razumikhin type theorems on general decay stability of stochastic functional differential equations with infinite delay to the best of the authors knowledge there are no results based on the so called razumikhin technique via a general decay stability for any type of stochastic differential equations in the present paper the razumikhin approach is applied to the study of both pth moment and almost sure stability on a general decay for stochastic functional differential equations with infinite delay the obtained results are extended to stochastic differential equations with infinite delay and distributed infinite delay some comments on how the considered approach could be extended to stochastic functional differential equations with finite delay are also given an example is presented to illustrate the usefulness of the theory

uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

what is the fractional laplacian a comparative review with new results abstract the fractional laplacian in r d which we write as 2 with 0 2 has multiple equivalent characterizations moreover in bounded domains boundary conditions must be incorporated in these characterizations in mathematically distinct ways and there is currently no consensus in the literature as to which definition of the fractional laplacian in bounded domains is most appropriate for a given application the riesz or integral definition for example admits a nonlocal boundary condition where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional laplacian in contrast the spectral definition requires only the standard local boundary condition these differences among others lead us to ask the question what is the fractional laplacian beginning from first principles we compare several commonly used definitions of the fractional laplacian theoretically through their stochastic interpretations as well as their analytical properties then we present quantitative comparisons using a sample of state of the art methods we discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems radial basis function collocation for the riesz fractional laplacian and nonharmonic lifting for the spectral fractional laplacian in our numerical studies we aim to compare different definitions on bounded domains using a collection of benchmark problems we consider the fractional poisson equation with both zero and nonzero boundary conditions where the fractional laplacian is defined according to the riesz definition the spectral definition the directional definition and the horizon based nonlocal definition we verify the accuracy of the numerical methods used in the approximations for each operator and we focus on identifying differences in the boundary behaviors of solutions to equations posed with these different definitions through our efforts we aim to further engage the research community in open problems and assist practitioners in identifying the most appropriate definition and computational approach to use for their mathematical models in addressing anomalous transport in diverse applications

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

spectral properties of the cauchy process on half line and interval we study the spectral properties of the transition semigroup of the killed one dimensional cauchy process on the half line 0 a nd the interval 1 1 this process is related to the square root of one dimensional laplacian a d 2 dx 2 with a dirichlet exterior condition on a complement of a domain and to a mixed steklov problem in the half plane for the half line an explicit formula for generalized eigenfunctions of a is derived and then used to construct a spectral representation of a explicit formulas for the transition density of the killed cauchy process on the half line or the heat kernel of a in 0 and for the distribution of the first exit time from the half line follow the formula for is also used to construct approximations to eigenfunctions of a in the interval for the eigenvalues n of a in the interval the asymptotic formula n n 2 8 o 1 n is derived and all eigenvalues n are proved to be simple finally efficient numerical methods of estimation of eigenvalues n are applied to obtain lower and upper numerical bounds for the first few eigenvalues up to the ninth decimal point

the martin boundary in non lipschitz domains the martin boundary with respect to the laplacian and with respect to uniformly elliptic operators in divergence form can be identified with the euclidean boundary in c domains where x bx loglog 1 x logloglog 1 x b small a counterexample shows that this result is very nearly sharp

the αth moment stability for the stochastic pantograph equation in this paper we investigate the ath moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the razumikhin technique especially the linear stochastic pantograph equations and the semi implicit euler method applying them are considered the convergence result of the semi implicit euler method is obtained the stability conditions of the analytic solution of those equations and the numerical method are given finally some experiments are given

estimates and structure of α harmonic functions we prove a uniform boundary harnack inequality for nonnegative harmonic functions of the fractional laplacian on arbitrary open set d this yields a unique representation of such functions as integrals against measures on dc satisfying an integrability condition the corresponding martin boundary of d is a subset of the euclidean boundary determined by an integral test

analysis on exponential stability of hybrid pantograph stochastic differential equations with highly nonlinear coefficients this paper discusses exponential stability of solutions for highly nonlinear hybrid pantograph stochastic differential equations psdes two criteria are proposed to guarantee exponential stability of the solution the first criterion is a khasminskii type condition involving general lyapunov functions the second is developed on coefficients of the equation in virtue of m matrix techniques based on the second criterion robust stability of a perturbed hybrid psde is also investigated the theory shows how much an exponentially stable hybrid psde can tolerate to remain stable

