Michael Scheutzow
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View article: Rivers under Additive Noise
Rivers under Additive Noise Open
We consider the deterministic and stochastic versions of a first order non-autonomous differential equation which allows us to discuss the persistence of rivers (“fleuves”) under additive noise.
View article: Rivers under Noise
Rivers under Noise Open
We consider the deterministic and stochastic versions of a first order non-autonomous differential equation which allows us to discuss the persistence of rivers ("fleuves") under noise.
View article: Compressibility and Stochastic Stability of Monotone Markov Chain
Compressibility and Stochastic Stability of Monotone Markov Chain Open
For a stochastically monotone Markov chain taking values in a Polish space, we present a number of conditions for existence and for uniqueness of its stationary regime, as well as for closeness of its transient trajectories. In particular,…
View article: Comment on: Criteria for Strong and Weak Random Attractors
Comment on: Criteria for Strong and Weak Random Attractors Open
In the article ’Criteria for Strong and Weak Random Attractors’ necessary and sufficient conditions for strong attractors and weak attractors are studied. In this note we correct two of its theorems on strong attractors.
View article: Expansion and attraction of RDS: long time behavior of the solution to singular SDE
Expansion and attraction of RDS: long time behavior of the solution to singular SDE Open
We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of th…
View article: Correction to: Criteria for Strong and Weak Random Attractors
Correction to: Criteria for Strong and Weak Random Attractors Open
In the article 'Criteria for Strong and Weak Random Attractors' necessary and sufficient conditions for strong attractors and weak attractors are studied. In this note we correct two of its theorems on strong attractors.
View article: Expansion and attraction of RDS: long time behavior of the solution to singular SDE
Expansion and attraction of RDS: long time behavior of the solution to singular SDE Open
We provide a framework for studying the expansion rate of the image of a bounded set under a flow in Euclidean space and apply it to stochastic differential equations (SDEs for short) with singular coefficients. If the singular drift of th…
View article: Existence of invariant probability measures for functional McKean-Vlasov SDEs
Existence of invariant probability measures for functional McKean-Vlasov SDEs Open
We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previ…
View article: A Wong-Zakai theorem for SDEs with singular drift
A Wong-Zakai theorem for SDEs with singular drift Open
We study stochastic differential equations (SDEs) with multiplicative Stratonovich-type noise of the form $ dX_t = b(X_t) dt + σ(X_t)\circ d W_t, X_0=x_0\in\mathbb{R}^d, t\geq0,$ with a possibly singular drift $b\in L^{p}(\mathbb{R}^d)$, $…
View article: The perfection of local semi-flows and local random dynamical systems with applications to SDEs
The perfection of local semi-flows and local random dynamical systems with applications to SDEs Open
We provide a rather general perfection result for crude local semi-flows taking values in a Polish space showing that a crude semi-flow has a modification which is a (perfect) local semi-flow which is invariant under a suitable metric dyna…
View article: Pinning of interfaces in a random medium with zero mean
Pinning of interfaces in a random medium with zero mean Open
We consider two related models for the propagation of a curvature sensitive interface in a time independent random medium. In both cases we suppose that the medium contains obstacles that act on the propagation of the interface with an inh…
View article: Existence of invariant probability measures for functional McKean-Vlasov SDEs
Existence of invariant probability measures for functional McKean-Vlasov SDEs Open
We show existence of an invariant probability measure for a class of functional McKean-Vlasov SDEs by applying Kakutani's fixed point theorem to a suitable class of probability measures on a space of continuous functions. Unlike some previ…
View article: Convergence of Markov chain transition probabilities
Convergence of Markov chain transition probabilities Open
Consider a discrete time Markov chain with rather general state space which has an invariant probability measure $μ$. There are several sufficient conditions in the literature which guarantee convergence of all or $μ$-almost all transition…
View article: Sharpness of Lenglart’s domination inequality and a sharp monotone version
Sharpness of Lenglart’s domination inequality and a sharp monotone version Open
We prove that the best so far known constant $c_p=\frac{p^{-p}}{1-p},\, p\in(0,1)$ of a domination inequality, which originates to Lenglart, is sharp. In particular, we solve an open question posed by Revuz and Yor. Motivated by the applic…
