alternative proof for the localization of sinai s walk we give an alternative proof of the localization of sinai s random walk in random environment under weaker hypothesis than the ones used by sinai moreover we give estimates that are stronger than the one of sinai on the localization neighborhood and on the probability for the random walk to stay inside this neighborhood

the problem of the most visited site in random environment we prove that the process of the most visited site of sinai s simple random walk in random environment is transient the rate of escape is characterized via an integral criterion our method also applies to a class of recurrent diffusion processes with random potentials it is interesting to note that the corresponding problem for the usual symmetric bernoulli walk or for brownian motion remains open

nonlinear lie derivations of triangular algebras in this paper we prove that every nonlinear lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero

rough paths and 1d sde with a time dependent distributional drift application to polymers motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions like the kardar parisi zhang kpz equation we reconsider the unique solvability of one dimensional stochastic differential equations the drift of which is a distribution by means of rough paths theory existence and uniqueness are established in the weak sense when the drift reads as the derivative of a holder continuous function regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed which makes the structure of the solutions more explicit than within the earlier framework of dirichlet processes

uniqueness of diffusion generators for two types of particle systems with singular interactions abstract for two types of stochastic particle systems in r d we show non explosion in finite time by proving that their respective generators are l1 unique where is their respective invariant in these cases even symmetrizing measure we also prove the much harder l2 uniqueness in both models

the limits of sinai s simple random walk in random environment we study the sample path asymptotics of a class of recurrent diffusion processes with random potentials including examples of sinai s simple random walk in random environment and brox s diffusion process with brownian potential the main results consist of several integral criteria which completely characterize all the possible levy classes therefore providing a very precise image of the almost sure asymptotic behaviors of these processes

invariant distributions and scaling limits for some diffusions in time varying random environments we consider a family of one dimensional diffusions in dynamical wiener mediums which are random perturbations of the ornstein uhlenbeck diffusion process we prove quenched and annealed convergences in distribution and under weigh ted total variation norms we find two kind of stationary probability measures which are either the standard normal distribution or a quasi invariant measure depending on the environment and which is naturally connected to a random dynamical system we apply these results to the study of a model of time inhomogeneous brox s diffusions which generalizes the diffusion studied by brox ann probab 14 4 1206 1218 1986 and those investigated by gradinaru and offret ann inst henri poincare probab stat 2011 we point out two distinct diffusive behaviours and we give the speed of convergences in the quenched situations

strong solutions of stochastic equations with singular time dependent drift we prove existence and uniqueness of strong solutions to stochastic equations in domains with unit diffusion and singular time dependent drift b up to an explosion time we only assume local l q l p integrability of b in g with d p 2 q 1 we also prove strong feller properties in this case if b is the gradient in x of a nonnegative function blowing up as g x g we prove that the conditions 2d t k 2d t ke 0 2 imply that the explosion time is infinite and the distributions of the solution have sub gaussian tails

solving the kpz equation we introduce a new concept of solution to the kpz equation which is shown to extend the classical cole hopf solution this notion provides a factorisation of the cole hopf solution map into a universal measurable map from the probability space into an explicitly described auxiliary metric space composed with a new solution map that has very good continuity properties the advantage of such a formulation is that it essentially provides a pathwise notion of a solution together with a very detailed approximation theory in particular our construction completely bypasses the cole hopf transform thus laying the groundwork for proving that the kpz equation describes the uctuations of systems in the kpz universality class as a corollary of our construction we obtain very detailed new regularity results about the solution as well as its derivative with respect to the initial condition other byproducts of the proof include an explicit approximation to the stationary solution of the kpz equation a well posedness result for the fokker planck equation associated to a particle diusing in a rough space time dependent potential and a new periodic homogenisation result for the heat equation with a space time periodic potential one ingredient in our construction is an example of a non gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge