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

the 2 n 1 2 ray algorithm a new simplicial algorithm to compute economic equilibria a new variable dimension simplicial restart algorithm is introduced to compute economic equilibria the number of rays along which the algorithm can leave the starting point differs from the thusfar known algorithms more precisely the new algorithm has one ray to each of the 2n 1 2 faces of then dimensional price simplex whereas the existing algorithms haven 1 rays either to each facet or to each vertex of the unit simplex the path of points followed by the algorithm can be interpreted as a globally and universally convergent price adjustment process the process is also economically meaningful and therefore it is a good alternative for the well known walras tatonnement process computational results show that the algorithm is competitive with the most efficient simplicial algorithms developed thusfar

note on the path following approach of equilibrium programming recently zangwill and garcia introduced a general formulation of equilibrium problems to prove the existence of an equilibrium they discussed a path following procedure in this note we consider the application to the exchange economy problem an economic equilibrium may be found by applying a simplicial variable dimension algorithm developed by van der laan and talman

variable dimension algorithms basic theory interpretations and extensions of some existing methods in this paper we establish a basic theory for variable dimension algorithms which were originally developed for computing fixed points by van der laan and talman we introduce a new concept primal dual pair of subdivided manifolds and by utilizing it we propose a basic model which will serve as a foundation for constructing a wide class of variable dimension algorithms our basic model furnishes interpretations to several existing methods lemke s algorithm for the linear complementarity problem its extension to the nonlinear complementarity problem a variable dimension algorithm on conical subdivisions and merrill s algorithm we shall present a method for solving systems of equations as an application of the second method and an efficient implementation of the fourth method to which our interpretation leads us a method for constructing triangulations with an arbitrary refinement factor of mesh size is also proposed

symmetrization symmetric stable processes and riesz capacities let x t be a symmetric stable process killed on exiting an open subset d of r n we prove a theorem that describes the behavior of its transition probabilities under polarization we show that this result implies that the probability of hitting a given set b in the complement of d in the first exit moment from d increases when d and b are polarized it can also lead to symmetrization theorems for hitting probabilities green functions and riesz capacities one such theorem is the following among all compact sets k in r n with given volume the balls have the least capacity 0 2

hardy spaces for α harmonic functions in regular domains we investigate hardy spaces h p for singular harmonic functions in bounded domains with regular boundaries we show the correspondence between these spaces and suitable l p spaces and measure spaces

potential theory of truncated stable processes for any alpha in 0 2 a truncated symmetric stable process is a symmetric levy process in mathbb r d with a levy density given by c x d alpha 1 x 1 for some constant c in this paper we study the potential theory of truncated symmetric stable processes in detail we prove a harnack inequality for nonnegative harmonic functions of these processes we also establish a boundary harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains we give an example of a non convex domain for which the boundary harnack principle fails

intrinsic ultracontractivity and the dirichlet laplacian we show that if d x y 0 x 1 f x y 1 where f is negative upper semicontinuous and bounded below then the semigroup associated with the dirichlet laplacian on d is intrinsically ultracontractive analogs in higher dimensions are proved we construct domains of infinite area with this property

razumikhin type theorems on p th moment exponential stability of impulsive stochastic delay differential equations this note develops razumikhin type theorems on pth moment exponential stability for general impulsive stochastic delay differential equations some applications are also given to some special types of impulsive stochastic delay differential equations two examples illustrate the effectiveness of the proposed results

estimates on green functions and poisson kernels for symmetric stable processes one of the most basic and most important subfamily of levy processes is symmetric stable processes a symmetric stable process x on rn is a levy process whose transition density p t x y relative to the lebesgue measure is uniquely determined by its fourier transform rn e ix p t x dx e t here must be in the interval 0 2 when 2 we get a brownian motion running with a time clock twice as fast as the standard one brownian motion plays a central role in modern probability theory and has numerous important applications in other scientific areas as well as in many other branches of mathematics thus it has been intensively studied in this paper symmetric stable processes are referred to the case when 0 2 unless otherwise specified in the last few years there has been an explosive growth in the study of physical and economic systems that can be successfully modeled with the use of stable processes stable processes are now widely used in physics operations research queuing theory mathematical finance and risk estimation in some physics literatures symmetric stable processes are called levy flights and they have been applied to a wide range of very complex physics issues such as turbulent diffusion vortex dynamics anomalous diffusion in rotating flows and molecular spectral fluctuations in mathematical finance stable processes can be used to model stock returns in incomplete market for these and more applications of stable processes please see the interesting book 14 by janicki and weron and the references therein and the recent article 15 by klafter shlesinger and zuomofen in order to make precise predictions about natural phenomena and to better cope with these

