global regularity for a class of quasi linear local and nonlocal elliptic equations on extension domains abstract we consider the solvability of quasi linear elliptic equations with either local or nonlocal neumann robin or wentzell boundary conditions defined in the generalized sense on a bounded w 1 p extension domain whose boundary is an upper d set for an appropriate d 0 then we extend the fine regularity theory for weak solutions of the elliptic equations with the above boundary conditions known for bounded lipschitz domains to bounded w 1 p extension domains whose boundaries are upper d sets by showing that such weak solutions are globally holder continuous consequently we generalize substantially the class of bounded domains where weak solutions of boundary value problems of type neumann robin or wentzell may be uniformly continuous up to the boundary

some potential theory for reflecting brownian motion in holder and lipschitz domains bounds are found on the transition densities and green functions for brownian motion with normal reflection in holder and lipschitz domains for lipschitz domains reflecting brownian motion and boundary local time are constructed a harnack inequality valid up to the boundary is proved a probabilistic solution to the neumann problem is given and the kuramochi boundary is identified

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

quasi linear boundary value problems with generalized nonlocal boundary conditions abstract we investigate a quasi linear boundary value problem of the form div u p 2 u 0 involving a general boundary map and mixed neumann boundary conditions on a bounded lipschitz domain we show existence uniqueness and holder continuity of the weak solution of this mixed boundary value problem and obtain maximum principles for this class of mixed equations as a consequence we obtain uniform continuity up to the boundary to solutions associated with a class of electrical models described by maxwell s equations with nonlocal boundary conditions an extension to boundary value problems with generalized nonlocal robin boundary conditions is also achieved

solutions of δu 4u2 with neumann s conditions using the brownian snake we consider a brownian snake ws s 0 with underlying process a reflected brownian motion in a bounded domain d we construct a continuous additive functional ls s 0 of the brownian snake which counts the time spent by the end points s of the brownian snake paths on d the random measure z sdls is supported by d then we represent the solution v of u 4u2 in d with weak neumann boundary condition 0 by using exponential moment of z under the excursion measure of the brownian snake we then derive an integral equation for v for small it is then possible to describe negative solution of u 4u2 in d with weak neumann boundary condition in contrast to the exit measure of the brownian snake out of d the measure z is more regular in particular we show it is absolutely continuous with respect to the surface measure on d for dimension 2 and 3

stochastic differential equations with reflecting boundary conditions on resout des equations differentielles stochastiques a conditions aux limites reflechissantes par une approche directe basee sur le probleme de skorokhod

on taylor series expansion methods for the second kind integral equations in this paper we comment on the recent papers by yuhe ren et al 1999 1 and maleknejad et al 2006 7 concerning the use of the taylor series to approximate a solution of the fredholm integral equation of the second kind as well as a solution of a system of fredholm equations the technique presented in yuhe ren et al 1999 1 takes advantage of a rapidly decaying convolution kernel k s t as s t increases however it does not apply to equations having other types of kernels we present in this paper a more general taylor expansion method which can be applied to approximate a solution of the fredholm equation having a smooth kernel also it is shown that when the new method is applied to the fredholm equation with a rapidly decaying kernel it provides more accurate results than the method in yuhe ren et al 1999 1 we also discuss an application of the new taylor series method to a system of fredholm integral equations of the second kind

on the stochastic regularity of distorted brownian motions we systematically develop general tools to apply fukushima s absolute continuity condition these tools comprise methods to obtain a hunt process on a locally compact separa ble metric state space whose transition function has a density w r t the reference measure and methods to estimate drift potentials comfortably we then apply our results to distorted brownian motions and construct weak solutions to singular stochastic differential equations i e equations with possibly unbounded and discontinuous drift and reflect ion terms which may be the sum of countably many local times the solutions can start from any point of the explicitly specified state space we consider different kind of weights like muckenhoupt a2 weights and weights with moderate growth at singularities as well as different kind of multiple boundary conditions our approach leads in particular to the construction and explicit identification of countably skew reflected and normally reflected brownian motions with singu lar drift in bounded and unbounded multi dimensional domains