some parabolic pdes whose drift is an irregular random noise in space a new class of random partial differential equations of parabolic type is considered where the stochastic term consists of an irregular noisy drift not necessarily gaussian for which a suitable interpretation is provided after freezing a realization of the drift stochastic process we study existence and uniqueness in some appropriate sense of the associated parabolic equation and a probabilistic interpretation is investigated

analysis and geometry on configuration spaces the gibbsian case using a natural riemannian geometry like structure on the configuration space over rd we prove that for a large class of potentials the corresponding canonical gibbs measures on 1 can be completely characterized by an integration by parts formula that is if is the gradient of the riemannian structure on one can define a corresponding divergence div such that the canonical gibbs measures are exactly those measures for which and div are dual operators onl2 one consequence is that for such the corresponding dirichlet forms e are defined in addition each of them is shown to be associated with a conservative diffusion process on with invariant measure the corresponding generators are extensions of the operator div the diffusions can be characterized in terms of a martingale problem and they can be considered as a brownian motion on perturbed by a singular drift another main result of this paper is the following if is a canonical gibbs measure then it is extreme or a pure phase if and only if the corresponding weak sobolev spacew1 2 on is irreducible as a consequence we prove that for extreme canonical gibbs measures the above mentioned diffusions are time ergodic in particular this holds for tempered grand canonical gibbs measures ruelle measures provided that the activity constant is small enough we also include a complete discussion of the free case i e 0 where the underlying space rdis even replaced by a riemannian manifoldx

commuting traces of biadditive mappings commutativity preserving mappings and lie mappings biadditive mappings b r r r where r is a prime ring with certain additional properties satisfying b x x x xb x x for all x r are characterized as an application we determine the structures of commutativity preserving mappings lie isomorphisms and lie derivations of certain prime rings

receding horizon unbiased fir filters for continuous time state space models without a priori initial state information a receding horizon unbiased finite impulse response filter rhuff is proposed for continuous time state space models linearity unbiasedness finite impulse response fir structure and independence of the initial state information will be required in advance in addition to a performance index of minimum variance the proposed rhuff is obtained by directly minimizing the performance index with the unbiasedness constraint the proposed rhuff is represented first in a standard fir form and then in an iterative form it is shown that the rhuff is equivalent to the existing receding horizon rh kalman fir filter the former is more systematic and logical while the latter is heuristic due to the handling of infinite covariance of the initial state information

fir filters and recursive forms for continuous time invariant state space models an fir finite impulse response filter and an fir smoother are introduced for continuous time invariant state space models it is shown in this note that finite impulse responses of the fir filter and smoother can be easily determined by solving a simple riccati type matrix differential equation on a finite interval especially for systems with stationary processes finite impulse responses of the fir filter and smoother become time invariant and can be computed from simpler equations for fast computational purposes recursive forms of the fir filter and smoother are derived by using adjoint variables in this case all gains for recursive forms are shown to be constant

stochastic differential equations for dirichlet processes we consider the stochastic differential equation dxt a xt dwt b xt dt where w is a one dimensional brownian motion we formulate the notion of solution and prove strong existence and pathwise uniqueness results when a is in c1 2 and b is only a generalized function for example the distributional derivative of a holder function or of a function of bounded variation when b aa that is when the generator of the sde is the divergence form operator a result on non existence of a strong solution and non pathwise uniqueness is given as well as a result which characterizes when a solution is a semimartingale or not we also consider extensions of the notion of stratonovich integral

the structure of lie derivations on c algebras we prove that every lie derivation on a c algebra is in standard form that is it can be

design of rls fir filter using covariance information in linear continuous time stochastic systems this paper addresses a new design method of recursive least squares rls and finite impulse response fir filter using covariance information in linear continuous time stochastic systems the signal process is observed with additive white noise it is assumed that the white observation noise is independent of the signal process the auto covariance function of the signal is expressed in the semi degenerate kernel form the rls fir filter uses the following information 1 the auto covariance function of the signal expressed in the semi degenerate kernel form 2 the variance of the white observation noise process 3 the observed values

