an approximation to δ couplings on graphs we discuss a general parametrization for vertices of quantum graphs and show in particular how the s and couplings at an n edge vertex can be approximated by means of n 1 couplings of the type provided the latter are properly scaled

on the existence of positive solutions for semilinear elliptic equations with neumann boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in unbounded lipschitz domainsd d d 3 having compact boundary with nonlinear neumann boundary conditions on the boundary ofd for this we use an implicit probabilistic representation schauder s fixed point theorem and a recently proved sobolev inequality forw 1 2 d special cases include equations arising from the study of pattern formation in various models in mathematical biology and from problems in geometry concerning the conformal deformation of metrics

potential theory for hyperbolic spdes keywords potential theory nonlinear hyperbolic stochastic partial differential equations multiparameter process gaussian process hitting probability reference prob article 2004 005view record in web of science record created on 2008 12 01 modified on 2017 05 12

some dimension results for super brownian motion the dawson watanabe super brownian motion has been intensively studied in the last few years in particular there has been much work concerning the hausdorff dimension of certain remarkable sets related to super brownian motion we contribute to this study in the following way let yt t 0 be a super brownian motion on d d 2 andh be a borel subset of d we determine the hausdorff dimension of t 0 suppyt h o improving and generalizing a result of krone we also obtain a new proof of a result of tribe which gives whend 4 the hausdorff dimension of cup t in rm b suppy t as a function of the dimension ofb

markov properties of multiparameter processes and capacities we study a class of multiparameter symmetric markov processes we prove that this class is stable by subordination in bochner s sense we show then that for these processes a probabilistic and an analytic potential theory correspond to each other in particular additive functionals are associated with finite energy measures hitting probabilities are estimated by capacities quasicontinuity corresponds to path continuity in the last section examples show that many earlier results as well as new ones in this domain can be obtained by our method

fundamental solution of the heat and schrodinger equations with point interaction abstract we find explicit formulae for the fundamental solution for the heat and time dependent schrodinger equations with point interaction in dimension n 3 we discuss in detail the small time asymptotics of these fundamental solutions the case n 2 has the peculiarity of involving logarithmic terms in the expansion

polarity of points for gaussian random fields we show that for a wide class of gaussian random fields points are polar in the critical dimension examples of such random fields include solutions of systems of linear stochastic partial differential equations with deterministic coefficients such as the stochastic heat equation or wave equation with space time white noise or colored noise in spatial dimensions k 1k 1 our approach builds on a delicate covering argument developed by m talagrand ann probab 23 1995 767 775 probab theory related fields 112 1998 545 563 for the study of fractional brownian motion and uses a harmonizable representation of the solutions of these stochastic pdes

harmonic functions and exit boundary of superdiffusion abstract all positive harmonic functions in an arbitrary domain of a euclidean space can be described in terms of the so called exit boundary this was established in 1941 by r s martin a probabilistic approach to the martin theory is due to doob and hunt it was extended later to harmonic functions associated with a wide class of markov processes the subject of this paper are harmonic functions associated with a superdiffusion x we call them x harmonic the results of evans and perkins imply the existence and uniqueness of an integral representation of positive x harmonic functions through extreme functions an outstanding problem is to find all extremal functions they are in 1 1 correspondence with the points of the exit boundary an interest in x harmonic functions is motivated in part by the fact that each of them provides a way of conditioning a superprocess path properties of conditioned superprocesses corresponding to various special x harmonic functions were investigated by a number of authors important classes of x harmonic functions are related to positive solutions of semilinear partial differential equations almost nothing is known about their decomposition into extreme elements a progress in this direction may create new tools for investigating solutions of the equations the goal of this paper is to summarize all known facts about x harmonic functions to present the results of various authors in a more general setting in a unified form and to outline a program of further work

path properties of superprocesses with a general branching mechanism we first consider a super brownian motion x with a general branching mechanism using the brownian snake representation with subordination we get the hausdorff dimension of supp x t the topological support of x t and more generally the hausdorff dimension of u t b supp x t we also provide estimations on the hitting probability of small balls for those random measures we then deduce that the support is totally disconnected in high dimension eventually considering a super stable process with a general branching mechanism we prove that in low dimension this random measure is absolutely continuous with respect to the lebesgue measure

