Related papers: Spontaneous stochasticity and renormalization grou…
High-dimensional dynamical systems projected onto a reduced-order model cease to be deterministic and are best described by probability distributions in state space. Their equations of motion map onto an evolution operator with a…
We consider a data-driven formulation of the classical discrete-time stochastic control problem. Our approach exploits the natural structure of many such problems, in which significant portions of the system are uncontrolled. Employing the…
In this paper we propose a finite-dimensional and deterministic approach to the study of invariant sets of certain nonautonomous differential inclusions naturally arising in the context of random and control dynamical systems, as well as in…
The current series of papers is concerned with stochastic stability of monotone dynamical systems by identifying the basic dynamical units that can survive in the presence of noise interference. In the first of the series, for the…
We review the mathematical formalism underlying the modelling of stochasticity in biological systems. Beginning with a description of the system in terms of its basic constituents, we derive the mesoscopic equations governing the dynamics…
We consider the most general scale invariant radial Hamiltonian allowing for anisotropic scaling between space and time. We formulate a renormalisation group analysis of this system and demonstrate the existence of a quantum phase…
The stochastic scenario of relaxation in the complex systems is presented. It is based on a general probabilistic formalism of limit theorems. The nonexponential relaxation is shown to result from the asymptotic self-similar properties in…
Invariant manifolds facilitate the understanding of nonlinear stochastic dynamics. When an invariant manifold is represented approximately by a graph for example, the whole stochastic dynamical system may be reduced or restricted to this…
We consider an infinite-dimensional stochastic clustering model on $\mathbb{R}$. In discrete time, each point of a unit-intensity simple point process moves halfway toward either of its left or right neighbors, chosen uniformly at random.…
A system of stochastic differential equations for the velocity and density of a classical self-gravitating matter is investigated by means of the field theoretic renormalization group. The existence of two types of large-scale scaling…
Stochastic gradient descent is an optimisation method that combines classical gradient descent with random subsampling within the target functional. In this work, we introduce the stochastic gradient process as a continuous-time…
Intrinsic stochasticity can induce highly non-trivial effects on dynamical systems, including stochastic and coherence resonance, noise induced bistability, noise-induced oscillations, to name but a few. In this paper we revisit a mechanism…
This paper deals with the multivalued non-autonomous random dynamical system generated by the non-autonomous stochastic wave equations on unbounded domains, which has a non-Lipschitz nonlinearity with critical exponent in the three…
Discrete-time stochastic systems are an essential modelling tool for many engineering systems. We consider stochastic control systems that are evolving over continuous spaces. For this class of models, methods for the formal verification…
Stochastic approximation is a framework unifying many random iterative algorithms occurring in a diverse range of applications. The stability of the process is often difficult to verify in practical applications and the process may even be…
Using dynamic renormalization group we study the transport in driven diffusive systems in the presence of quenched random drift velocity with long-range correlations along the transport direction. In dimensions $d\mathopen< 4$ we find fixed…
The empirical measure of an interacting particle system is a purely atomic random probability measure. In the limit as the number of particles grows to infinity, we show for McKean-Vlasov systems with common noise that this measure becomes…
Leveraging recent work on data-driven methods for constructing a finite state space Markov process from dynamical systems, we address two problems for obtaining further reduced statistical representations. The first problem is to extract…
We distinguish a mechanical representation of the world in terms of point masses with positions and momenta and the chemical representation of the world in terms of populations of different individuals, each with intrinsic stochasticity,…
We consider a 2-dimensional stochastic differential equation in polar coordinates depending on several parameters. We show that if these parameters belong to a specific regime then the deterministic system explodes in finite time, but the…