absolute continuity of symmetric diffusions let x and y be symmetric diffusion processes with a common state space and let p m resp q be the law of x resp y with its symmetry measure m resp as initial distribution we study the consequences of the absolute continuity condition q t 1 t the density p also serves to link the dirichlet forms of x and y in a simple way our identification of l relies on notions of even and odd for additive functionals these notions complement fukushima s decomposition and the forward backward martingale decomposition of lyons and zheng

controlling rough paths abstract we formulate indefinite integration with respect to an irregular function as an algebraic problem which has a unique solution under some analytic constraints this allows us to define a good notion of integral with respect to irregular paths with holder exponent greater than 1 3 e g samples of brownian motion and study the problem of the existence uniqueness and continuity of solution of differential equations driven by such paths we recover young s theory of integration and the main results of lyons theory of rough paths in holder topology

the infinitesimal generator of the stochastic burgers equation we develop a martingale approach for a class of singular stochastic pdes of burgers type including fractional and multi component burgers equations by constructing a domain for their infinitesimal generators it was known that the domain must have trivial intersection with the usual cylinder test functions and to overcome this difficulty we import some ideas from paracontrolled distributions to an infinite dimensional setting in order to construct a domain of controlled functions using the new domain we are able to prove existence and uniqueness for the kolmogorov backward equation and the martingale problem we also extend the uniqueness result for energy solutions of the stochastic burgers equation of gubinelli and perkowski j am math soc 31 2 427 471 2018 to a wider class of equations as applications of our approach we prove that the stochastic burgers equation on the torus is exponentially l 2 ergodic and that the stochastic burgers equation on the real line is ergodic

slow movement of random walk in random environment on a regular tree we consider a recurrent random walk in random environment on a regular tree under suitable general assumptions upon the distribution of the environment we show that the walk exhibits an unusual slow movement the order of magnitude of the walk in the first n steps is log n 3

a characterization of first order phase transitions for superstable interactions in classical statistical mechanics we give bounds on finite volume expectations for a set of boundary conditions containing the support of any tempered gibbs state and prove a theorem connecting the behavior of gibbs states to the differentiability of the pressure for continuum statistical mechanical systems with long range superstable potentials convergence of grand canonical gibbs states is also studied

generalized integration and stochastic odes stochastic forward integrals for processes more general than semimartingales are shown to exist generalized forms of ito wentzell formula and covariation formula are proved and one dimensional stochastic equations driven by finite quadratic variation processes and semimartingales are solved this generalized stochastic calculus is motivated by applications to uniqueness and dependence on parameters for stochastic equations with nonregular drift

moderate deviations for diffusions with brownian potentials we present precise moderate deviation probabilities in both quenched and annealed settings for a recurrent diffusion process with a brownian potential our method relies on fine tools in stochastic calculus including kotani s lemma and lamperti s representation for exponential functionals in particular our result for quenched moderate deviations is in agreement with a recent theorem of comets and popov probab theory related fields 126 2003 571 609 who studied the corresponding problem for sinai s random walk in random environment

capacity and quantum mechanical tunneling we connect the notion of capacity of sets in the theory of symmetric markov process and dirichlet forms with the notion of tunneling through the boundary of sets in quantum mechanics in particular we show that for diffusion processes the notion appropriate to a boundary without tunneling is more refined than simply capacity zero we also discuss several examples in d

the structure of nonlinear lie derivation on von neumann algebras national natural science foundation of china 11071201 youth national natural science foundation of china 11001230 youth talented natural science foundation of fujian 2008f3103 fundamental research funds for the central universities 2010121001

multidimensional stochastic differential equations with distributional drift this paper investigates a time dependent multidimensional stochastic differential equation with drift being a distribution in a suitable class of sobolev spaces with negative derivation order this is done through a careful analysis of the corresponding kolmogorov equation whose coefficient is a distribution