exact packing measure of the range of psi super brownian motions we consider super processes whose spatial motion is the d dimensional brownian motion and whose branching mechanism psi is critical or subcritical such processes are called psi super brownian motions if d 2 varvec gamma varvec gamma 1 where varvec gamma in 1 2 is the lower index of psi at infty then the total range of the psi super brownian motion has an exact packing measure whose gauge function is g r log log 1 r varphi 1 1 r log log 1 r 2 where varphi psi prime circ psi 1 more precisely we show that the occupation measure of the psi super brownian motion is the g packing measure restricted to its total range up to a deterministic multiplicative constant only depending on d and psi this generalizes the main result of duquesne ann probab 37 6 2431 2458 2009 that treats the quadratic branching case for a wide class of psi the constant 2 varvec gamma varvec gamma 1 is shown to be equal to the packing dimension of the total range

self adjoint domains symplectic geometry and limit circle solutions abstract everitt and markus characterized the domains of self adjoint operator realizations of very general even and odd order symmetric ordinary differential equations in terms of lagrangian subspaces of symplectic spaces recently for the even order case with real coefficients wang sun and zettl constructed limit circle lc solutions and hao wang sun and zettl characterized the self adjoint domains in terms of lc solutions these lc solutions are higher order analogues of the celebrated titchmarsh weyl limit circle solutions in the second order case this lc characterization has been used to obtain information about the discrete continuous and essential spectra of these operators in this paper we investigate the connection between these two very different kinds of characterizations and thus add the methods of symplectic geometry to the techniques available for the investigation of the spectrum of self adjoint operators

the spectrum of differential operators and square integrable solutions abstract we give a comprehensive account of the relationship between the square integrable solutions for real values of the spectral parameter and the spectrum of self adjoint even order ordinary differential operators with real coefficients and arbitrary deficiency index d and we solve an open problem stated by weidmann in his well known 1987 monograph according to a well known result if one endpoint is regular and for some real value of the spectral parameter the number of linearly independent square integrable solutions is less than d then is in the essential spectrum of every self adjoint realization of the equation weidmann extends this result to the two singular endpoint case provided an additional condition is satisfied here we prove this result without the additional condition

boundary blow up in nonlinear elliptic equations of bieberbach rademacher type the research of the first author was carried out at victoria university melbourne with the support of the australian government through detya the second author has been supported by grant 2 cex06 11 18 2006

branching particle systems and superprocesses we start from a model of a branching particle system with immigration and with death rate and branching mechanism depending on time and location then we consider a limit case when the mass of particles and their life times are small and their density is high this way we construct a measure valued process x t which we call a superprocess replacing the underlying markov process xi t by the corresponding historical process xi leq t we construct a measure valued process m t in functional spaces which we call a historical superprocess the moment functions for superprocesses are evaluated linear positive additive functionals are studied they are used to construct a continuous analog of a random tree obtained by stopping every particle at a time depending on its path say at the first exit time from a domain a related special markov property for superprocesses is proved which is useful for applications to certain nonlinear partial differential equations the concluding section is devoted to a survey of the literature and the terminology on markov processes used in the paper is explained in the appendix

lower bounds for densities of uniformly elliptic random variables on wiener space in this article we generalize the lower bound estimates for uniformly elliptic diffusion processes obtained by kusuoka and stroock we define the concept of uniform elliptic random variable on wiener space and show that with this definition one can prove a lower bound estimate of gaussian type for its density we apply our results to the case of the stochastic heat equation under the hypothesis of unifom ellipticity of the diffusion coefficient

regular transition functions and regular superprocesses the class of regular markov processes is very close to the class of right processes studied by meyer getoor and others we say that a transition function p is regular if it is the transition function of a well defined regular markov process a characterization of regular transition functions is given which implies that if p is regular then the dawson watanabe and the flemingviot supertransition functions over p belong to the same class

on states of exit measures for superdiffusions we consider the exit measures of l superdiffusions 1 2 from a bounded smooth domain d in r d by using analytic results about solutions of the corresponding boundary value problem we study the states of the exit measures abraham and le gall investigated earlier this problem for a special case l and 2 also as an application of these analytic results we give a different proof for the critical hausdorff dimension of boundary polarity established earlier by le gall under more restrictive assumptions and by dynkin and kuznetsov for general situations