relative fatou s theorem for δ α 2 harmonic functions in bounded κ fat open sets recently it was shown in p kim fatou s theorem for censored stable processes stochastic process appl 108 1 2003 63 92 that fatou s theorem for transient censored stable processes in a bounded c1 1 open set is true here we give a probabilistic proof of relative fatou s theorem for 2 harmonic functions equivalently for symmetric stable processes in bounded fat open set where 0 2 that is if u is positive 2 harmonic function in a bounded fat open set d and h is singular positive 2 harmonic function in d then nontangential limits of u h exist almost everywhere with respect to the martin representing measure of h this extends the result of bogdan and dyda k bogdan b dyda relative fatou theorem for harmonic functions of rotation invariant stable processes in smooth domain studia math 157 1 2003 83 96 it is also shown that under the gaugeability assumption relative fatou s theorem is true for operators obtained from the generator of the killed stable process in bounded fat open set d through nonlocal feynman kac transforms as an application relative fatou s theorem for relativistic stable processes is also true if d is bounded c1 1 open set

a fatou theorem for α harmonic functions in lipschitz domains we study harmonic functions in lipschitz domains we prove a fatou theorem when the boundary function is bounded and lp holder continuous of order with p 1

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

stopping times of bessel processes let x x alpha be a bessel process with parameter alpha starting at x geq 0 gordon 3 obtained l p inequalities which relate stopping times to stopping places for the case alpha 1 x 0 and p frac 1 2 rosenkrantz and sawyer 5 extended them to alpha 0 x 0 and p geq 1 burkholder 1 obtained results for alpha a positive integer x geq 0 and p 0 here we consider arbitrary starting points x alpha 0 and p 0 the l p inequalities are valid for alpha geq 2 with p 0 and also for 0 2 alpha 2 examples are constructed to show that for 0 alpha 2 with p leq 2 alpha 2 the l p inequalities cannot hold

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

lifetimes of conditioned diffusions we investigate when an upper bound on expected lifetimes of conditioned diffusions associated with elliptic operators in divergence and non divergence form can be found the critical value of the parameter is found for each of the following classes of domains l p domains p n 1 uniformly regular twistedl p domains p n 1 and twisted holder domains 1 3 a related parabolic boundary harnack principle is proved

higher order pdes and symmetric stable processes we describe a connection between the semigroup of a symmetric stable process with rational index and higher order partial differential equations as an application we obtain a variational formula for the eigenvalues associated with the process killed upon leaving a bounded open set d the variational formula is more user friendly than the classical rayleigh ritz formula we illustrate this by obtaining upper bounds on the eigenvalues in terms of dirichlet eigenvalues of the laplacian on d these results generalize some work of banuelos and kulczycki on the cauchy process along the way we prove an operator inequality for the operators associated with the transition densities of brownian motion and the brownian motion killed upon leaving d

robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay this paper establishes the existence and uniqueness theorem of neutral stochastic functional differential equations with infinite delay and examines the almost sure stability of this solution with general decay rate this result may be used to examine almost sure robust stability to illustrate our idea more carefully we carefully discuss a scalar stochastic integro differential equation with neutral type and its asymptotic stability including the exponential stability and the polynomial stability

almost sure exponential stability of numerical solutions for stochastic delay differential equations using techniques based on the continuous and discrete semimartingale convergence theorems this paper investigates if numerical methods may reproduce the almost sure exponential stability of the exact solutions to stochastic delay differential equations sddes the important feature of this technique is that it enables us to study the almost sure exponential stability of numerical solutions of sddes directly this is significantly different from most traditional methods by which the almost sure exponential stability is derived from the moment stability by the chebyshev inequality and the borel cantelli lemma

computing equilibria in finance economies the general equilibrium model with incomplete asset markets is ideally suited for the study of problems in cross sectional asset pricing and portfolio theory in this paper we develop a homotopy algorithm to approximate equilibria in these models since the algorithm is tailor made for so called finance economies the number of nonlinear equations that has to be solved for and therefore the computing time is an order of magnitude smaller than that of existing general purpose algorithms the algorithm is shown to be generically convergent we implement the algorithm using hompack to illustrate its performance we present various numerical examples and report running times