on some random walks on z in random medium we consider random walks on z in a stationary random medium defined by an ergodic dynamical system in the case when the possible jumps are l 1 0 1 for some fixed integer l we provide a recurrence criterion expressed in terms of the sign of the maximal liapounov exponent of a certain random matrix and give an algorithm of calculation of that exponent next we characterize the existence of the absolutely continuous invariant measure for the markov chain of the environments viewed from the particle and also characterize in the transient cases the existence of a nonzero drift to study the validity of the central limit theorem we consider the notion of harmonic coordinates introduced by kozlov we characterize the existence of both the invariant measure and the harmonic coordinates and show in the recurrent case that the existence of those two objects is equivalent to the validity of an invariance principle we give sufficient conditions for the validity of the central limit theorem in the transient cases finally we consider the previous results in the context of a random medium defined by an irrational rotation on the circle and study their realization in terms of regularity and diophantine approximation

rates of convergence of diffusions with drifted brownian potentials we are interested in the asymptotic behaviour of a diffusion process with drifted brownian potential the model is a continuous time analogue to the random walk in random environment studied in the classical paper of kesten kozlov and spitzer we not only recover the convergence of the diffusion process which was previously established by kawazu and tanaka but also obtain all the possible convergence rates an interesting feature of our approach is that it shows a clear relationship between drifted brownian potentials and bessel processes

probability distribution of the free energy of the continuum directed random polymer in 1 1 dimensions we consider the solution of the stochastic heat equation tz d 1 2 x zz p w with delta function initial condition z t d0 x dixd0 whose logarithm with appropriate normalization is the free energy of the con tinuum directed polymer or the hopf cole solution of the kardar parisi zhang equation with narrow wedge initial conditions we obtain explicit formulas for the one dimensional marginal distributions the crossover distributions which interpolate between a standard gaussian dis tribution small time and the gue tracy widom distribution large time the proof is via a rigorous steepest descent analysis of the tracy widom formula for the asymmetric simple exclusion process with antishock initial data which is shown to converge to the continuum equations in an appropriate weakly asymmetric limit the limit also describes the crossover behavior between the symmetric and asymmetric exclusion processes 2010 wiley periodicals inc

existence of strong solutions for ito s stochastic equations via approximations given strong uniqueness for an ito s stochastic equation with discontinuous coefficients we prove that its solution can be constructed on any probability space by using for example euler s polygonal approximations stochastic equations in d and in domains in d are considered

a one dimensional diffusion process in a wiener medium we consider a one dimensional diffusion process in a random medium which is itself generated by a wiener process the asymptotic behaviour is investigated using the self similarity of the medium process the results are analogous to the lattice case

limit law for transition probabilities and moderate deviations for sinai s random walk in random environment we consider a one dimensional random walk in random environment in the sinai s regime our main result is that logarithms of the transition probabilities after a suitable rescaling converge in distribution as time tends to infinity to some functional of the brownian motion we compute the law of this functional when the initial and final points agree also among other things we estimate the probability of being at time t at distance at least z from the initial position when z is larger than ln2t but still of logarithmic order in time

recursive limited memory filtering and scattering theory corresp redheffer scattering theory is reviewed in a generalized setting as a method to derive recursive solutions of linear two point boundary value problems tpbvp over arbitrarily varying intervals the results can be used to derive a complete solution for the problem of limited memory or sliding window estimation when a usual state space model for the signal is available recursive limited memory filters are derived for both continuous and discrete time signals

