obtuse cones and gram matrices with nonnegative inverse abstract we study some properties of gram matrices with nonnegative inverse which lead to the constructions of obtuse cones these constructions can find applications in optimization problems e g in methods of convex feasibility problems or of convex minimization problems

the critical parameter for the heat equation with a noise term to blow up in finite time consider the stochastic partial differential equation u t u xx u gamma w where x in mathbf i equiv 0 j w w t x is 2 parameter white noise and we assume that the initial function u 0 x is nonnegative and not identically 0 we impose dirichlet boundary conditions on u in the interval i we say that u blows up in finite time with positive probability if there is a random time t 0 it was known that if gamma 3 2 then there is a positive probability that u blows up in finite time

optimal information bit decoding of linear block codes we consider the problem of decoding a linear block code used over a binary symmetric channel with crossover probability p when the goal of the decoding is to minimize the probability of an information bit error in general there will be several different decoding strategies each corresponding to the optimal rule over some range p spl isin p sub i 1 p sub i each of these strategies can be implemented by standard array with the appropriate choice of coset leaders that are not in general of minimum weight we give optimal rules for selecting coset leaders in the first region 0 p sub 1 and the region p sub max 1 2 for any linear mapping between information bits and codewords the determination of the optimal rule has been an open problem for small p the rule for p p sub max is important because for many codes when p is fixed the optimal rule for any coset corresponds to either the optimal rule for small p or the optimal rule for p near 1 2 the optimal decoding strategy varies with the generator matrix used in general but it is demonstrated that the optimal generator matrix is separable when d sup spl perp 2 the results are shown to extend in a natural way to the case where the goal is to minimize the probability that arbitrary collections of information bits are in error we examine some specific codes and compare the performance under optimal decoding to the performance of previously suggested strategies

impacts of noise on a class of partial differential equations abstract this paper is concerned with effects of noise on the solutions of partial differential equations we first provide a sufficient condition to ensure the existence of a unique positive solution for a class of stochastic partial differential equations then we prove that noise could induce singularities finite time blow up of solutions finally we show that a stochastic allen cahn equation does not have finite time singularities and the unique solution exists globally

long time existence for signed solutions of the heat equation with a noise term let be the circle 0 j with the ends identified we prove long time existence for the following equation

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

blow up of solutions for semilinear stochastic delayed reaction diffusion equations with levy noise abstract this paper is concerned with a class of semilinear stochastic delayed reaction diffusion equations driven by levy noise in a separable hilbert space we establish sufficient conditions to ensure the existence of a unique positive solution moreover we study blow up of solutions in finite time in mean l p norm sense several examples are given to illustrate applications of the theory

asymptotic couplings by reflection and applications for nonlinear monotone spdes abstract asymptotic couplings by reflection are constructed for a class of nonlinear monotone spdes stochastic partial differential equations as applications gradient holder estimates as well as exponential convergence are derived for the associated markov semigroup the main results are illustrated by stochastic generalized porous media equations stochastic p laplacian equations and stochastic generalized fast diffusion equations we emphasize that the gradient estimate is studied for the first time for these equations

non homogeneous generalized newtonian fluids we show the existence of weak solutions to the system describing the motion of incompressible non homogeneous generalized newtonian fluids if the extra stress tensor s d possesses p structure with p ge frac 3d 2 d 2 and variable viscosity the limiting process in the equation of motion is justified by a variational argument which is new in this context

continuous dependence on the coefficients and global existence for stochastic reaction diffusion equations abstract we prove convergence of the solutions x n of semilinear stochastic evolution equations on a banach space b driven by a cylindrical brownian motion in a hilbert space h d x n t a n x t f n t x n t d t g n t x n t d w h t x n 0 n assuming that the operators a n converge to a and the locally lipschitz functions f n and g n converge to the locally lipschitz functions f and g in an appropriate sense moreover we obtain estimates for the lifetime of the solution x of the limiting problem in terms of the lifetimes of the approximating solutions x n we apply the results to prove global existence for reaction diffusion equations with multiplicative noise and a polynomially bounded reaction term satisfying suitable dissipativity conditions the operator governing the linear part of the equation can be an arbitrary uniformly elliptic second order elliptic operator

