a probabilistic approach to spectral analysis of growth fragmentation equations abstract the growth fragmentation equation describes a system of growing and dividing particles and arises in models of cell division protein polymerisation and even telecommunications protocols several important questions about the equation concern the asymptotic behaviour of solutions at large times at what rate do they converge to zero or infinity and what does the asymptotic profile of the solutions look like does the rescaled solution converge to its asymptotic profile at an exponential speed these questions have traditionally been studied using analytic techniques such as entropy methods or splitting of operators in this work we present a probabilistic approach we use a feynman kac formula to relate the solution of the growth fragmentation equation to the semigroup of a markov process and characterise the rate of decay or growth in terms of this process we then identify the malthus exponent and the asymptotic profile in terms of a related markov process and give a spectral interpretation in terms of the growth fragmentation operator and its dual

ergodicity of the 2d navier stokes equations with degenerate stochastic forcing the stochastic 2d navier stokes equations on the torus driven by degenerate noise are studied we characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic in particular our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in l 2 0 t 2 unlike previous works this class is independent of the viscosity and the strength of the noise the two main tools of our analysis are the asymptotic strong feller property introduced in this work and an approximate integration by parts formula the first when combined with a weak type of irreducibility is shown to ensure that the dynamics is ergodic the second is used to show that the first holds un

integer programs for logic constraint satisfaction abstract logic constraint satisfaction problems are in general np hard and a general deterministic polynomial time algorithm is not known since several logic constraint problems can be reduced in polynomial time to the satisfaction of a conjunctive normal form cnf sat this case is very important we present here a technique to transform a cnf sat problem in an integer optimization problem that can be solved by linear programming the size of the obtained integer program has a polynomial growth in comparison with the original problem size

existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions abstract in this paper we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions which can be applied to a large class of spde with finite delays e g d dimensional stochastic fractional navier stokes equations with delays d dimensional stochastic reaction diffusion equations with delays d dimensional stochastic porous media equations with delays moreover under local monotonicity conditions for the nonlinear terms we obtain the existence and uniqueness of strong solutions to spde with delays

ergodic and mixing properties of the boussinesq equations with a degenerate random forcing we establish the existence uniqueness and attraction properties of an ergodic invariant measure for the boussinesq equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales the central challenge is to establish time asymptotic smoothing properties of the markovian dynamics corresponding to this system towards this aim we encounter a lie bracket structure in the associated vector fields with a complicated dependence on solutions this leads us to develop a novel hormander type condition for infinite dimensional systems demonstrating the sufficiency of this condition requires new techniques for the spectral analysis of the malliavin covariance matrix

ergodicity for functional stochastic differential equations and applications abstract in this paper using the remote start or dissipative method we investigate ergodicity for several kinds of functional stochastic equations including functional stochastic differential equations sdes with variable delays neutral functional sdes functional sdes driven by jump processes and semi linear functional stochastic partial differential equations spdes using the ergodicity derived we then treat a couple of applications in stochastic approximation and optimization problems

ergodicity of the 2d navier stokes equations with random forcing we consider the navier stokes equation on a two dimensional torus with a random force acting at discrete times and analytic in space for arbitrarily small viscosity coefficient we prove the existence and uniqueness of the invariant measure for this system as well as exponential mixing in time

exponential ergodicity of stochastic burgers equations driven by α stable processes in this work we prove the strong feller property and the exponential ergodicity of stochastic burgers equations driven by 2 subordinated cylindrical brownian motions with 1 2 to prove the results we truncate the nonlinearity and use the derivative formula for sdes driven by stable noises established in zhang in stoch process appl 123 4 1213 1228 2013

quantitative harris type theorems for diffusions and mckean vlasov processes we consider d valued diffusion processes of type dxt b xt dt dbt assuming a geometric drift condition we establish contractions of the transitions kernels in kantorovich l1 wasserstein distances with explicit constants our results are in the spirit of hairer and mattingly s extension of harris theorem in particular they do not rely on a small set condition instead we combine lyapunov functions with reflection coupling and concave distance functions we retrieve constants that are explicit in parameters which can be computed with little effort from one sided lipschitz conditions for the drift coefficient and the growth of a chosen lyapunov function consequences include exponential convergence in weighted total variation norms gradient bounds bounds for ergodic averages and kantorovich contractions for nonlinear mckean vlasov diffusions in the case of sufficiently weak but not necessarily bounded nonlinearities we also establish quantitative bounds for sub geometric ergodicity assuming a sub geometric drift condition

