Stochastic evolution equations with singular drift and gradient noise via curvature and commutation conditions

We prove existence and uniqueness of solutions to a nonlinear stochastic evolution equation on the $d$-dimensional torus with singular $p$-Laplace-type or total variation flow-type drift with general sublinear doubling nonlinearities and Gaussian gradient Stratonovich noise with divergence-free coefficients. Assuming a weak defective commutator bound and a curvature-dimension condition, the well-posedness result is obtained in a stochastic variational inequality setup by using resolvent and Dirichlet form methods and an approximative It\^{o}-formula.Comment: 26 pages, 58 references. Essential changes to Version 4: Examples revised. Accepted for publication in Stochastic Processes and their Application

Existence and Uniqueness of Invariant Measures for Stochastic Evolution Equations with Weakly Dissipative Drifts

In this paper, a new decay estimate for a class of stochastic evolution equations with weakly dissipative drifts is established, which directly implies the uniqueness of invariant measures for the corresponding transition semigroups. Moreover, the existence of invariant measures and the convergence rate of corresponding transition semigroup to the invariant measure are also investigated. As applications, the main results are applied to singular stochastic $p$-Laplace equations and stochastic fast diffusion equations, which solves an open problem raised by Barbu and Da Prato in [Stoc. Proc. Appl. 120(2010), 1247-1266].Comment: http://www.math.washington.edu/~ejpecp/ECP/viewarticle.php?id=2308&layout=abstrac

Invariant measures for monotone SPDE's with multiplicative noise term

We study diffusion processes corresponding to infinite dimensional semilinear stochastic differential equations with local Lipschitz drift term and an arbitrary Lipschitz diffusion coefficient. We prove tightness and the Feller property of the solution to show existence of an invariant measure. As an application we discuss stochastic reaction diffusion equations.Comment: 10 page

A spatio-temporal entropy-based approach for the analysis of cyber attacks (demo paper)

Computer networks are ubiquitous systems growing exponentially with a predicted 50 billion devices connected by 2050. This dramatically increases the potential attack surface of Internet networks. A key issue in cyber defense is to detect, categorize and identify these attacks, the way they are propagated and their potential impacts on the systems affected. The research presented in this paper models cyber attacks at large by considering the Internet as a complex system in which attacks are propagated over a network. We model an attack as a path from a source to a target, and where each attack is categorized according to its intention. We setup an experimental testbed with the concept of honeypot that evaluates the spatiotemporal distribution of these Internet attacks. The preliminary results show a series of patterns in space and time that illustrate the potential of the approach, and how cyber attacks can be categorized according to the concept and measure of entropy

The stochastic Klausmeier system and a stochastic Schauder-Tychonoff type theorem

On the one hand, we investigate the existence and pathwise uniqueness of a nonnegative martingale solution to the stochastic evolution system of nonlinear advection-diffusion equations proposed by Klausmeier with Gaussian multiplicative noise. On the other hand, we present and verify a general stochastic version of the Schauder-Tychonoff fixed point theorem, as its application is an essential step for showing existence of the solution to the stochastic Klausmeier system. The analysis of the system is based both on variational and semigroup techniques. We also discuss additional regularity properties of the solution.Comment: 52 pages, 74 reference

Singular limits for stochastic equations

We study singular limits of stochastic evolution equations in the interplay of disappearing strength of the noise and increasing roughness of the noise, so that the noise in the limit would be too rough to define a solution to the limiting equations. Simultaneously, the limit is singular in the sense that the leading order differential operator may vanish. Although the noise is disappearing in the limit, additional deterministic terms appear due to renormalization effects. We give an abstract framework for the main error estimates, that first reduce to bounds on a residual and in a second step to bounds on the stochastic convolution. Moreover, we apply it to a singularly regularized Allen-Cahn equation and the Cahn-Hilliard/Allen-Cahn homotopy.Comment: 23 pages, 36 reference

Improved regularity for the stochastic fast diffusion equation

We prove that the solution to the singular-degenerate stochastic fast-diffusion equation with parameter $m\in (0,1)$, with zero Dirichlet boundary conditions on a bounded domain in any spatial dimension, and driven by linear multiplicative Wiener noise, exhibits improved regularity in the Sobolev space $W^{1,m+1}_0$ for initial data in $L^{2}$.Comment: 7 pages, 29 reference

Stability and moment estimates for the stochastic singular Φ\Phi-Laplace equation

We study stability, long-time behavior and moment estimates for stochastic evolution equations with additive Wiener noise and with singular drift given by a divergence type quasilinear diffusion operator which may not necessarily exhibit a homogeneous diffusivity. Our results cover the singular $p$-Laplace and, more generally, singular $\Phi$-Laplace equations with zero Dirichlet boundary conditions. We obtain improved moment estimates and quantitative convergence rates of the ergodic semigroup to the unique invariant measure, classified in a systematic way according to the degree of local degeneracy of the potential at the origin. We obtain new concentration results for the invariant measure and establish maximal dissipativity of the associated Kolmogorov operator. In particular, we recover the results for the curve shortening flow in the plane by Es-Sarhir, von Renesse and Stannat, NoDEA 16(9), 2012.Comment: 23 pages, 54 reference

Structural relaxation in orthoterphenyl: a schematic mode coupling theory model analysis

Depolarized light scattering spectra of orthoterphenyl showing the emergence of the structural relaxation below the oscillatory microscopic excitations are described by solutions of a schematic mode--coupling--theory model