ornstein uhlenbeck processes with singular drifts integral estimates and girsanov densities we consider a perturbation of a hilbert space valued ornstein uhlenbeck process by a class of singular nonlinear non autonomous maximal monotone time dependent drifts the only further assumption on the drift is that it is bounded on balls in the hilbert space uniformly in time first we introduce a new notion of generalized solutions for such equations which we call pseudo weak solutions and prove that they always exist and obtain pathwise estimates in terms of the data of the equation then we prove that their laws are absolutely continuous with respect to the law of the original ornstein uhlenbeck process in particular we show that pseudo weak solutions always have continuous sample paths in addition we obtain integrability estimates of the associated girsanov densities some of our results concern non random equations as well while probabilistic results are new even in finite dimensional autonomous settings

refined asymptotics for the blowup of ut δu up this work is concerned with positive blowing up solutions of the semilinear heat equation ut u up in rn our main contribution is a sort of center manifold analysis for the equation in similarity variables leading to refined asymptotics for u in a backward space time parabola near any blowup point we also explore a connection between the asymptotics of u and the local geometry of the blowup set

large deviations for a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the non gaussian stochastic reaction diffusion equation srde partial t nu varepsilon mathscr l nu varepsilon f x nu varepsilon varepsilon sigma x nu varepsilon ddot w tx as a random perturbation of the deterministic rde partial t nu 0 mathscr l nu 0 f x nu 0 here the space variable takes values on the unit circle s 1 and mathscr l is a strongly elliptic second order operator with constant coefficients the functions f and sigma are sufficiently regular so that there is a unique solution to the above srde for any continuous initial condition we also assume that there are positive constants m and m such that m leq sigma x y leq m for all x in s 1 and all y in mathbb r the perturbation ddot w tx is the formal derivative of a brownian sheet it is known that if the initial condition is continuous then the solution will also be continuous and moreover if the initial condition is assumed to be holder continuous of exponent kappa for some 0 kappa frac 1 2 then the solution will be holder continuous of exponent kappa 2 as a function of t x in this paper we establish the large deviations principle for nu varepsilon in the holder norm of exponent kappa 2 when the initial condition is holder continuous of exponent kappa for any 0 kappa frac 1 2 and when the initial condition is assumed only to be continuous we establish the large deviations principle for nu varepsilon in the supremum norm moreover we prove that these large deviations principles are uniform with respect to the initial condition

on the effect of a noise on the solutions of the focusing supercritical nonlinear schrodinger equation we investigate the influence of a random perturbation of white noise type on the finite time blow up of solutions of a focusing supercritical nonlinear schrodinger equation we prove that contrary to the deterministic case any initial data gives birth to a solution which develops singularities moreover the singularities appear immediately we use a stochastic generalization of the variance identity and a control argument

successive approximations to solutions of stochastic differential equations abstract in the present paper we shall investigate under what conditions the sequence of stochastic processes constructed by the successive approximations converges uniformly to solutions of a stochastic differential equation of ito type and shall present the local or global existence and uniqueness theorem for solutions of the above mentioned equation under more general conditions we note that lemma 3 in this paper is a generalization of gard s lemma and guarantees the existence of functions which satisfy the conditions of theorems 2 and 3 in this paper respectively theorem 3 includes as a special case a generalization of yamada s theorem which is proved by the method of the successive approximations

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

boundary behavior and interior holder regularity of the solution to nonlinear stochastic partial differential equation driven by space time white noise abstract we present unique solvability result in weighted sobolev spaces of the equation u t a u x x b u x c u u 1 b t 0 x 0 1 given with initial data u 0 u 0 and zero boundary condition here 0 1 2 b is a space time white noise and the coefficients a b c and are random functions depending on t x we also obtain various interior holder regularities and boundary behaviors of the solution for instance if the initial data is in appropriate l p space then for any small e 0 and t almost surely 1 1 2 u c t x 1 4 2 e 1 2 e 0 t 0 1 1 2 where x is the distance from x to the boundary taking one gets the maximal holder exponents in time and space which are 1 4 2 e and 1 2 e respectively also letting 1 2 one gets better decay or behavior near the boundary

random attractors for a class of stochastic partial differential equations driven by general additive noise abstract the existence of random attractors for a large class of stochastic partial differential equations spde driven by general additive noise is established the main results are applied to various types of spde as e g stochastic reaction diffusion equations the stochastic p laplace equation and stochastic porous media equations besides classical brownian motion we also include space time fractional brownian motion and space time levy noise as admissible random perturbations moreover cases where the attractor consists of a single point are also investigated and bounds for the speed of attraction are obtained