analysis and geometry on configuration spaces abstract in this paper foundations are presented to a new systematic approach to analysis and geometry for an important class of infinite dimensional manifolds namely configuration spaces more precisely a differential geometry is introduced on the configuration space x over a riemannian manifold x this geometry is non flat even if x r d it is obtained as a natural lifting of the riemannian structure on x in particular a corresponding gradient divergence div and laplace beltrami operator h div gamma are constructed the associated volume elements i e all measures on x w r t which and div gamma become dual operators on l 2 x are identified as exactly the mixed poisson measures with mean measure equal to a multiple of the volume element dx on x in particular all these measures obey an integration by parts formula w r t vector fields on x the corresponding dirichlet forms e on l 2 x are therefore defined each is shown to be associated with a diffusion process which is thus the brownian motion on x and which is sub sequently identified as the usual independent infinite particle process on x the associated heat semigroup t t t 0 is calculated explicitly it is also proved that the diffusion process when started with is time ergodic or equivalently e gamma is irreducible or equivalently t t t 0 is ergodic if and only if is poisson measure z dx with intensity z dx for some z 0 furthermore it is shown that the laplace beltrami operator h div gamma gamma on l 2 x z dx is unitary equivalent to the second quantization of the laplacian x on x on the corresponding fock space n 0 l 2 x z dx n as another direct consequence of our results we obtain a representation of the lie algebra of compactly supported vector fields on x on poisson space finally generalizations to the case where dx is replaced by an absolutely continuous measure and also to interacting particle systems on x are described in particular the case where the mixed poisson measures are replaced by gibbs measures of ruelle type on x

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

rough stochastic pdes in this article we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic pdes of burgers type that exhibit too high spatial roughness for classical analytical methods to apply in fact the class of spdes that we consider is genuinely ill posed in the sense that different approximations to the nonlinearity may converge to different limits using rough path theory a pathwise notion of solution to these spdes is formulated and we show that this yields a well posed problem that is stable under a large class of perturbations including the approximation of the rough driving noise by a mollified version and the addition of hyperviscosity we also show that under certain structural assumptions on the coefficients the spdes under consideration generate a reversible markov semigroup with respect to a diffusion measure that can be given explicitly

a receding horizon kalman fir filter for discrete time invariant systems a receding horizon kalman fir filter is presented that combines the kalman filter and the receding horizon strategy when the horizon initial state is assumed to be unknown the suggested filter is a fir filter form which has many good inherent properties it can always be defined irrespective of singularity problems caused by unknown information about the horizon initial state the suggested filter can be represented in either an iterative form or a standard fir form it is also shown that the suggested filter possesses the unbiasedness property and the remarkable deadbeat property irrespective of any horizon initial condition the validity of the suggested filter is illustrated by numerical examples

discrete schemes for processes in random media we present discrete schemes for processes in random media we prove two results the first one is the convergence of sinai s random walks in random environments to the brox model the second one is the convergence of random walks in media with random gates to a continuous process in a poisson potential the proofs are based on the following idea we consider the discrete media as random potentials for continuous models

convergence of dirichlet forms and associated schrodinger operators abstract we study the convergence of local dirichlet forms and associated schrodinger operators conditions are given for n in order that the schrodinger operators n n associated with dirichlet forms u x 2 n 2 x dx generate contraction semigroups which converge strongly in l2 r d moreover conditions are given which permit one to identify the limit contraction semigroups and schrodinger operators

a receding horizon kalman fir filter for linear continuous time systems a receding horizon kalman finite impulse response fir filter is suggested for continuous time systems combining the kalman filter with the receding horizon strategy in the suggested filter the horizon initial state is assumed to be unknown it can always be obtained irrespective of unknown information on the horizon initial state the filter may be the first stochastic fir form for continuous time systems that may have many good inherent properties the suggested filter can be represented in an iterative form and also in a standard fir form the suggested filter turns out to be a remarkable deadbeat observer the validity of the suggested filter is illustrated by numerical examples

ring derivations on standard operator algebras abstract let a be a standard operator algebra on an infinite dimensional banach space x we prove that every ring derivation d a b x is of the form d a ta at for some t b x

patterns in sinai s walk sinai s random walk in random environment shows interesting patterns on the exponential time scale we characterize the patterns that appear on infinitely many time scales after appropriate rescaling a functional law of iterated logarithm the curious rate function captures the difference between one sided and two sided behavior

lyapunov functions for random walks and strings in random environment we study two typical examples of countable markov chains in random environment using the lyapunov functions method random walk and random string in random environment in each case we construct an explicit lyapunov function investigating the behavior of this function we get the classification for recurrence transience ergodicity we obtain new results for random strings in random environment though we simply review well known results for random walks using our approach