on the resistance of the sierpinski carpet let f text n be the nth stage in the construction of the sierpinski carpet let r text n be the electrical resistance of f text n when the left and right sides are each short circuited and a voltage is applied between them we prove that there exists a constant rho such that textstyle frac 1 4 rho n leq r n leq 4 rho n the motivation for this result came from the problem of establishing a the existence and b the value of the spectral dimension of the sierpinski carpet in this and a subsequent paper we settle a and give bounds for b

energy inequalities for cutoff functions and some applications let x d m be a metric measure space with a local regular dirichlet form we establish necessary and sufficient conditions for upper heat kernel bounds with subdiffusive space time exponent to hold this characterization is stable under rough isometries that is it is preserved under bounded perturbations of the dirichlet form further we give a criterion for stochastic completeness in terms of a sobolev inequality for cutoff functions as an example we show that this criterion applies to an anomalous diffusion on a geodesically incomplete fractal space where the well established criterion in terms of volume growth fails

functional central limit theorem for brownian particles in domains with robin boundary condition abstract we rigorously derive non equilibrium space time fluctuation for the particle density of a system of reflected diffusions in bounded lipschitz domains in r d the particles are independent and are killed by a time dependent potential which is asymptotically proportional to the boundary local time we generalize the functional analytic framework introduced by kotelenez 20 21 to deal with time dependent perturbations our proof relies on dirichlet form method rather than the machineries derived from kotelenez s sub martingale inequality our result holds for any symmetric reflected diffusion for any bounded lipschitz domain and for any dimension d 1

coupling and harnack inequalities for sierpinski carpets uniform harnack inequalities for harmonic functions on the preand graphical sierpinski carpets are proved using a probabilistic coupling argument various results follow from this including the construction of brownian motion on sierpinski carpets embedded in rd d 3 estimates on the fundamental solution of the heat equation and sobolev and poincare inequalities the sierpinski carpets scs we will study are generalizations of the cantor set let f0 0 1 be the unit cube in r d 2 centered at z0 1 2 1 2 let k a be integers with 1 a k and a k even divide f0 into k equal subcubes remove a central block of a subcubes and let f1 be what remains thus f1 f0 k a 2k k a 2k now repeat this operation on each of the k a remaining subcubes to obtain f2 iterating we obtain a decreasing sequence of closed sets fn then f n 0 fn is a sierpinski carpet and has hausdorff dimension df df f log k a log k when d 2 k 3 and a 1 we get the usual sierpinski carpet let fn kfn 0 and define the pre sierpinski carpet by f n 1 fn see 10 the graphical sierpinski carpet is the graph g v e with vertex set v z0 z f and edge set e x y v x y 1 thus int f is a domain in r with a large scale structure which mimics the small scale structure of f we are interested in the behavior of solutions of the laplace and heat equations on f f and g one reason for this is applications to transport phenomena in disordered media see 6 another is the new type of behavior of the heat kernel on these spaces let w be brownian motion on f with normal reflection on f and let q t x y be the transition density of w so that q solves the heat equation on f with neumann boundary conditions on f 1991 mathematics subject classification primary 60b99 secondary 60j35

skorokhod decomposition of reflected diffusions on bounded lipschitz domains with singular non reflection part let overline g subset mathbb r d be a compact set with interior g let l1 g dx 0 dx a e on g and m dx let a aij be symmetric and globally uniformly strictly elliptic on g let be such that mathcal e r f g frac 1 2 sum i j 1 d int g a ij partial i f partial j g dm f g in c infty overline g is closable in l2 g m with closure r d r the latter is fulfilled if satisfies the hamza type condition or i l1loc g dx 1 i d conservative non symmetric diffusion processes xt related to the extension of a generalized dirichlet form mathcal e r f g sum i 1 d int g rho 1 overline b i partial i f g dm f g in d mathcal e r b where rho 1 overline b 1 overline b d in l 2 g mathbb r d m satisfies sum i 1 d int g overline b i partial i f dx 0 quad rm for all f in c infty overline g are constructed and analyzed if g is a bounded lipschitz domain h1 1 g and aij d r a skorokhod decomposition for xt is given this happens through a local time that is uniquely associated to the smooth measure 1 tr 0 d where tr denotes the trace and the surface measure on g