spectral gaps in wasserstein distances and the 2d stochastic navier stokes equations we develop a general method to prove the existence of spectral gaps for markov semigroups on banach spaces unlike most previous work the type of norm we consider for this analysis is neither a weighted supremum norm nor an lp type norm but involves the derivative of the observable as well and hence can be seen as a type of 1 wasserstein distance this turns out to be a suitable approach for infinite dimensional spaces where the usual harris or doeblin conditions which are geared toward total variation convergence often fail to hold in the first part of this paper we consider semigroups that have uniform behavior which one can view its the analog of doeblin s condition we then proceed to study situations where the behavior is not so uniform but the system has a suitable lyapunov structure leading to a type of harris condition we finally show that the latter condition is satisfied by the two dimensional stochastic navier stokes equations even in situations where the forcing is extremely de generate using the convergence result we show that the stochastic navier stokes equations invariant measures depend continuously on the viscosity and the structure of the forcing

uniqueness of the invariant measure for a stochastic pde driven by degenerate noise we consider the stochastic ginzburg landau equation in a bounded domain we assume the stochastic forcing acts only on high spatial frequencies the low lying frequencies are then only connected to this forcing through the non linear cubic term of the ginzburg landau equation under these assumptions we show that the stochastic pde has a unique invariant measure the techniques of proof combine a controllability argument for thelow lying frequencies with an infinite dimensional version of the malliavin calculus to show positivity and regularity of the invariant measure this then implies the uniqueness of that measure

a condition for the equivalence of coupling and shift coupling it is proved in this paper that a weak parabolic harnack inequality for a markov semigroup implies the existence of a coupling and a shift coupling for the corresponding process with equal chances of success this implies equality of the tail and invariant sigma fields for the diffusion as well as equality of the class of bounded parabolic functions and the class of bounded harmonic functions

sufficient conditions for the eventual strong feller property for degenerate stochastic evolutions abstract the strong feller property is an important quality of markov semigroups which helps for example in establishing uniqueness of invariant measure unfortunately degenerate stochastic evolutions such as stochastic delay equations do not possess this property however the eventual strong feller property is sufficient in establishing uniqueness of invariant probability measure in this paper we provide operator theoretic conditions under which a stochastic evolution equation with additive noise possesses the eventual strong feller property the results are used to establish uniqueness of invariant probability measure for stochastic delay equations and stochastic partial differential equations with delay with an application in neural networks

chasing balls through martingale fields we consider the way sets are dispersed by the action of stochastic flows derived from martingale fields under fairly general continuity and ellipticity conditions the following dichotomy result is shown any nontrivial connected set x either contracts to a point under the action of the flow or its diameter grows linearly in time with speed at least a positive deterministic constant a the linear growth may further be identified again almost surely with a much stronger behavior which we call ball chasing if is any path with lipschitz constant smaller than a the ball of radius e around t contains points of the image of x for an asymptotically positive fraction of times t if the ball grows as the logarithm of time there are individual points in x whose images eventually remain in the ball

gibbsian dynamics and ergodicity for the stochastically forced navier stokes equation we study stationary measures for the two dimensional navier stokes equation with periodic boundary condition and random forcing we prove uniqueness of the stationary measure under the condition that all determining modes are forced the main idea behind the proof is to study the gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time history of the low modes

exponential mixing properties of stochastic pdes through asymptotic coupling we consider parabolic stochastic partial differential equations driven by white noise in time we prove exponential convergence of the transition probabilities towards a unique invariant measure under suitable conditions these conditions amount essentially to the fact that the equation transmits the noise to all its determining modes several examples are investigated including some where the noise does not act on every determining mode directly

generalized couplings and convergence of transition probabilities we provide sufficient conditions for the uniqueness of an invariant measure of a markov process as well as for the weak convergence of transition probabilities to the invariant measure our conditions are formulated in terms of generalized couplings we apply our results to several spdes for which unique ergodicity has been proven in a recent paper by glatt holtz mattingly and richards and show that under essentially the same assumptions the weak convergence of transition probabilities actually holds true