weak solution of a stochastic 3d cahn hilliard navier stokes model driven by jump noise abstract we investigate a stochastic 2d and 3d cahn hilliard navier stokes system with a multiplicative noise of levy type the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard system for the order phase parameter we prove that the system the existence of weak martingale solution for both 2d and 3d cases the proof of the existence is based on a classical galerkin approximation as well as some compactness methods in the 2d case we prove the pathwise uniqueness of the weak solution

uniqueness and blow up for a stochastic viscous dyadic model we consider the dyadic model with viscosity and additive gaussian noise as a simplified version of the stochastic navier stokes equations with the purpose of studying uniqueness and emergence of singularities we prove pathwise uniqueness and absence of blow up in the intermediate intensity of the non linearity morally corresponding to the 3d case and blow up for stronger intensity moreover blow up happens with probability one for regular initial data

strong solutions for a stochastic model of two dimensional second grade fluids driven by levy noise abstract in this paper we consider stochastic second grade fluids driven by levy noise on a bounded domain of r 2 global existence and uniqueness of strong probabilistic solutions are established which improves the existing corresponding results for martingale solutions in literature

long time existence for the heat equation with a noise term we consider the equationu t u xx u w forx on a finite interval with dirichlet boundary conditions w is spacetime white noise the initial condition is continuous and nonnegative we show existence and uniqueness for all time provided 1 3 2

well posedness of stochastic partial differential equations with lyapunov condition abstract in this paper we show the existence and uniqueness of strong solutions for a large class of spde where the coefficients satisfy the local monotonicity and lyapunov condition one sided linear growth condition moreover some new invariance result and stronger regularity estimate are also established for the solutions as examples the main result is applied to stochastic tamed 3d navier stokes equations stochastic generalized curve shortening flow singular stochastic p laplace equations stochastic fast diffusion equations stochastic burgers type equations and stochastic reaction diffusion equations

large deviations for the invariant measure of a reaction diffusion equation with non gaussian perturbations in this paper we establish a large deviations principle for the invariant measure of the non gaussian stochastic partial differential equation spde t v e v e f x v e e x v e ddot w tx here is a strongly elliptic second order operator with constant coefficients h dh xx h and the space variablex takes values on the unit circles 1 the functionsf and are of sufficient regularity to ensure existence and uniqueness of a solution of the stochastic pde and in particular we require that 0 m m wherem andm are some finite positive constants the perturbationw is a brownian sheet it is well known that under some simple assumptions the solutionv 2 is ac k s 1 valued markov process for each 0 1 2 wherec s 1 is the banach space of real valued continuous functions ons 1 which are holder continuous of exponent we prove under some further natural assumptions onf and which imply that the zero element ofc s 1 is a globally exponentially stable critical point of the unperturbed equation t 0 0 f x 0 that e has a unique stationary distributionv k on c s 1 c k s 1 when the perturbation parameter e is small enough some further calculations show that as e tends to zero v k tends tov k 0 the point mass centered on the zero element ofc s 1 the main goal of this paper is to show that in factv k is governed by a large deviations principle ldp our starting point in establishing the ldp forv k is the ldp for the process e which has been shown in an earlier paper our methods of deriving the ldp forv k based on the ldp for e are slightly non standard compared to the corresponding proofs for finite dimensional stochastic differential equations since the state spacec s 1 is inherently infinite dimensional

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

well posedness and invariant measures for a class of stochastic 3d navier stokes equations with damping driven by jump noise abstract a class of stochastic 3d navier stokes equation with damping driven by a noise of levy type is considered in this paper the existence and uniqueness of the solution for the problem 1 1 1 2 are proved for 2 with any 0 and 1 4 as 2 the main result covered various types of spde such as stochastic 3d navier stokes equations with damping stochastic tamed 3d navier stokes equations stochastic three dimensional brinkman forchheimer extended darcy model by using the exponential stability of solutions the existence of a unique invariant measures is proved

optimal control of semilinear stochastic evolution equations the aim of this paper is to initiate a semigroup theory based approach to characterization of optimal markov controls for controlled semilinear stochastic evolution equations it may be recalled that markov controls are those that depend only on the current state at each time for finite dimensional controlled stochastic differential equations with a nondegenerate diffusion matrix this task is traditionally achieved through the hamilton jacobi bellman equation of dynamic programming associated with the problem and an accompanying verification theorem the latter states that an optimal markov control can be explicitly obtained by the pointwise minimization of a hamiltonian derivable from the solution of the hjb equation moreover any optimal markov control is obtainable in this manner