boundary singularities of solutions to elliptic viscous hamilton jacobi equations abstract we study the boundary value problem with measures for e1 u g u 0 in a bounded domain in r n satisfying e2 u on and prove that if g l 1 1 t 2 n 1 n d t is nondecreasing e1 e2 can be solved with any positive bounded measure when g r r q with q 1 we prove that any positive function satisfying e1 admits a boundary trace which is an outer regular borel measure not necessarily bounded when g r r q with 1 q q c n 1 n we prove the existence of a positive solution with a general outer regular borel measure as boundary trace and characterize the boundary isolated singularities of positive solutions when g r r q with q c q 2 we prove that a necessary condition for solvability is that must be absolutely continuous with respect to the bessel capacity c 2 q q q we also characterize boundary removable sets for moderate and sigma moderate solutions

gaussian type lower bounds for the density of solutions of sdes driven by fractional brownian motions in this paper we obtain gaussian type lower bounds for the density of solutions to stochastic differential equations sdes driven by a fractional brownian motion with hurst parameter h in the one dimensional case with additive noise our study encompasses all parameters h 0 1 while the multidimensional case is restricted to the case h 1 2 we rely on a mix of pathwise methods for stochastic differential equations and stochastic analysis tools

some properties of the range of super brownian motion we consider a super brownian motion x its canonical measures can be studied through the path valued process called the brownian snake we obtain the limiting behavior of the volume of the neighborhood for the range of the brownian snake and as a consequence we derive the analogous result for the range of super brownian motion and for the support of the integrated super brownian excursion then we prove the support of xt is capacity equivalent to 0 1 2 in d d 3 and the range of x as well as the support of the integrated super brownian excursion are capacity equivalent to 0 1 4 in d d 5

blow up boundary solutions of some semilinear fractional equations in the unit ball for 0 we are interested in blow up solutions u c b of the fractional problem in the unit ball b we distinguish particularly two orders of singularity at the boundary solutions exploding at the same rate than 2 1 denotes the euclidean distance and those higher singular than 2 1 as a consequence it will be shown that the classical keller osserman condition cannot be readopted in the fractional setting

intersection equivalence of brownian paths and certain branching processes we show that sample paths of brownian motion and other stable processes intersect the same sets as certain random cantor sets constructed by a branching process with this approach the classical result that two independent brownian paths in four dimensions do not intersect reduces to the dying out of a critical branching process and estimates due to lawler 1982 for the long range intersection probability of several random walk paths reduce to kolmogrov s 1938 law for the lifetime of a critical branching process extensions to random walks with long jumps and applications to hausdorff dimension are also derived

a local time correspondence for stochastic partial differential equations it is frequently the case that a white noise driven parabolic and or hyperbolic stochastic partial differential equation spde can have random field solutions only in spatial dimension one here we show that in many cases where the spatial operator is the l generator of a levy process x a linear spde has a random field solution if and only if the symmetrization of x possesses local times this result gives a probabilistic reason for the lack of existence of random field solutions in dimensions strictly bigger than one in addition we prove that the solution to the spde is holder continuous in its spatial variable if and only if the said local time is holder continuous in its spatial variable we also produce examples where the random field solution exists but is almost surely unbounded in every open subset of space time our results are based on first establishing a quasi isometry between the linear l space of the weak solutions of a family of linear spdes on one hand and the dirichlet space generated by the symmetrization of x on the other hand we study mainly linear equations in order to present the local time correspondence at a modest technical level however some of our work has consequences for nonlinear spdes as well we demonstrate this assertion by studying a family of parabolic spdes that have additive nonlinearities for those equations we prove that if the linearized problem has a random field solution then so does the nonlinear spde moreover the solution to the linearized equation is holder continuous if and only if the solution to the nonlinear equation is and the solutions are bounded and unbounded together as well finally we prove that in the cases that the solutions are unbounded they almost surely blow up at exactly the same points date october 7 2007 2000 mathematics subject classification primary 60h15 60j55 secondary 35r60 35d05