tamed em scheme of neutral stochastic differential delay equations abstract in this paper we investigate the convergence of the tamed euler maruyama em scheme for a class of neutral stochastic differential delay equations the strong convergence results of the tamed em scheme are presented under global and local non lipschitz conditions respectively moreover under the global lipschitz condition we provide the convergence rate of tamed em scheme which could be the same as the convergence rate of classical em scheme one half

reflection couplings and contraction rates for diffusions we consider contractivity for diffusion semigroups w r t kantorovich l 1 wasserstein distances based on appropriately chosen concave functions these distances are inbetween total variation and usual wasserstein distances it is shown that by appropriate explicit choices of the underlying distance contractivity with rates of close to optimal order can be obtained in several fundamental classes of examples where contractivity w r t standard wasserstein distances fails applications include overdamped langevin diffusions with locally non convex potentials products of these processes and systems of weakly interacting diffusions both of mean field and nearest neighbour type

geometric ergodicity for piecewise contracting processes with applications for tropical stochastic lattice models stochastic lattice models are increasingly prominent as a way to capture highly intermittent unresolved features of moist tropical convection in climate science and as continuum mesoscopic models in material science stochastic lattice models consist of suitably discretized continuum partial differential equations interacting with markov jump processes at each lattice site with transition rates depending on the local value of the continuum equation they are a special case of piecewise deterministic markov processes but often have an infinite state space and unbounded transition rates here a general theorem on geometric ergodicity for piecewise deterministic contracting processes is developed with full generality to apply to stochastic lattice models a highly nontrivial application to the stochastic skeleton model for the madden julian oscillation thual et al 2013 is developed here where there is an infinite state space with unbounded and also degenerate transition rates geometric ergodicity for the stochastic skeleton model guarantees exponential convergence to a unique invariant measure that defines the statistical tropical climate of the model another application of the general framework is developed here for stochastic lattice models designed to capture intermittent fluctuation in the simplest tropical climate models other straightforward applications to models motivated by material science are mentioned briefly here 2015 wiley periodicals inc

numerical simulation of polynomial speed convergence phenomenon we provide a hybrid method that captures the polynomial speed of convergence and polynomial speed of mixing for markov processes the hybrid method that we introduce is based on the coupling technique and renewal theory we propose to replace some estimates in classical results about the ergodicity of markov processes by numerical simulations when the corresponding analytical proof is difficult after that all remaining conclusions can be derived from rigorous analysis then we apply our results to seek numerical justification for the ergodicity of two 1d microscopic heat conduction models the mixing rate of these two models are expected to be polynomial but very difficult to prove in both examples our numerical results match the expected polynomial mixing rate well

exponential convergence to non equilibrium stationary states in classical statistical mechanics we continue the study of a model for heat conduction 6 consisting of a chain of non linear oscillators coupled to two hamiltonian heat reservoirs at different temperatures we establish existence of a liapunov function for the chain dynamics and use it to show exponentially fast convergence of the dynamics to a unique stationary state ingredients of the proof are the reduction of the infinite dimensional dynamics to a finite dimensional stochastic process as well as a bound on the propagation of energy in chains of anharmonic oscillators

exponential convergence for the stochastically forced navier stokes equations and other partially dissipative dynamics we prove that the two dimensional navier stokes equations possess an exponentially attracting invariant measure this result is in fact the consequence of a more general harris like ergodic theorem applicable to many dissipative stochastic pdes and stochastic processes with memory a simple iterated map example is also presented to help build intuition and showcase the central ideas in a less encumbered setting to analyze the iterated map a general doeblin like theorem is proven one of the main features of this paper is the novel coupling construction used to examine the ergodic theory of the non markovian processes

stability in distribution of stochastic functional differential equations abstract in this paper we investigate the stability in distribution for a class of stochastic functional differential equations sfdes which include stochastic differential delay equations sddes although stability in distribution has been studied by several authors recently there is so far no stability in distribution criterion on sfdes where the terms involved the delay components are highly nonlinear not bounded by linear functions in this paper we will establish the sufficient criteria on the stability in distribution for a class of highly nonlinear sfdes two examples will be given to illustrate our new results we also explain the reason why the existing stability in distribution criteria are not applicable by these two examples