the existence and uniqueness of energy solutions to local non lipschitz stochastic evolution equations abstract let h v and k be separable hilbert spaces in this paper we consider the existence and uniqueness of energy solutions to the following stochastic evolution equation d x t a t x t f t x t d t g t x t d w t t 0 t x 0 x 0 h where a t v v is a linear bounded operator with coercivity monotone condition and hemicontinuity f 0 h h and g 0 h l 2 0 k h are measurable functions and satisfy the local non lipschitz condition proposed by the author t taniguchi successive approximations to solutions of stochastic differential equations j differential equations 96 1992 152 169

some non existence results for a class of stochastic partial differential equations abstract consider the following stochastic partial differential equation t u t x l u t x u t x f t x t 0 and x r d the operator l is the generator of a strictly stable process and f is the random forcing term which is assumed to be gaussian under some additional conditions most notably on and the initial condition we show non existence of global random field solutions our results complement those of p chow in 5 and 6 among others

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

a note on the computation of the bit error rate for binary block codes abstract the bit error rate of binary codes on the binary symmetric channel with raw error probability p is a performance index of primary importance in many applications it is the probability that decoded information symbols are in error after complete decoding in this paper the class of linear block codes having a transitive group of automorphisms is considered it is shown that a linear homogeneous differential transformation of the weight enumerator enables us to obtain a closed expression for the bit error rate in terms of p the application of the method is general enough to allow the computation of bit error rates for all cyclic codes

pathwise mild solutions for quasilinear stochastic partial differential equations abstract stochastic partial differential equations spdes have become a key modeling tool in applications yet there are many classes of spdes where the existence and regularity theory for solutions is not completely developed here we contribute to this aspect and prove the existence of mild solutions for a broad class of quasilinear cauchy problems including among others cross diffusion systems as a key application our solutions are local in time and are derived via a fixed point argument in suitable function spaces the key idea is to combine in a suitable way the classical theory of deterministic quasilinear parabolic partial differential equations pdes with recent theory of evolution semigroups we also show how to apply our theory to the shigesada kawasaki teramoto skt model furthermore we provide examples of blow up and ill posed operators which can occur after finite time showing that solutions can only be local in time for general quasilinear spdes while they might be global in time for special subclasses of problems

stochastic tamed 3d navier stokes equations existence uniqueness and ergodicity in this paper we prove the existence of a unique strong solution to a stochastic tamed 3d navier stokes equation in the whole space as well as in the periodic boundary case then we also study the feller property of solutions and prove the existence of invariant measures for the corresponding feller semigroup in the case of periodic conditions moreover in the case of periodic boundary and degenerated additive noise using the notion of asymptotic strong feller property proposed by hairer and mattingly ann math 164 993 1032 2006 we prove the uniqueness of invariant measures for the corresponding transition semigroup

lower estimates of transition densities and bounds on exponential ergodicity for stochastic pde s a formula for the transition density of a markov process defined by an infinite dimensional stochastic equation is given in terms of the ornstein uhlenbeck bridge and a useful lower estimate on the density is provided as a consequence uniform exponential ergodicity and v ergodicity are proved for a large class of equations we also provide computable bounds on the convergence rates and the spectral gap for the markov semigroups defined by the equations the bounds turn out to be uniform with respect to a large family of nonlinear drift coefficients examples of finite dimensional stochastic equations and semilinear parabolic equations are given

random attractors for locally monotone stochastic partial differential equations abstract we prove the existence of random dynamical systems and random attractors for a large class of locally monotone stochastic partial differential equations perturbed by additive levy noise the main result is applicable to various types of spde such as stochastic burgers type equations stochastic 2d navier stokes equations the stochastic 3d leray model stochastic power law fluids the stochastic ladyzhenskaya model stochastic cahn hilliard type equations stochastic kuramoto sivashinsky type equations stochastic generalized porous media equations and stochastic generalized p laplace equations

on the abstract cauchy problem of parabolic type in spaces of continuous functions on etudie l existence l unicite et les proprietes de regularite des solutions du probleme de cauchy u t au r f t o t t u o u 0 ou a d a ce e est un operateur lineaire clos dans l espace de banach e