three point boundary value problems for second order ordinary differential equations in banach spaces in this paper by using the sadovskii fixed point theorem we study the existence of at least one solution for the second order three point boundary value problem u t f t u t u t email protected 0

markov processes on the lipschitz boundary for the neumann and robin problems abstract we investigate the markov process on the boundary of a bounded lipschitz domain associated to the neumann and robin boundary value problems we first construct l p semigroups of sub markovian contractions on the boundary generated by the boundary conditions and we show that they are induced by the transition functions of the forthcoming processes as in the smooth boundary case the process on the boundary is obtained by the time change with the inverse of a continuous additive functional of the reflected brownian motion the robin problem is treated with a kato type l p perturbation method using the revuz correspondence an exceptional polar set occurs on the boundary we make the link with the dirichlet forms approach

traps for reflected brownian motion consider an open set d r d d 2 and a closed ball b d let e x tb denote the expectation of the hitting time of b for reflected brownian motion in d starting from x d we say that d is a trap domain if supx e x tb a domain d is not a trap domain if and only if the reflecting brownian motion in d is uniformly ergodic we fully characterize the simply connected planar trap domains using a geometric condition and give a number of less complete results for d 2

regularity and well posedness of some quasi linear elliptic and parabolic problems with nonlinear general wentzell boundary conditions on nonsmooth domains abstract we show that on a bounded domain r n with lipschitz continuous boundary if 2 n n 2 p n 1 then weak solutions of the quasi linear elliptic equation p u u p 2 u 1 x u f in with the general wentzell boundary conditions p u u p 2 u u p 2 u 2 x u g weakly on are globally holder continuous on here p and p denote the p laplace operator and the p laplace beltrami operator on the boundary respectively the functions 1 x for x and 2 x for x are continuous on r and satisfy a certain growth condition we also obtain that a realization of the operator p u 1 x u in c with the boundary conditions p u 1 x u p u u p 2 u 2 x u 0 on generates a strongly continuous semigroup of nonlinear operators on c

traces of symmetric markov processes and their characterizations time change is one of the most basic and very useful transformations for markov processes the time changed process can also be regarded as the trace of the original process on the support of the revuz measure used in the time change in this paper we give a complete characterization of time changed processes of an arbitrary symmetric markov process in terms of the beurling deny decomposition of their associated dirichlet forms and of feller measures of the process in particular we determine the jumping and killing measure or equivalently the levy system for the time changed process we further discuss when the trace dirichlet form for the time changed process can be characterized as the space of finite douglas integrals defined by feller measures finally we give a probabilistic characterization of feller measures in terms of the excursions of the base process

on reflecting diffusion processes and skorokhod decompositions letg be ad dimensional bounded euclidean domain h1 g the set off in l2 g such that f defined in the distribution sense is in l2 g reflecting diffusion processes associated with the dirichlet spaces h1 g on l2 g dx are considered in this paper where a aij is a symmetric bounded uniformly ellipticd d matrix valued function such thata ij h1 g for eachi j and h1 g is a positive bounded function ong which is bounded away from zero a skorokhod decomposition is derived for the continuous reflecting markov processes associated with h1 g having starting points ing under a mild condition which is satisfied when g has finite d 1 dimensional lower minkowski content

on the hot spots conjecture of j rauch rodrigo banuelos research partially supported by nsf grant dms 9400854 krzysztof burdzy s research partially supported by nsf grant dms 9322689