View article: Well-posedness, stability and sensitivities for stochastic delay equations: A generalized coupling approach
Well-posedness, stability and sensitivities for stochastic delay equations: A generalized coupling approach Open
We develop a new generalized coupling approach to the study of stochastic delay equations with Hölder continuous coefficients, for which analytical PDE-based methods are not available. We prove that such equations possess unique weak solut…
View article: Couplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting
Couplings via Comparison Principle and Exponential Ergodicity of SPDEs in the Hypoelliptic Setting Open
We develop a general framework for studying ergodicity of order-preserving Markov semigroups. We establish natural and in a certain sense optimal conditions for existence and uniqueness of the invariant measure and exponential convergence …
View article: Noise-induced strong stabilization
Noise-induced strong stabilization Open
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 …
View article: Infinite pinning
Infinite pinning Open
In this work, we address the occurrence of infinite pinning in a random medium. We suppose that an initially flat interface starts to move through the medium due to some constant driving force. The medium is assumed to contain random obsta…
View article: Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions
Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions Open
Spatially structured neural networks driven by jump diffusion noise with monotone coefficients, fully path dependent delay and with a disorder parameter are considered. Well-posedness for the associated McKean–Vlasov equation and a corresp…
View article: Generalized couplings and ergodic rates for SPDEs and other Markov models
Generalized couplings and ergodic rates for SPDEs and other Markov models Open
We establish verifiable general sufficient conditions for exponential or subexponential ergodicity of Markov processes that may lack the strong Feller property. We apply the obtained results to show exponential ergodicity of a variety of n…
View article: Couplings, generalized couplings and uniqueness of invariant measures
Couplings, generalized couplings and uniqueness of invariant measures Open
We provide sufficient conditions for uniqueness of an invariant probability measure of a Markov kernel in terms of (generalized) couplings. Our main theorem generalizes previous results which require the state space to be Polish. We provid…
View article: A Stochastic Gronwall Lemma and Well-Posedness of Path-Dependent SDEs Driven by Martingale Noise
A Stochastic Gronwall Lemma and Well-Posedness of Path-Dependent SDEs Driven by Martingale Noise Open
We show existence and uniqueness of solutions of stochastic path-dependent differential equations driven by cadlag martingale noise under joint local monotonicity and coercivity assumptions on the coefficients with a bound in terms of the …
View article: A prey-predator model with three interacting species
A prey-predator model with three interacting species Open
In this paper we consider a class of discrete time prey-predator models with three interacting species defined on the two-dimensional simplex. For some choices of parameters of the operator describing the evolution of the relative frequenc…
View article: A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces
A dynamical theory for singular stochastic delay differential equations I: Linear equations and a Multiplicative Ergodic Theorem on fields of Banach spaces Open
We show that singular stochastic delay differential equations (SDDEs) induce cocycle maps on a field of Banach spaces. A general Multiplicative Ergodic Theorem on fields of Banach spaces is proved and applied to linear SDDEs. In Part II of…
View article: Minimal forward random point attractors need not exist
Minimal forward random point attractors need not exist Open
It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in [1] that if there exist more than one pullback or weak random attractor which attracts a given family of (possibly random) sets,…
View article: Connectedness of random set attractors
Connectedness of random set attractors Open
We examine the question whether random set attractors for continuous-time random dynamical systems on a connected state space are connected. In the deterministic case, these attractors are known to be connected. In the probabilistic setup,…
View article: Minimal forward random point attractors need not exist
Minimal forward random point attractors need not exist Open
It is well-known that random attractors of a random dynamical system are generally not unique. It was shown in recent work by Hans Crauel and the author that if there exist more than one pullback or weak random attractor which attracts a g…