on evaluating the donsker varadhan i function let x t be a feller process on a complete separable metric space a and consider the occupation measure l t omega cdot int t 0 chi cdot x s ds the i function is defined for mu in mathscr p a the set of probability measures on a by i mu inf u in mathscr d int a lu u d mu where l mathscr d is the generator of the process and mathscr d subset mathscr d consists of the strictly positive functions in mathscr d the i function determines the asymptotic rate of decay of p 1 t l t omega cdot in g for g subset mathscr p a the first difficulty encountered in evaluating i mu is that the domain mathscr d is generally not known explicitly in this paper we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain we then apply this general result to diffusion processes with boundaries

the cauchy process and the steklov problem let xt be a cauchy process in rd d 1 we investigate some of the fine spectral theoretic properties of the semigroup of this process killed upon leaving a domain d we establish a connection between the semigroup of this process and a mixed boundary value problem for the laplacian in one dimension higher known as the mixed steklov problem using this we derive a variational characterization for the eigenvalues of the cauchy process in d this characterization leads to many detailed properties of the eigenvalues and eigenfunctions for the cauchy process inspired by those for brownian motion our results are new even in the simplest geometric setting of the interval 1 1 where we obtain more precise information on the size of the second and third eigenvalues and on the geometry of their corresponding eigenfunctions such results although trivial for the laplacian take considerable work to prove for the cauchy processes and remain open for general symmetric stable processes along the way we present other general properties of the eigenfunctions such as real analyticity which even though well known in the case of the laplacian are not available for more general symmetric stable processes

dirichlet heat kernel estimates for fractional laplacian with gradient perturbation suppose that d 2 and 1 2 let d be a bounded c1 1 open set in rd and b an rd valued function on rd whose components are in a certain kato class of the rotationally symmetric stable process in this paper we derive sharp two sided heat kernel estimates for lb 2 b in d with zero exterior condition we also obtain the boundary harnack principle for lb in d with explicit decay rate

two sided eigenvalue estimates for subordinate processes in domains let x xt t 0 be a symmetric markov process in a state space e and d an open set of e denote by x d the subprocess of x killed upon leaving d let s st t 0 be a subordinator with laplace exponent that is independent of x the processes x xst t 0 and x d x dt t 0 are called the subordinate processes of x and x d respectively under some mild conditions we show that if n n 1 and n n 1 denote the eigenvalues of the generators of the subprocess of x killed upon leaving d and of the process x d respectively then

conditional gauge theorem for non local feynman kac transforms feynman kac transforms driven by discontinuous additive functionals are studied in this paper for a large class of markov processes general gauge and conditional gauge theorems are established for such transforms furthermore the l 2 infinitesimal generator of the schrodinger semigroup given by a non local feynman kac transform is determined in terms of its associated bilinear form

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

new criteria on exponential stability of neutral stochastic differential delay equations neutral stochastic differential delay equations nsddes have recently been studied intensively see e g v b kolmanovskii v r nosov stability and periodic modes of control systems with aftereffect nauka moscow 1981 x mao exponential stability in mean square of neutral stochastic differential functional equations systems control lett 26 1995 245 251 x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 x mao stochastic differential equations and their applications horwood publishing chichester 1997 more recently mao asymptotic properties of neutral stochastic differential delay equations stochastics and stochastics rep 68 2000 273 295 provided with some useful criteria on the exponential stability for nsddes however the criteria there require not only the coefficients of the nsddes to obey the linear growth condition but also the time delay to be a constant one of our aims in this paper is to remove these two restrictive conditions moreover the key condition on the diffusion operator associated with the underlying nsdde will take a much more general form our new stability criteria not only cover many highly non linear nsddes with variable time delays but they can also be verified much more easily than the known criteria

a new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices a new simplicial variable dimension restart algorithm is introduced to solve the nonlinear complementarity problem on the product spaces of unit simplices the triangulation which underlies the algorithm differs from the triangulations ofs used thus far moreover the number of rays along which the algorithm can leave the arbitrarily chosen starting point is much larger more precisely there is a ray leading from the starting point to each vertex ofs in cases is the product ofn one dimensional unit simplices the alogrithm is similar to the octahedral algorithm onrn having 2n rays also the accuracy of an approximate solution in the terminal simplex of the algorithm is in general better than for the other algorithms ons computational results will show that the number of iterations for the new algorithm is much less the examples concern the computation of equilibria in noncooperative games exchange economies and trade models