orbifolds as configuration spaces of systems with gauge symmetries in systems like yang mills or gravity theory which have a symmetry of gauge type neither phase space nor configuration space is a manifold but rather an orbifold with singular points corresponding to classical states of non generically higher symmetry the consequences of these symmetries for quantum theory are investigated first a certain orbifold configuration space is identified then the schrodinger equation on this orbifold is considered as a typical case the schrodinger equation on double cones over riemannian manifolds is discussed in detail as a problem of selfadjoint extensions a marked tendency towards concentration of the wave function around the singular points in configuration space is observed which generically even reflects itself in the existence of additional bound states and can be interpreted as a quantum mechanism of symmetry enhancement

a phase diagram for a stochastic reaction diffusion system in this paper a stochastic reaction diffusion system is considered which models the spread of a finite population reacting with a non renewable resource in the presence of individual based noise a two parameter phase diagram is established to describe the large time evolution distinguishing between certain death or possible life of the population

singular solutions of some nonlinear elliptic equations abstract we give a classification of isolated singularities of any solution of the equation u u u q 1 0 if 0 is the singular point we prove that there exist two types of singularities when 1 n n 2 and that we have essentially the following in 0 i either u x l q n x 2 q 1 with l q n 2 q 1 2q q 1 n 1 q 1 ii or u x c x where c is any constant and the fundamental solution of the laplacian in r n

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

mobius structure of the spectral space of schrodinger operators with point interaction the schrodinger operator with point interaction in one dimension has a u 2 family of self adjoint extensions we study the spectrum of the operator and show that i the spectrum is uniquely determined by the eigenvalues of the matrix u u 2 that characterizes the extension and that ii the space of distinct spectra is given by the orbifold t2 z2 which is a mobius strip with boundary we employ a parametrization of u 2 that admits a direct physical interpretation and furnishes a coherent framework to realize the spectral duality and anholonomy recently found this allows us to find that iii physically distinct point interactions form a three parameter quotient space of the u 2 family

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

level sets of additive levy processes we provide a probabilistic interpretation of a class of natural capacities on euclidean space in terms of the level sets of a suitably chosen multiparameter additive levy process x we also present several probabilistic applications of the aforementioned potential theoretic connections they include areas such as intersections of levy processes and level sets as well as hausdorff dimension computations

large deviations and laws of the iterated logarithm for the local times of additive stable processes we study the upper tail behaviors of the local times of the additive stable processes let x 1 t x p t be independent d dimensional symmetric stable processes with stable index 0 2 and consider the additive stable process x t 1 t p x 1 t 1 x p t p under the condition d ap we obtain a precise form of the large deviation principle for the local time x 0 t p t 0 t 0 x x 1 s 1 x p sp ds 1 ds p of the multiparameter process x t 1 t p and for its supremum norm sup x r d x 0 t p our results apply to the law of the iterated logarithm and our approach is based on fourier analysis moment computation and time exponentiation

jordan curves in the level sets of additive brownian motion keywords additive brownian motion brownian sheet level set jordan curve reference prob article 2001 002doi 10 1090 s0002 9947 01 02811 2view record in web of science record created on 2008 12 01 modified on 2017 05 12

multiple points of the brownian sheet in critical dimensions it is well known that an n parameter d dimensional brownian sheet has no k multiple points when k 1 d 2kn and does have such points when k 1 d 2kn we complete the study of the existence of k multiple points by showing that in the critical cases where k 1 d 2kn there are a s no k multiple points

geography of the level sets of the brownian sheet we describe geometric properties of w wherew is a standard real valued brownian sheet in the neighborhood of the first hitp of the level set w along a straight line or smooth monotone curvel in such a neighborhood we use a decomposition of the formw s t b s b t x s t whereb s andb t are particular diffusion processes andx s t is comparatively small to show thatp is not on the boundary of any connected component of w rather components of this set form clusters nearp an integral test for thorn shaped neighborhoods ofl with tip atp that do not meet w is given we then analyse the position and size of clusters and individual connected components of w near such a thorn giving upper bounds on their height width and the space between clusters this provides a local picture of the level set our calculations are based on estimates of the length of excursions ofb andb and an accounting of the error termx

