range of scientific disciplines, such as structural dynamics and neuroscience. Building data-driven models requires…

In this paper, a non-autonomous stochastic logistic system is considered. An interesting result on the effect of stochastically perturbation for the dynamic behavior are obtained. That is, under certain conditions the stochastic system have…

In many real-world scenarios, the underlying random fluctuations are non-Gaussian, particularly in contexts where heavy-tailed data distributions arise. A typical example of such non-Gaussian phenomena calls for L\'evy noise, which…

This thesis develops exact analytical tools to study strongly correlated stochastic systems, with a focus on extreme value statistics, gap statistics, and full counting statistics in multi-particle processes. A central contribution is the…

Many systems such as autonomous vehicles and quadrotors are subject to parametric uncertainties and external disturbances. These uncertainties can lead to undesired performance degradation and safety issues. Therefore, it is important to…

Donsker-type functional limit theorems are proved for empirical processes arising from discretely sampled increments of a univariate L\'evy process. In the asymptotic regime the sampling frequencies increase to infinity and the limiting…

In the Heliosphere, power-law particle distributions are observed e.g. upstream of interplanetary shocks, which can result from superdiffusive transport. This non-Gaussian transport regime may result from intermittent magnetic field…

Stochastic processes offer a flexible mathematical formalism to model and reason about systems. Most analysis tools, however, start from the premises that models are fully specified, so that any parameters controlling the system's dynamics…

Differential equations are important mechanistic models that are integral to many scientific and engineering applications. With the abundance of available data there has been a growing interest in data-driven physics-informed models.…

Multiple types of fluctuations impact the collective dynamics of power grids and thus challenge their robust operation. Fluctuations result from processes as different as dynamically changing demands, energy trading, and an increasing share…