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We develop a general theory dealing with stochastic models for dynamical systems that are governed by various nonlinear, ordinary or partial differential, equations. In particular, we address the problem how flows in the random medium…
We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative…
We consider the linear stochastic wave equation with spatially homogenous Gaussian noise, which is fractional in time with index $H>1/2$. We show that the necessary and sufficient condition for the existence of the solution is a relaxation…
The two-variable Langevin equations, modeling the Brownian motion of a particle moving in a potential and leading to the Maxwell-Boltzmann distribution of the corresponding Fokker-Planck equation, are shown to give rise to types of…
The influence of dissipation on the fluctuation statistics of the total energy is investigated through both a phenomenological and a stochastic model for dissipative energy-transfer through a cascade of states. In equilibrium the states…
In this paper, we consider three-dimensional nonlinear stochastic wave equations driven by the Gaussian noise which is white in time and has some spatial correlations. Using the Malliavin-Stein's method, we prove the Gaussian fluctuation…
We propose a coarse-graining procedure for describing the superhorizon dynamics of inflationary tensor modes. Our aim is to formulate a stochastic description for the statistics of spin-2 modes which seed the background of gravitational…
We study a non Markovian three state model, subjected to an external periodic signal. This model is intended to describe an excitable systems with periodical driving. In the limit of a small amplitude of the external signal we derive…
In this article, we propose a generalized model for nonequilibrium vibrational energy distribution functions. The model can be used, in place of equilibrium (Boltzmann) distribution functions, when deriving reaction rate constants for…
Various approaches to stochastic processes exist, noting that key properties such as measurability and continuity are not trivially satisfied. We introduce a new theory for Gaussian processes using improper linear functionals. Using a…
We study a $d$-dimensional wave equation model ($2\leq d\leq 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter…
We consider a stochastic partial differential equation with logarithmic (or negative power) nonlinearity, with one reflection at 0 and with a constraint of conservation of the space average. The equation, driven by the derivative in space…
The asymptotic behavior of a nonlinear oscillator subject to a multiplicative Ornstein-Uhlenbeck noise is investigated. When the dynamics is expressed in terms of energy-angle coordinates, it is observed that the angle is a fast variable as…
We propose a curvaton model in which the initial condition of the curvaton oscillation is determined by its attractor behavior during inflation. Assuming a chaotic inflation model, we find that the initial condition determined by the…
We derive a stochastic wave equation for an inflaton in an environment of an infinite number of fields. We study solutions of the linearized stochastic evolution equation in an expanding universe. The Fokker-Planck equation for the inflaton…
We study a class of stochastic time-fractional equations on $\mathbb{R}^d$ driven by a centered Gaussian noise, involving a Caputo time derivative of order $\beta>0$, a fractional (power) Laplacian of order $\alpha>0$, and a…
A fourth-order and a second-order nonlinear diffusion models in spectral space are proposed to describe gravitational wave turbulence in the approximation of strongly local interactions. We show analytically that the model equations satisfy…
A class of non-canonical inflationary models is identified, where the leading-order contribution to the non-Gaussianity of the curvature perturbation is determined by the sound speed of the fluctuations in the inflaton field. Included in…
This work is devoted to the effective macroscopic dynamics of a weakly damped stochastic nonlinear wave equation with a random dynamical boundary condition. The white noises are taken into account not only in the model equation defined on a…
The influence of small random perturbations on a deterministic dynamical system with a locally stable equilibrium is considered. The perturbed system is described by the It\^{o} stochastic differential equation. It is assumed that the noise…