boundary harnack principle for symmetric stable processes in this paper we study potential theoretic properties of the symmetric stable processes 0 2 establishing the boundary harnack principle for ratios of harmonic functions on any open sets identifying the martin boundary with the euclidean boundary for open sets with a certain interior fatness property and extending earlier results on intrinsic ultracontractivity and the conditional gauge theorem to certain open sets

the first exit time of a two dimensional symmetric stable process from a wedge let t be the first exit time of a symmetric stable process with parameter 0 2 from a wedge of angle 2 0 0 such that for starting points x in the wedge e x t p p we characterize p and obtain upper and lower bounds

razumikhin type theorems on the moment stability of the exact and numerical solutions for the stochastic pantograph differential equations abstract in this paper we establish two different th moment type stability criterions on the exact and numerical solutions of the stochastic pantograph differential equations one is established by the virtue of lyapunov method and the other is used by the continuous and discrete razumikhin technique by comparing the two stability criterions we can easily see that the conditions constructed by the razumikhin type technique are better than those constructed by the virtue of lyapunov method using the conditions constructed for the th moment type stability we study the stability of the euler maruyama method and the backward euler maruyama method respectively for a special class of the stochastic pantograph differential equations finally examples are given to illustrate the consistence with the theoretical results on the th moment type stability

two sided green function estimates for killed subordinate brownian motions a subordinate brownian motion is a levy process that can be obtained by replacing the time of the brownian motion by an independent subordinator the infinitesimal generator of a subordinate brownian motion is where is the laplace exponent of the subordinator in this paper we consider a large class of subordinate brownian motions without diffusion component and with comparable to a regularly varying function at infinity this class of processes includes symmetric stable processes relativistic stable processes sums of independent symmetric stable processes sums of independent relativistic stable processes and much more we give sharp two sided estimates on the green functions of these subordinate brownian motions in any bounded fat open set d w hend is a bounded c 1 1 open set we establish an explicit form of the estimates in terms of the distance to the boundary as a consequence of such sharp green function estimates we obtain a boundary harnack principle in c 1 1 open sets with explicit rate of decay

razumikhin type exponential stability criteria of neutral stochastic functional differential equations the paper discusses both pth moment and almost sure exponential stability of solutions to neutral stochastic functional differential equations and neutral stochastic differential delay equations by using the razumikhin type technique the main goal is to find sufficient stability conditions that could be verified more easily then by using the usual method with lyapunov functionals the analysis is based on paper x mao razumikhin type theorems on exponential stability of neutral stochastic functional differential equations siam j math anal 28 2 1997 389 401 referring to mean square and almost sure exponential stability

on the pth moment exponential stability criteria of neutral stochastic functional differential equations the paper discusses the pth moment exponential stability for a general class of neutral stochastic functional differential equations of the ito type this investigation can be very complicated even in many special cases by using usual methods based on lyapunov functionals in this paper we present criteria which are relatively easy to verify the pth moment exponential stability of the solutions of such equations

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

existence uniqueness almost sure polynomial stability of solution to a class of highly nonlinear pantograph stochastic differential equations and the euler maruyama approximation abstract in the first part of this paper the existence uniqueness and almost sure polynomial stability of solutions for pantograph stochastic differential equations are considered under nonlinear growth conditions the results are obtained using the idea from mao and rassias 2005 10 where stochastic differential equation with constant delay are considered however the presence of the unbounded delay in stochastic pantograph differential equations required certain modification of that idea moreover the convergence in probability of the appropriate euler maruyama solution is proved under the same nonlinear growth conditions adding the linear growth condition we show that the almost sure polynomial stability of the euler maruyama solution implies the almost sure polynomial stability of the exact solution this part of the paper represents the extension of the idea from wu et al 2010 17 the main novelty in this part of the paper is also related to the treatment of the unbounded delay in pantograph stochastic differential equations

martin boundary and integral representation for harmonic functions of symmetric stable processes abstract martin boundaries and integral representations of positive functions which are harmonic in a bounded domain d with respect to brownian motion are well understood unlike the brownian case there are two different kinds of harmonicity with respect to a discontinuous symmetric stable process one kind are functions harmonic in d with respect to the whole process x and the other are functions harmonic in d with respect to the process x d killed upon leaving d in this paper we show that for bounded lipschitz domains the martin boundary with respect to the killed stable process x d can be identified with the euclidean boundary we further give integral representations for both kinds of positive harmonic functions also given is the conditional gauge theorem conditioned according to martin kernels and the limiting behaviors of the h conditional stable process where h is a positive harmonic function of x d in the case when d is a bounded c 1 1 domain sharp estimate on the martin kernel of d is obtained