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

markovianity and ergodicity for a surface growth pde the paper analyzes a model in surface growth where the uniqueness of weak solutions seems to be out of reach we prove existence of a weak martingale solution satisfying energy inequalities and having the markov property furthermore under nondegeneracy conditions on the noise we establish that any such solution is strong feller and has a unique invariant measure

global l2 solutions of stochastic navier stokes equations this paper concerns the cauchy problem in r d for the stochastic navier stokes equation 1 u u u u p f u u p g u o w u 0 u 0 div u 0 driven by white noise w under minimal assumptions on regularity of the coefficients and random forces the existence of a global weak martingale solution of the stochastic navier stokes equation is proved in the two dimensional case the existence and pathwise uniqueness of a global strong solution is shown a wiener chaos based criterion for the existence and uniqueness of a strong global solution of the navier stokes equations is established

existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators abstract in this paper we establish the existence and uniqueness of solutions for nonlinear evolution equations on a banach space with locally monotone operators which is a generalization of the classical result for monotone operators in particular we show that local monotonicity implies pseudo monotonicity the main results are applied to pde of various types such as porous medium equations reaction diffusion equations the generalized burgers equation the navier stokes equation the 3d leray model and the p laplace equation with non monotone perturbations

weighted stochastic sobolev spaces and bilinear spdes driven by space time white noise in this paper we develop basic elements of malliavin calculus on a weightedl2 this class of generalized wiener functionals is a hilbert space it turns out to be substantially smaller than the space of hida distributions while large enough to accommodate solutions of bilinear stochastic pdes as an example we consider a stochastic advection diffusion equation driven by space time white noise in rd it is known that ford 1 this equation has no solutions inl2 in contrast it is shown in the paper that in an appropriately weightedl2 there is a unique solution to the stochastic advection diffusion equation for anyd 1 in addition we present explicit formulas for the hermite fourier coefficients in the wiener chaos expansion of the solution

noise prevents singularities in linear transport equations a stochastic linear transport equation with multiplicative noise is considered and the question of no blow up is investigated the drift is assumed only integrable to a certain power opposite to the deterministic case where smooth initial conditions may develop discontinuities we prove that a certain sobolev degree of regularity is maintained which implies holder continuity of solutions the proof is based on a careful analysis of the associated stochastic flow of characteristics

particular formulae for the moore penrose inverse of a columnwise partitioned matrix abstract an essential part of cegielski s obtuse cones and gram matrices with non negative inverse linear algebra appl 335 2001 167 181 considerations of some properties of gram matrices with nonnegative inverses which are pointed out to be crucial in constructing obtuse cones consists in developing some particular formulae for the moore penrose inverse of a columnwise partitioned matrix a a1 a2 under the assumption that it is of full column rank in the present paper these results are generalized and extended the generalization consists in weakening the assumption mentioned above to the requirement that the ranges of a1 and a2 are disjoint while the extension consists in introducing the conditions referring to the class of all generalized inverses of a

particular formulae for the moore penrose inverses of the partitioned bounded linear operators abstract for any two complex hilbert spaces h and k let bl h k be the set of bounded linear operators from h to k and h k be the direct sum of h and k given three hilbert spaces h 1 h 2 h 3 and two operators a 1 bl h 1 h 3 a 2 bl h 2 h 3 a partitioned bounded linear operator a a 1 a 2 bl h 1 h 2 h 3 can be induced where a h 1 h 2 a 1 h 1 a 2 h 2 for h i h i i 1 2 in this note we study the moore penrose inverse a of such a partitioned bounded linear operator a and generalize a recent result of j k baksalary and o m baksalary particular formulae for the moore penrose inverse of a columnwise partitioned matrix linear algebra appl 421 2007 16 23 from finite matrices to hilbert space operators

the solutions to some operator equations abstract in this paper we study the solvability of the operator equation axb bx a c in the general setting of the adjointable operators between hilbert c modules based on the generalized inverses of the associated operators we propose the necessary and sufficient conditions for the existence of a solution to this equation and obtain the general expression of the solution in the solvable case we apply the results to the study of the real positive and positive solutions to the operator equation axb c in the case that the underlying space is finite dimensional or the range of b is contained in that of a we propose new necessary and sufficient conditions for the existence of a positive solution to the operator equation axb c and derive new formula in each case for the general positive solution to this operator equation