isolated singularities of some semilinear elliptic equations abstract let be an open subset of r n n 3 containing 0 we consider the solutions of u x g u x f x in 0 where g is nondecreasing and f is bounded and we study the possible singularities at 0 when u x o x 1 n we prove that u is isotropic near 0 and show that either it is a c1 function in removable singularity or x n 2 u x c c 0 weak singularity or x n 2 u x strong singularity we also characterize the g s for which solutions with a weak singularity exist and improve a previous removability result of h brezis and l veron arch rational mech anal 23 1979 153 166

hausdorff dimension of the contours of symmetric additive levy processes let x 1 x n denote n independent symmetric levy processes on r d the corresponding additive levy process is defined as the following n parameter random field on r d mathfrak x t x 1 t 1 cdots x n t n quad t in bf r n khoshnevisan and xiao ann probab 30 1 62 100 2002 have found a necessary and sufficient condition for the zero set mathfrak x 1 0 of mathfrak x to be non trivial with positive probability they also provide bounds for the hausdorff dimension of mathfrak x 1 0 which hold with positive probability in the case that mathfrak x 1 0 can be non void here we prove that the hausdorff dimension of mathfrak x 1 0 is a constant almost surely on the event mathfrak x 1 0 neq varnothing moreover we derive a formula for the said constant this portion of our work extends the well known formulas of horowitz israel j math 6 176 182 1968 and hawkes j lond math soc 8 517 525 1974 both of which hold for one parameter levy processes more generally we prove that for every nonrandom borel set f in 0 n the hausdorff dimension of mathfrak x 1 0 cap f is a constant almost surely on the event mathfrak x 1 0 cap f neq varnothing this constant is computed explicitly in many cases

measure valued branching diffusions with spatial interactions a strong equation driven by a historical brownian motion is used to construct and characterize measure valued branching diffusions in which the spatial motions obey an ito equation with drift and diffusion depending on the position of an individual and the entire population

brownian sheet and capacity the main goal of this paper is to present an explicit capacity estimate for hitting probabilities of the brownian sheet as applications we determine the escape rates of the brownian sheet and also obtain a local intersection equivalence between the brownian sheet and the additive brownian motion other applications concern quasisure properties in wiener space

blow up boundary solutions of semilinear elliptic problems we consider the following semilinear elliptic equation u p x f u in u 0 u 0 in 1 where rn n 3 is a smooth domain bounded or possibly unbounded with compact possibly empty boundary we assume throughout this paper that p is a non negative function such that p c0 if is bounded and p c loc otherwise the nonlinearity f is assumed to ful ll f 1 f c1 0 f 0 f 0 0 and f 0 on 0 and the keller osserman condition see 7 13

hitting probabilities for systems of non linear stochastic heat equations with multiplicative noise we consider a system of d non linear stochastic heat equations in spatial dimension 1 driven by d dimensional space time white noise the non linearities appear both as additive drift terms and as multipliers of the noise using techniques of malliavin calculus we establish upper and lower bounds on the one point density of the solution u t x and upper bounds of gaussian type on the two point density of u s y u t x in particular this estimate quantifies how this density degenerates as s y t x from these results we deduce upper and lower bounds on hitting probabilities of the process u t x t in mathbb r x in 0 1 in terms of respectively hausdorff measure and newtonian capacity these estimates make it possible to show that points are polar when d 7 and are not polar when d 5 we also show that the hausdorff dimension of the range of the process is 6 when d 6 and give analogous results for the processes t mapsto u t x and x mapsto u t x finally we obtain the values of the hausdorff dimensions of the level sets of these processes

on a problem of erdos and taylor let s n n 0 be a centered d dimensional random walk d 3 and consider the so called future infima process j n df inf n s this paper is concerned with obtaining precise integral criteria for a function to be in the levy upper class of j this solves an old problem of erdos and taylor who posed the problem for the simple symmetric random walk on z d d 3 these results are obtained by a careful analysis of the future infima of transient bessel processes and using strong approximations our results belong to a class of ciesielski taylor theorems which relate d and d 2 dimensional bessel processes

wiener s test for super brownian motion and the brownian snake we prove a wiener type criterion for super brownian motion and the brownian snake if f is a borel subset of d and x d we provide a necessary and sufficientcondition for super brownian motion started at x to immediately hit the set f equivalently this condition is necessary and sufficient for the hitting time of f by thebrownian snake with initial point x to be 0 a key ingredient of the proof isan estimate showing that the hitting probability of f is comparable up to multiplicative constants to the relevant capacity of f this estimate which is of independent interest refines previous results due to perkins and dynkin an important role is played by additivefunctionals of the brownian snake which are investigated here via the potentialtheory of symmetric markov processes as a direct application of our probabilisticresults we obtain a necessary and sufficient condition for the existence in a domain d of a positivesolution of the equation u u 2 which explodes at a given point of d