isometric study of wasserstein spaces the real line recently kloeckner described the structure of the isometry group of the quadratic wasserstein space mathcal w 2 r n it turned out that the case of the real line is exceptional in the sense that there exists an exotic isometry flow following this line of investigation we compute isom ler wp r the isometry group of the wasserstein space wp r for all p in 1 infty setminus 2 we show that mathcal w 2 r is also exceptional regarding the parameter p wp r is isometrically rigid if and only if p neq 2 regarding the underlying space we prove that the exceptionality of p 2 disappears if we replace r by the compact interval 0 1 surprisingly in that case wp zo is isometrically rigid if and only if p neq1 moreover wo zo admits isometries that split mass and isom ler wo zo cannot be embedded into isom ler wor

lyapunov exponents of linear stochastic functional differential equations ii examples and case studies we give several examples and examine case studies of linear stochastic functional differential equations the examples fall into two broad classes regular and singular according to whether an underlying stochastic semi flow exists or not in the singular case we obtain upper and lower bounds on the maximal exponential growth rate overline lambda 1 sigma of the trajectories expressed in terms of the noise variance roughly speaking we show that for small sigma overline lambda 1 sigma behaves like sigma 2 2 while for large sigma it grows like log sigma in the regular case it is shown that a discrete oseledec spectrum exists and upper estimates on the top exponent lambda 1 are provided these estimates are sharp in the sense that they reduce to known estimates in the deterministic or nondelay cases

ergodic theory of infinite dimensional systems with applications to dissipative parabolic pdes we consider a class of randomly perturbed dynamical systems satisfying conditions which reflect the properties of general nonlinear dissipative parabolic pdes results on invariant measures and their exponential mixing properties are proved and applications to 2d navier stokes systems are included

a useful version of the central limit theorem for a general class of markov chains abstract in the paper we propose certain conditions relatively easy to verify which ensure the central limit theorem for some general class of markov chains to justify the usefulness of our criterion we further verify it for a particular discrete time markov dynamical system from the application point of view the examined system provides a useful tool in analysing the stochastic dynamics of gene expression in prokaryotes

asymptotic coupling and a general form of harris theorem with applications to stochastic delay equations there are many markov chains on infinite dimensional spaces whose one step transition kernels are mutually singular when starting from different initial conditions we give results which prove unique ergodicity under minimal assumptions on one hand and the existence of a spectral gap under conditions reminiscent of harris theorem the first uses the existence of couplings which draw the solutions together as time goes to infinity such asymptotic couplings were central to mattingly and sinai in comm math phys 219 3 523 565 2001 mattingly in comm math phys 230 3 461 462 2002 hairer in prob theory relat field 124 345 380 2002 bakhtin and mattingly in commun contemp math 7 553 582 2005 on which this work builds as in bakhtin and mattingly 2005 the emphasis here is on stochastic differential delay equations harris celebrated theorem states that if a markov chain admits a lyapunov function whose level sets are small in the sense that transition probabilities are uniformly bounded from below then it admits a unique invariant measure and transition probabilities converge towards it at exponential speed this convergence takes place in a total variation norm weighted by the lyapunov function a second aim of this article is to replace the notion of a small set by the much weaker notion of a d small set which takes the topology of the underlying space into account via a distance like function d with this notion at hand we prove an analogue to harris theorem where the convergence takes place in a wasserstein like distance weighted again by the lyapunov function this abstract result is then applied to the framework of stochastic delay equations in this framework the usual theory of harris chains does not apply since there are natural examples for which there exist no small sets except for sets consisting of only one point this gives a solution to the long standing open problem of finding natural conditions under which a stochastic delay equation admits at most one invariant measure and transition probabilities converge to it

stability analysis of polynomial fuzzy models via polynomial fuzzy lyapunov functions in this paper the stability of continuous time polynomial fuzzy models by means of a polynomial generalization of fuzzy lyapunov functions is studied fuzzy lyapunov functions have been fruitfully used in the literature for local analysis of takagi sugeno models a particular class of the polynomial fuzzy ones based on a recent taylor series approach which allows a polynomial fuzzy model to exactly represent a nonlinear model in a compact set of the state space it is shown that a refinement of the polynomial lyapunov function so as to make it share the fuzzy structure of the model proves advantageous conditions thus obtained are tested via available sos software

on unique ergodicity in nonlinear stochastic partial differential equations we illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes examples exhibiting parabolic and hyperbolic structure are studied in detail in the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the krylov bogolyubov procedure and compactness fails