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Constructing numerical models of noisy partial differential equations is very delicate. Our long term aim is to use modern dynamical systems theory to derive discretisations of dissipative stochastic partial differential equations. As a…
In this paper, we study a class of nonlinear space-time fractional stochastic kinetic equations in $\mathbb{R}^d$ with Gaussian noise which is white in time and homogeneous in space. This type of equation constitutes an extension of the…
This paper is centered around the approximation of dynamical systems by means of Gaussian processes. To this end, trajectories of such systems must be collected to be used as training data. The measurements of these trajectories are…
We investigate instabilities in a stochastic mathematical model of cochlear dynamics. The cochlea is modeled as a spatio-temporal dynamical system made up of a spatially distributed array of coupled oscillators, together with the cochlear…
The amplitude of fluctuation-induced patterns might be expected to be proportional to the strength of the driving noise, suggesting that such patterns would be difficult to observe in nature. Here, we show that a large class of…
This chapter presents specific aspects of Gaussian process modeling in the presence of complex noise. Starting from the standard homoscedastic model, various generalizations from the literature are presented: input varying noise variance,…
This article focuses on the general theory of open quantum systems in the Gaussian regime and explores a number of diverse ramifications and consequences of the theory. We shall first introduce the Gaussian framework in its full generality,…
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 this paper we present a general framework in which one can rigorously study the effect of spatio-temporal noise on traveling waves, stationary patterns and oscillations that are invariant under the action of a finite-dimensional set of…
The non-linear dynamics of long-wavelength cosmological fluctuations may be phrased in terms of an effective classical, but stochastic evolution equation. The stochastic noise represents short-wavelength modes that continually redshift into…
We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by…
In this chapter we review stochastic modelling methods in climate science. First we provide a conceptual framework for stochastic modelling of deterministic dynamical systems based on the Mori-Zwanzig formalism. The Mori-Zwanzig equations…
Spatial patterning and synchronization are pervasive features of plankton communities, yet the mechanisms that allow such patterns to persist coherently under environmental noise remain unresolved. In vertically structured aquatic…
We report on the effect of spatially correlated noise on the velocities of self propelled particles. Correlations in the random forces acting on self propelled particles can induce directed collective motion, i.e. swarming. Even with…
A stochastic averaging technique based on energy-dependent frequency is extended to dynamical systems with triple-well potential driven by colored noise. The key procedure is the derivation of energy-dependent frequency according to the…
We investigate conditional McKean-Vlasov equations driven by time-space white noise, motivated by the propagation of chaos in an N-particle system with space-time Ornstein-Uhlenbeck dynamics. The framework builds on the stochastic calculus…
Complex dynamical systems which are governed by anomalous diffusion often can be described by Langevin equations driven by L\'evy stable noise. In this article we generalize nonlinear stochastic differential equations driven by Gaussian…
Traffic flow oscillations, including traffic waves, are a common yet incompletely understood feature of congested traffic. Possible mechanisms include traffic flow instabilities, indifference regions or finite human perception thresholds…
Nonlinear instabilities are responsible for spontaneous pattern formation in a vast number of natural and engineered systems ranging from biology to galaxies build-up. We propose a new instability mechanism leading to pattern formation in…
Nonlinear systems driven by noise and periodic forces with more than one frequency exhibit the phenomenon of Ghost Stochastic Resonance (GSR) found in a wide and disparate variety of fields ranging from biology to geophysics. The common…