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

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

a bi hamiltonian formulation for triangular systems by perturbations a bi hamiltonian formulation is proposed for triangular systems resulting from perturbations around solutions from which infinitely many symmetries and conserved functionals of triangular systems can be explicitly constructed provided that one operator of the hamiltonian pair is invertible through our formulation four examples of triangular systems are exhibited which also show that bi hamiltonian systems in both lower dimensions and higher dimensions are many and varied two of four examples give local 2 1 dimensional bi hamiltonian systems and illustrate that multiscale perturbations can lead to higher dimensional bi hamiltonian systems

a study on orthogonal two direction vector valued wavelets and two direction wavelet packets abstract in this paper the notion of two direction vector valued multiresolution analysis and the two direction orthogonal vector valued wavelets are introduced the definition for two direction orthogonal vector valued wavelet packets is proposed an algorithm for constructing a class of two direction orthogonal vector valued compactly supported wavelets corresponding to the two direction orthogonal vector valued compactly supported scaling functions is proposed by virtue of matrix theory and time frequency analysis method the properties of the two direction vector valued wavelet packets are investigated at last the direct decomposition relation for space l 2 r r is presented

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

integrable counterparts of the d kaup newell soliton hierarchy two integrable counterparts of the d kaup newell soliton hierarchy are constructed from a matrix spectral problem associated with the three dimensional special orthogonal lie algebra so 3 r an application of the trace identity presents hamiltonian or quasi hamiltonian structures of the resulting counterpart soliton hierarchies thereby showing their liouville integrability i e the existence of infinitely many commuting symmetries and conserved densities the involved hamiltonian and quasi hamiltonian properties are shown by computer algebra systems

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

nonlinear continuous integrable hamiltonian couplings based on a kind of special non semisimple lie algebras a scheme is presented for constructing nonlinear continuous integrable couplings variational identities over the corresponding loop algebras are used to furnish hamiltonian structures for the resulting continuous integrable couplings the application of the scheme is illustrated by an example of nonlinear continuous integrable hamiltonian couplings of the akns hierarchy of soliton equations

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

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

constructing nonlinear discrete integrable hamiltonian couplings beginning with lax pairs from special non semisimple matrix lie algebras we establish a scheme for constructing nonlinear discrete integrable couplings discrete variational identities over the associated loop algebras are used to build hamiltonian structures for the resulting integrable couplings we illustrate the application of the scheme by means of an enlarged volterra spectral problem and present an example of nonlinear discrete integrable hamiltonian couplings for the volterra lattice equations

the quadratic form identity for constructing the hamiltonian structure of integrable systems a usual loop algebra not necessarily the matrix form of the loop algebra is also made use of for constructing linear isospectral problems whose compatibility conditions exhibit a zero curvature equation from which integrable systems are derived in order to look for the hamiltonian structure of such integrable systems a quadratic form identity is created in the present paper whose special case is just the trace identity that is when taking the loop algebra the quadratic form identity presented in this paper is completely consistent with the trace identity

green function estimate for censored stable processes sharp two sided estimates for green functions of censored stable process y in a bounded c 1 1 open set d are obtained where 1 2 it is shown that the martin boundary and minimal martin boundary of y can all be identified with the euclidean boundary of d sharp two sided estimates for the martin kernel of y are also derived

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

two integrable couplings of the tu hierarchy and their hamiltonian structures the double integrable couplings of the tu hierarchy are worked out by use of vector loop algebras g 6 and g 9 respectively also the hamiltonian structures of the obtained system are given by the quadratic form identity

integrable evolution systems based on gerdjikov ivanov equations bi hamiltonian structure finite dimensional integrable systems and n fold darboux transformation a spectral problem and the associated gerdjikov ivanov gi hierarchy of nonlinear evolution equations is presented as a reduction the well known gi equation of derivative nonlinear schrodinger equations is obtained it is shown that the gi hierarchy is integrable in a liouville sense and possesses bi hamiltonian structure moreover the spectral problem can be nonlinearized as a finite dimensional completely integrable system under the bargmann constraint between the potentials and the eigenfunctions in particular an explicit n fold darboux transformation for the gi equation is constructed with the help of a gauge transformation of spectral problems and a reduction technique some explicit solitonlike solutions of the gi equation are given by applying its darboux transformation

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

conditioned brownian motion in planar domains we give an upper bound for the green functions of conditioned brownian motion in planar domains a corollary is the conditional gauge theorem in bounded planar domains

intrinsic ultracontractivity of the feynman kac semigroup for relativistic stable processes let x t be the relativistic stable process in r d a 0 2 d with infinitesimal generator h 0 a m 2 2 m we study intrinsic ultracontractivity iu for the feynman kac semigroup t t for this process with generator h 0 v v 0 v locally bounded we prove that if lim x v x then for every t 0 the operator t t is compact we consider the class v of potentials v such that v 0 lim x v x and v is comparable to the function which is radial radially nondecreasing and comparable on unit balls for v in the class v we show that the semigroup t t is iu if and only if lim x v x x if this condition is satisfied we also obtain sharp estimates of the first eigenfunction 1 for t t in particular when v x x 0 then the semigroup t t is iu if and only if 1 for 1 the first eigenfunction 1 x is comparable to

a direct method for integrable couplings of td hierarchy a direct method for establishing integrable couplings is proposed in this paper as an example illustration integrable couplings of td hierarchy are obtained by constructing a suitable transformation of lax pairs and a new lie algebra

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

semidirect sums of lie algebras and discrete integrable couplings a relation between semidirect sums of lie algebras and integrable couplings of lattice equations is established and a practicable way to construct integrable couplings is further proposed an application of the resulting general theory to the generalized toda spectral problem yields two classes of integrable couplings for the generalized toda hierarchy of lattice equations the construction of integrable couplings using semidirect sums of lie algebras provides a good source of information on complete classification of integrable lattice equations

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

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

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