Related papers: Absolute Continuity Theorem for Random Dynamical S…

We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability.…

We consider a random network of nonlinear maps exhibiting a wide range of local dynamics, with the links having normally distributed interaction strengths. The stability of such a system is examined in terms of the asymptotic fraction of…

We show that for entire maps of the form $z \mapsto \lambda \exp(z)$ such that the orbit of zero is bounded and such that Lebesgue almost every point is transitive, no absolutely continuous invariant probability measure can exist. This…

We prove a persistence result for noncompact normally hyperbolic invariant manifolds in the setting of Riemannian manifolds of bounded geometry. Bounded geometry of the ambient manifold is a crucial assumption required to control the…

For a dynamical system, it is known that the existence of a Lyapunov-type density function, called Lyapunov density or Rantzer's density function, implies convergence of Lebesgue almost all solutions to an equilibrium. Using the duality…

We are concerned with the absolute continuity of stationary distributions corresponding to some piecewise deterministic Markov process, being typically encountered in biological models. The process under investigation involves a…

We consider families of transformations in multidimensional Riemannian manifolds with non-uniformly expanding behavior. We give sufficient conditions for the continuous variation (in the $L^1$-norm) of the densities of absolutely continuous…

We prove that self-similar measures on the real line are absolutely continuous for almost all parameters in the super-critical region, in particular confirming a conjecture of S-M. Ngai and Y. Wang. While recently there has been much…

We analyse the so-called Marginal Instability of linear switching systems, both in continuous and discrete time. This is a phenomenon of unboundedness of trajectories when the Lyapunov exponent is zero. We disprove two recent conjectures of…

We consider piecewise deterministic Markov processes with degenerate transition kernels of the "house-of-cards"-type. We use a splitting scheme based on jump times to prove the absolute continuity, as well as some regularity, of the…

For a family of random intermittent dynamical systems with a superattracting fixed point we prove that a phase transition occurs between the existence of an absolutely continuous invariant probability measure and infinite measure depending…

We consider smooth random dynamical systems defined by a distribution with a finite moment of the norm of the differential, and prove that under suitable non-degeneracy conditions any stationary measure must be H\"older continuous. The…

One of the most elusive challenges within the area of topological data analysis is understanding the distribution of persistence diagrams. Despite much effort, this is still largely an open problem. In this paper, we present a series of…

In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these…

We study dynamical systems forced by a combination of random and deterministic noise and provide criteria, in terms of Lyapunov exponents, for the existence of random attractors with continuous structure in the fibres. For this purpose, we…

A general construction for $\sigma-$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f^n_*(\lambda)$ will imply the existence of a…

This paper concerns piecewise-smooth maps on $\mathbb{R}^d$ that are continuous but not differentiable on switching manifolds (where the functional form of the map changes). The stability of fixed points on switching manifolds is…

We prove a quenched almost sure invariance principle for certain classes of random distance expanding dynamical systems which do not necessarily exhibit uniform decay of correlations.

We give a condition for absolute continuity of self-similar measures in arbitrary dimensions. This allows us to construct the first explicit absolutely continuous examples of inhomogeneous self-similar measures in dimension one and two. In…

We consider a dynamical system which has the hyperbolic structure along an attracting invariant manifold $M$. The problem is whether every motion starting in a neighborhood of $M$ possesses an asymptotic phase, i.e. eventually approaches a…