alpha continuity properties of the symmetric alpha stable process let d be a domain of finite lebesgue measure in d and let x t d be the symmetric stable process killed upon exiting d each element of the set i i 1 of eigenvalues associated to x t d regarded as a function of a 0 2 is right continuous in addition if d is lipschitz and bounded then each i is continuous in and the set of associated eigenfunctions is precompact

reflected symmetric α stable processes and regional fractional laplacian in this paper we investigate the reflected symmetric stable processes and their generators we show that the generators are regional fractional laplacians on the closed region in the case of 1 2 their existence requires that open image in new window on the boundary among other things we obtain the integration by parts formula of the regional fractional laplacian and the semi martingale decomposition of the reflected symmetric stable processes

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

brief paper razumikhin type stability theorems for discrete delay systems in this paper by employing the razumikhin technique and lyapunov functions razumikhin type theorems that guarantee the uniform stability uniformly asymptotic stability and uniformly exponential stability for the general discrete delay systems are established respectively moreover razumikhin type uniformly exponential stability theorem gives the estimation of the convergence speed as theoretic application the razumikhin type uniformly exponential stability result is further studied and used to show some well known stability results for some kinds of discrete delay systems finally examples are also worked through to illustrate our results

almost sure exponential stability of numerical solutions for stochastic pantograph differential equations abstract the sufficient conditions of the almost sure exponential stability of the exact solution for the stochastic pantograph differential equation are considered with a khasminskii type condition the almost sure exponential stability of the numerical solutions by the euler maruyama method and the backward euler maruyama method is also discussed based on the discrete semimartingale convergence theorem we present the sufficient conditions for the stability of the euler maruyama method with one extra condition when compared with the exact solution we show that the backward euler maruyama method can share almost the same conditions for the almost sure exponential stability as the exact solution

equilibrium programming the path following approach and dynamics a general formulation of equilibrium is introduced which subsumes a large range of equilibrium models in economics mathematical programming game theory networks organizations etc the existence of a solution is proved by path following and without invocation of a fixed point theorem the path following procedure provides new means for establishing existence of equilibria for constructing them and for interpreting them a dynamic equilibrium model is also presented and the existence stability and computation of dynamic equilibrium paths are discussed moreover these paths can often be interpreted as adjustment processes

holder domains and the boundary harnack principle the constant c depends on l only through the constant cl defined in 1 below of course c also depends on d v and k the above result shows that the boundary harnack principle holds in all holder domains it was first proven for the range 1 2 1 by bass and burdzy 1990a theorems 3 5 and 4 5 it was subsequently extended by banuelos unpublished to the full range of provided the domain also satisfies a uniform capacity condition on the boundary in banuelos 1990 a class of domains called uniformly holder domains of order was introduced a boundary harnack principle may be obtained for such domains for all 0 1 by a variation of the proof of theorem 1 in order to make this note compact we refer

exit times from cones in ℝ n of brownian motion for a fairly general class of cones inn dimensions n 3 we determine the corresponding distributions of brownian first exit times asymptotic results may then be read off

intrinsic ultracontractivity and conditional gauge for symmetric stable processes abstract it is shown in this paper that the conditional gauge theorem holds for symmetric stable processes on bounded c 1 1 domains in r n where 0 n 2 two of the major tools used to prove this conditional gauge theorem are logarithmic sobolev inequality and intrinsic ultracontractivity

stability of martin boundary under non local feynman kac perturbations recently the authors showed that the martin boundary and the minimal martin boundary for a censored or resurrected stable process y in a bounded c 1 1 open set d with 1 2 can all be identified with the euclidean boundary d of d under the gaugeability assumption we show that the martin boundary and the minimal martin boundary for the schrodinger operator obtained from y through a non local feynman kac transform can all be identified with d in other words the martin boundary and the minimal martin boundary are stable under non local feynman kac perturbations moreover an integral representation of nonnegative excessive functions for the schrodinger operator is explicitly given these results in fact hold for a large class of strong markov processes as are illustrated in the last section of this paper as an application the martin boundary for censored relativistic stable processes in bounded c 1 1 smooth open sets is studied in detail