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In this paper,under an abstract setting we establish the existence of spatially inhomogeneous steady states and the asymptotic propagation properties for a large class of monotone evolution systems without spatial translation invariance.…
We give a overview of stochastic models of evolution that have found applications in genetics, ecology and linguistics for an audience of nonspecialists, especially statistical physicists. In particular, we focus mostly on neutral models in…
Moving boundary problems allow to model systems with phase transition at an inner boundary. Driven by problems in economics and finance, in particular modeling of limit order books, we consider a stochastic and non-linear extension of the…
Building on results obtained in [GVRS], we prove Local Stable and Unstable Manifold Theorems for nonlinear, singular stochastic delay differential equations. The main tools are rough paths theory and a semi-invertible Multiplicative Ergodic…
In the present work, we investigate the dynamics of the infinite-dimensional stochastic partial differential equation (SPDE) with multiplicative white noise. We derive the effective equation on the approximate slow manifold in detail by…
Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a…
Noise-induced transitions between metastable fixed points in systems evolving on multiple time scales are analyzed in situations where the time scale separation gives rise to a slow manifold with bifurcation. This analysis is performed…
The limiting behavior of stochastic evolution processes with small noise intensity $\epsilon$ is investigated in distribution-based approach. Let $\mu^{\epsilon}$ be stationary measure for stochastic process $X^{\epsilon}$ with small…
We construct stable manifolds for a class of degenerate evolution equations including the steady Boltzmann equation, establishing in the process exponential decay of associated kinetic shock and boundary layers to their limiting equilibrium…
We study the dynamics of a degenerate parabolic equation with a variable, generally non-smooth diffusion coefficient, which may vanish at some points or be unbounded. We show the existence of a global branch of nonnegative stationary…
We construct an example of a one-dimensional parabolic integro-differential equation with nonlocal diffusion which does not have asymptotically finite-dimensional dynamics in the corresponding state space. This example is more natural in…
A parameter estimation problem for a class of semilinear stochastic evolution equations is considered. Conditions for consistency and asymptotic normality are given in terms of growth and continuity properties of the nonlinear part.…
This paper presents theoretical advances in the application of the Stochastic Partial Differential Equation (SPDE) approach in geostatistics. We show a general approach to construct stationary models related to a wide class of linear SPDEs,…
The main result of the present paper is a statement on existence, uniqueness and regularity for mild solutions to a parabolic transport diffusion type equation that involves a non-smooth coefficient. We investigate related Cauchy problems…
The limiting stability of invariant probability measures of time homogeneous transition semigroups for autonomous stochastic systems has been extensively discussed in the literature. In this paper we initially initiate a program to study…
We study some classes of semi-linear differential equations including both well-posed and ill-posed cases that can generate cocycles (or cocycle correspondences with generating cocycles). Under exponential dichotomy condition with other…
We investigate the profound relation between the equations of biological evolution and quantum mechanics by writing a biologically inspired equation for the stochastic dynamics of an ensemble of particles. Interesting behavior is observed…
In this paper, we study the existence of random periodic solutions for nonlinear stochastic differential equations with additive white noise. We extend the input-to-state characteristic operator of the system to the non-autonomous…
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…
This paper is mainly concerned with a kind of fractional stochastic evolution equations driven by L\'evy noise in a bounded domain. We first state the well-posedness of the problem via iterative approximations and energy estimates. Then,…