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Dynamical systems theory provides powerful methods to extract effective macroscopic dynamics from complex systems with slow modes and fast modes. Here we derive and theoretically support a macroscopic, spatially discrete, model for a class…
Quantifying the speed of linguistic change is challenging due to the fact that the historical evolution of languages is sparsely documented. Consequently, traditional methods rely on phylogenetic reconstruction. In this paper, we propose a…
We generalize the spherical collapse model for the formation of bound objects to apply in a Universe with arbitrary positive cosmological constant. We calculate the critical condition for collapse of an overdense region and give exact…
In pattern-forming systems, competition between patterns with different wave numbers can lead to domain structures, which consist of regions with differing wave numbers separated by domain walls. For domain structures well above threshold…
In this paper, we study dimension reduction techniques for large-scale controlled stochastic differential equations (SDEs). The drift of the considered SDEs contains a polynomial term satisfying a one-sided growth condition. Such…
We deal with parameter estimation for a linear parabolic second-order stochastic partial differential equation in two space dimensions driven by two types of $Q$-Wiener processes based on high frequency data with respect to time and space.…
Stochastic mathematical models are essential tools for understanding and predicting complex phenomena. The purpose of this work is to study the exit times of a stochastic dynamical system-specifically, the mean exit time and the…
Conformal field theories with central charge $c\le1$ on random surfaces have been extensively studied in the past. Here, this discussion is extended from their equilibrium distribution to their critical dynamics. This is motivated by the…
We develop an approach to learn an interpretable semi-parametric model of a latent continuous-time stochastic dynamical system, assuming noisy high-dimensional outputs sampled at uneven times. The dynamics are described by a nonlinear…
Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established,…
In this paper, we develop a way of analyzing the random dynamics of stochastic evolution equations with a non-dense domain. Such problems cover several types of evolution equations. We are particularly interested in evolution equations with…
We study strictly parabolic stochastic partial differential equations on $\R^d$, $d\ge 1$, driven by a Gaussian noise white in time and coloured in space. Assuming that the coefficients of the differential operator are random, we give…
We investigate the longtime behavior of stochastic partial differential equations (SPDEs) with differential operators that depend on time and the underlying probability space. In particular, we consider stochastic parabolic evolution…
A semi-classical non-Hamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes non-Hamiltonian at random instants of transition of pure states to reduced ones, given by a…
We consider the stochastic Swift-Hohenberg equation on a large domain near its change of stability. We show that, under the appropriate scaling, its solutions can be approximated by a periodic wave, which is modulated by the solutions to a…
Discrete canonical evolution is a key tool for understanding the dynamics in discrete models of spacetime, in particular those represented by a triangular Regge lattice. We consider a finite-dimensional system whose evolution is realized by…
The main goal of this article is to study the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle (LDP) based on exponential equivalence arguments) for a class of stochastic partial differential…
We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or…
We formulate a general, high-dimensional kinetic theory describing the internal state (such as gene expression or protein levels) of cells in a stochastically evolving population. The resolution of our kinetic theory also allows one to…
Stochastic Differential Equations (SDEs) serve as a powerful modeling tool in various scientific domains, including systems science, engineering, and ecological science. While the specific form of SDEs is typically known for a given…