local and global well posedness of spde with generalized coercivity conditions abstract in this paper we establish the local and global existence and uniqueness of solutions for general nonlinear evolution equations with coefficients satisfying some local monotonicity and generalized coercivity conditions an analogous result is obtained for stochastic evolution equations in hilbert space with additive noise as applications the main results are applied to obtain simpler proofs in known cases as the stochastic 3d navier stokes equation the tamed 3d navier stokes equation and the cahn hilliard equation but also to get new results for stochastic surface growth pde and stochastic power law fluids

positive semi definite matrices of adjointable operators on hilbert c modules abstract let a be a c algebra h k be two hilbert a modules and b be an adjointable operator from h to k in this paper we prove that the moore penrose inverse of b exists if and only if b has closed range in addition some known results about the moore penrose inverses acting on hilbert spaces as well as in c algebras are extended in the context of hilbert c modules as an application a characterization of positive semi definite matrices of adjointable operators with respect to an orthogonal pair of a hilbert c module is given

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

singular initial conditions for the heat equation with a noise term we consider the equation u t u xx u w t 0 0 x j u 0 x u 0 x u t 0 u t j 0 where w w t x is two parameter white noise we show local existence and uniqueness for unbounded initial conditions satisfying certain conditions our results are motivated by earlier work which showed that for large y solutions of this equation can blow up one would wish to show that solutions can be extended beyond blowup and our results can be viewed as a step in that direction

time space white noise eliminates global solutions in reaction diffusion equations abstract we prove that perturbing the reaction diffusion equation u t u x x u p p 1 with time space white noise produces that solutions explodes with probability one for every initial datum opposite to the deterministic model where a positive stationary solution exists

explosive solutions of stochastic reaction diffusion equations in mean lp norm abstract the paper is concerned with the problem of non existence of global solutions for a class of stochastic reaction diffusion equations of ito type under some sufficient conditions on the initial state the nonlinear term and the multiplicative noise it is proven that in a bounded domain d r d there exist positive solutions whose mean l p norm will blow up in finite time for p 1 while if d r d the previous result holds in any compact subset of r d two examples are given to illustrate some application of the theorems

on the hermitian solutions to a system of adjointable operator equations we establish necessary and sufficient conditions for the existence of a hermitian solution to the system of equations a1x1 c1 x1b1 d1 a2x2 c2 x2b2 d2 a3x1a3 a4x2a4 c5 for adjointable operators between hilbert c modules and provide an expression for the general hermitian solution to the system we present necessary and sufficient conditions for the existence of a unique hermitian solution to the systems a1x1 c1 x1b1 d1 and a3x1a3 c5 of adjointable operators between hilbert c modules several examples are given to explain the use of the theory some of the findings of this paper extend some known results in the literature

stochastic nonlocal conservation laws on whole space abstract after introducing an entropy formulation for stochastic nonlocal conservation laws via an integral formula we develop a uniqueness theory by using a doubling variables technique and an existence theory by using a vanishing viscosity method we further consider the continuous dependence of the entropy solution on initial data

random perturbations of reaction diffusion equations the quasi deterministic approximation on considere des champs aleatoires u e t x u 1 e t x u n e t x definis comme les solutions d un systeme d equations aux derivees partielles u k e t l k u k e f k x u 1 e u n e e k t x les l k sont des operateurs elliptiques lineaires d ordre 2 k sont des champs gaussiens bruit blanc

encoding and decoding for the minimization of message symbol error rates in linear block codes given any fixed linear block code the error rates for the message symbols depend both on the encoding function and on the decoding map this research shows how to optimize the choice of a generator matrix and decoding map simultaneously to minimize the error rates for all message symbols the model used assumes that the distribution of messages is flat and that the distribution of error vectors defining the channel is independent of the message transmitted in addition it is shown that with proper choice of coset leaders standard array decoding is optimal in this circumstance the results generalize previously known results on unequal error protection and are sufficiently general to apply when a code is used for error detection only

non explosion boundedness and ergodicity for stochastic semilinear equations abstract sufficient conditions for non explosion boundedness and ergodicity of solutions of a class of semilinear stochastic equations are given applications of the results are based on a new regularity theorem for solutions of the linear problem

global existence large time behavior and life span of solutions of a semilinear parabolic cauchy problem we investigate the behavior of the solution u x t of where 1 1 n 2 xi 2 is the laplace operator p 1 is a constant t 0 and is a nonnegative bounded continuous function in r n the main results are for the case when the initial value has polynomial decay near x assuming 1 x 75a with a 0 various questions of global in time existence and nonexistence large time behavior or life span of the solution u x t are answered in terms of simple conditions on a p and the space dimension n