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

on the existence of resonances in the transmission probability for interactions arising from derivatives of dirac s delta function the scattering properties of regularizing finite range potentials constructed in the form of squeezed rectangles which approximate the first and second derivatives of the dirac delta function x are studied in the zero range limit particularly for a countable set of interaction strength values a non zero transmission through the point potential x defined as the weak limit in the standard sense of distributions of a special dipole like sequence of rectangles is shown to exist when the rectangles are squeezed to zero width a tripole sequence of rectangles which gives in the weak limit the distribution x is demonstrated to exhibit the total transmission for a countable sequence of the rectangle s width that tends to zero however this tripole sequence does not admit a well defined point interaction in the zero range limit making sense only for a finite range of the regularizing rectangular like potentials

lower bounds for the density of locally elliptic ito processes we give lower bounds for the density pt x y of the law of xt the solution of dxt xt dbt b xt dt x0 x under the following local ellipticity hypothesis there exists a deterministic differentiable curve xt 0 t t such that x0 x xt y and xt 0 for all t 0 t the lower bound is expressed in terms of a distance related to the skeleton of the diffusion process this distance appears when we optimize over all the curves which verify the above ellipticity assumption the arguments which lead to the above result work in a general context which includes a large class of wiener functionals for example ito processes our starting point is work of kohatsu higa which presents a general framework including stochastic pde s

on the supports of measure valued critical branching brownian motion on etudie la taille globale et la localisation des supports des etats du processus de markov a valeur de mesure qui est un analogue du mouvement brownien ramifie critique classique sur l espace euclidien a d dimensions

trace on the boundary for solutions of nonlinear differential equations let l be a second order elliptic differential operator in rd with no zero order terms and let e be a bounded domain in rd with smooth boundary e we say that a function h is l harmonic if lh 0 in e every positive l harmonic function has a unique representation h x j x y i dy e where k is the poisson kernel for l and z is a finite measure on e we call z the trace of h on me our objective is to investigate positive solutions of a nonlinear equation lu u in e for 1 2 we associate with every solution u a pair if i where f is a closed subset of e and zis a radon measure on 0 e f we call f zi the trace of u on me f is empty if and only if u is dominated by an l harmonic function we call such solutions moderate a moderate solution is determined uniquely by its trace in general many solutions can have the same trace we establish necessary and sufficient conditions for a pair f z to be a trace and we give a probabilistic formula for the maximal solution with a given trace

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

on an open problem of weidmann essential spectra and square integrable solutions in 1987 weidmann proved that for a symmetric differential operator and a real if there exist fewer square integrable solutions of y 0 than needed and if there is a self adjoint extension of such that is not its eigenvalue then belongs to the essential spectrum of however he posed an open problem of whether the second condition is necessary and it has been conjectured that the second condition can be removed in this paper we first set up a formula of the dimensions of null spaces for a closed symmetric operator and its closed symmetric extension at a point outside the essential spectrum we then establish a formula of the numbers of linearly independent square integrable solutions on the left and the right subintervals and on the entire interval for n th order differential operators the latter formula ascertains the above conjecture these results are crucial in criteria of essential spectra in terms of the numbers of square integrable solutions for real values of the spectral parameter

on essential spectra of singular linear hamiltonian systems abstract the paper is concerned with essential spectra of singular linear hamiltonian systems of arbitrary order with arbitrary equal defect indices several sufficient conditions for the essential spectral points are given in terms of the number of linearly independent square integrable solutions of the corresponding hamiltonian system and a sufficient and necessary condition for the essential spectral points is obtained for hamiltonian systems of even order with one singular endpoint an advantage of these results is that one can determine the essential spectral points of hamiltonian systems by the information of solutions obtained by numerous tools available in the fundamental theory of differential equations in addition two illustrative examples are provided to show how to get some information about the essential spectrum by our results