on the complete decoding of binary linear codes corresp complete decoding of systematic linear block codes by standard arrays with coset leaders not necessarily of minimal weight gives in several interesting cases on the binary symmetric channel with error probability p a smaller symbol error probability than decoding using the coset leaders of minimum weight the proposed choice of the coset leaders allows for the classes of uniformly packed and nearly perfect binary codes to achieve a lower information bit error probability at least for p sufficiently near zero

a study of a class of stochastic differential equations with non lipschitzian coefficients we study a class of stochastic differential equations with non lipschitz coefficients a unique strong solution is obtained and the non confluence of the solutions of stochastic differential equations is proved the dependence with respect to the initial values is investigated to obtain a continuous version of solutions the modulus of continuity of coefficients is assumed to be less than x y log open image in new window finally a large deviation principle of freidlin wentzell type is also established in the paper

on the existence and uniqueness of solution to a stochastic 2d cahn hilliard navier stokes model abstract we study in this article a stochastic version of a coupled cahn hilliard navier stokes model in a two dimensional bounded domain the model consists of the navier stokes equations for the velocity coupled with a cahn hilliard model for the order phase parameter we prove the existence and the uniqueness of a variational solution

convergence to equilibrium for classical and quantum spin systems the paper is devoted to stochastic equations describing the evolution of classical and quantum unbounded spin systems on discrete lattices and on euclidean spaces existence and asymptotic properties of the corresponding transition semigroups are studied in a unified way using the theory of dissipative operators on weighted hilbert and banach spaces this paper is an enlarged and rewritten version of the paper 7

moore penrose inverses of partitioned adjointable operators on hilbert c modules abstract let a be a c algebra h i i 1 2 3 be three hilbert a modules a 1 l h 1 h 3 and a 2 l h 2 h 3 where l h 1 h 3 resp l h 2 h 3 is the set of the adjointable operators from h 1 to h 3 resp h 2 to h 3 for such two operators a 1 and a 2 a 1 2 partitioned operator a a 1 a 2 can be induced by letting a h 1 h 2 a 1 h 1 a 2 h 2 for h i h i i 1 2 in this paper several formulae for the moore penrose inverse a of a are derived and an approach to constructing the weighted moore penrose inverse from the non weighted case is provided in particular the main result of udwadia and phohomsiri f e udwadia p phohomsiri recursive formulas for the generalized lm inverse of a matrix j optimiz theory app 131 2006 1 16 is generalized

stochastic 3d tamed navier stokes equations existence uniqueness and small time large deviation principles abstract in this paper we first give a direct approach to the existence and uniqueness of strong solutions to stochastic 3d tamed navier stokes equations in case of dirichlet boundary conditions for the stokes laplacian then we prove a small time large deviation principle for the solutions

martingale and stationary solutions for stochastic navier stokes equations we prove the existence of martingale solutions and of stationary solutions of stochastic navier stokes equations under very general hypotheses on the diffusion term the stationary martingale solutions yield the existence of invariant measures when the transition semigroup is well defined the results are obtained by a new method of compactness

blowup for the heat equation with a noise term in this paper we study blow up of the equation u t u xx u gamma dot w tx where dot w tx is a two dimensional white noise field and where dirichlet boundary conditions are enforced it is known that if 3 2 then the solution exists for all time in this paper we show that if is much larger than 3 2 then the solution blows up in finite time with positive probability we prove this by considering how peaks in the solution propagate if a peak of high mass forms we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass and these peaks evolve almost independently in this way we compare the evolution ofu to a branching process large peaks are regarded as particles in this branching process offspring are peaks which are higher by some factor we show that the expected number of offspring is greater than one when is much larger than 3 2 and thus the branching process survives with positive probability corresponding to blowup in finite time

markov selections for the 3d stochastic navier stokes equations we investigate the markov property and the continuity with respect to the initial conditions strong feller property for the solutions to the navier stokes equations forced by an additive noise first we prove by means of an abstract selection principle that there are markov solutions to the navier stokes equations due to the lack of continuity of solutions in the space of finite energy the markov property holds almost everywhere in time then depending on the regularity of the noise we prove that any markov solution has the strong feller property for regular initial conditions we give also a few consequences of these facts together with a new sufficient condition for well posedness