inequivalent quantizations of the n 3 calogero model with scale and mirror s3 symmetry we study the inequivalent quantizations of the n 3 calogero model by separation of variables in which the model decomposes into the angular and the radial parts our inequivalent quantizations respect the mirror s3 invariance which realizes the symmetry under the cyclic permutations of the particles and the scale invariance in the limit of vanishing harmonic potential we find a two parameter family of novel quantizations in the angular part and classify the eigenstates in terms of the irreducible representations of the s3 group the scale invariance restricts the quantization in the radial part uniquely except for the eigenstates coupled to the lowest two angular levels for which two types of boundary conditions are allowed independently from all upper levels it is also found that the eigenvalues corresponding to the singlet representations of the s3 are universal parameter independent in the family whereas those corresponding to the doublets of the s3 are dependent on one of the parameters th

estimates for the density of a nonlinear landau process abstract the aim of this paper is to obtain estimates for the density of the law of a specific nonlinear diffusion process at any positive bounded time this process is issued from kinetic theory and is called landau process by analogy with the associated deterministic fokker planck landau equation it is not markovian its coefficients are not bounded and the diffusion matrix is degenerate nevertheless the specific form of the diffusion matrix and the nonlinearity imply the non degeneracy of the malliavin matrix and then the existence and smoothness of the density in order to obtain a lower bound for the density the known results do not apply however our approach follows the main idea consisting in discretizing the interval time and developing a recursive method to this aim we prove and use refined results on conditional malliavin calculus the lower bound implies the positivity of the solution of the landau equation and partially answers to an analytical conjecture we also obtain an upper bound for the density which again leads to an unusual estimate due to the bad behavior of the coefficients

evolution equation of a stochastic semigroup with white noise drift we study the existence and uniqueness of the solution of a functionvalued stochastic evolution equation based on a stochastic semigroup whose kernel p s t y x is brownian in s and t the kernel p is supposed to be measurable with respect to the increments of an underlying wiener process in the interval s t the evolution equation is then anticipative and choosing the skorohod formulation we establish existence and uniqueness of a continuous solution with values in l r as an application we prove the existence of a mild solution of the stochastic parabolic equation dut xudt v dt x ru f t x u w dt x where v and w are brownian in time with respect to a common ltration in this case p is the formal backward heat kernel of x v dt x rx supported by the dgicyt grant no pb96 0087 ysupported by nsf international opportunities postdoctoral fellowship no int9600278 and nsf nato postdoctoral fellowship no dge 9633937 ams subject classi cation primary 60h15 secondary 60h07

resonant tunnelling through short range singular potentials a three parameter family of point interactions constructed from sequences of symmetric barrier well barrier and well barrier well rectangles is studied in the limit when the rectangles are squeezed to zero width but the barrier height and the well depth become infinite the zero range limit the limiting generalized potentials are referred to as the second derivative of dirac s delta function x with a renormalized coupling constant 0 or simply as like point interactions as a result a whole family of self adjoint extensions of the one dimensional schrodinger operator is shown to exist which results in full and partial resonant tunnelling through this class of singular potentials the resonant tunnelling occurs for countable sets of interaction strength values in the space which are the roots of several transcendental equations the comparison with the previous results for like point interactions is also discussed

critical brownian sheet does not have double points we derive a decoupling formula for the brownian sheet which has the following ready consequence an n parameter brownian sheet in r d has double points if and only if d 4n in particular in the critical case where d 4n the brownian sheet does not have double points this answers an old problem in the folklore of the subject we also discuss some of the geometric consequences of the mentioned decoupling and establish a partial result concerning k multiple points in the critical case k d 2n d

kolmogorov s test for the brownian snake we present a kolmogorov s test for the brownian snake this result was conjectured by le gall in 1998 it has to be compared with kolmogorov s test for super brownian motion by dhersin and le gall

density minoration of a strongly non degenerated random variable abstract we obtain a lower bound for the density of a d dimensional random variable on the wiener space under exponential moment condition of the divergence of covering vector fields