stationary distributions for diffusions with inert drift consider reflecting brownian motion in a bounded domain in mathbb r d that acquires drift in proportion to the amount of local time spent on the boundary of the domain we show that the stationary distribution for the joint law of the position of the reflecting brownian motion and the value of the drift vector has a product form moreover the first component is uniformly distributed on the domain and the second component has a gaussian distribution we also consider more general reflecting diffusions with inert drift as well as processes where the drift is given in terms of the gradient of a potential

solutions and green s functions for some linear second order three point boundary value problems in this paper we consider the green s functions for a second order linear ordinary differential equation with some three point boundary conditions we give exact expressions of the unique solutions for the linear three point boundary problems by the green s functions as applications we study the iterative solutions for some nonlinear singular second order three point boundary value problems

regularity of solutions of linear second order elliptic and parabolic boundary value problems on lipschitz domains for a linear strictly elliptic second order differential operator in divergence form with bounded measurable coefficients on a lipschitz domain we show that solutions of the corresponding elliptic problem with robin and thus in particular with neumann boundary conditions are holder continuous up to the boundary for sufficiently lp regular right hand sides from this we deduce that the parabolic problem with robin or wentzell robin boundary conditions is well posed on c

scaling coupling of reflecting brownian motions and the hot spots problem we introduce a new type of coupling of reflecting brownian motions in smooth planar domains called scaling coupling we apply this to obtain monotonicity properties of antisymmetric second neumann eigenfunctions of convex planar domains with one line of symmetry in particular this gives the proof of the hot spots conjecture for some known types of domains and some new ones

reaction diffusion on metric graphs from 3d to 1d abstract we provide stochastic foundations for the analysis of a class of reaction diffusion systems using as an example the known temporal analysis of products tap experiments showing how to effectively obtain explicit solutions to the associated equations by approximating the 3 dimensional domain of diffusion u the reactor by 1 dimensional network models in a typical tap experiment a pulse of reactant gas of species a is injected into u which is filled with chemically inert material permeable to gas diffusion particles of catalyst are placed amid this inert medium forming active sites where the reaction a b may occur on part of the boundary of u designated as the exit a mixture of a and b can escape we study the problem of determining the molar fraction of product gas in the mixture after u is fully evacuated this fraction is identified with the reaction probability that is the probability of a single diffusing molecule reacting before leaving u specifically we are interested in how this probability depends on the reaction rate constant k after giving a stochastic formulation of the problem and the boundary value problem whose solution is this probability we study a class of domains called fat graphs comprising a network of thin tubes with active sites at junctures the main result of the paper is that in the limit as the thin tubes approach curves in 3 dimensional space reaction probability converges to functions of the point of gas injection that can be computed explicitly in terms of a rescaled parameter k by this 3d to 1d reduction the simpler processes on metric graphs can be used as model systems for more realistic 3 dimensional configurations this is illustrated with analytic and numerical examples one example in particular illustrates an important application of our method finding the catalyst configuration that maximizes reaction probability at a given reaction rate constant

effective conductivity and skew brownian motion we consider heat conduction in a periodic body which is composed of finitely many different components the effective conductivity is represented in terms of skew brownian motion the representation formula is a fluctuation dissipation relation the dissipation term in this formula is related to the transmission of heat through the surface separating the different components of the body it is described by the skew reflections of brownian motion at these surfaces the problems caused by the discontinuity of the microscopic conductivity are handled in the framework of dirichlet forms

switched diffusion processes and systems of elliptic equations a dirichlet space approach the switched diffusion process associated with a weakly coupled system of elliptic equations is studied via a dirichlet space approach and is applied to prove the existence theorem of the cauchy initial problem for the system a representation theorem for the solution of the dirichlet boundary value problem and a generalised skorohod decomposition for the reflecting switched diffusion process are obtained

uniqueness of schrodinger operators restricted in a domain abstract let s 2 v be the schrodinger s operator defined on c 0 d where d is a open domain in r d by means of the asymptotic behavior of v near the boundary d we give the necessary and sufficient conditions to the essential markovian self adjointness of s for the nonnegative potential v and to the uniqueness of s in l 1 d for general v

strong feller properties for distorted brownian motion with reflecting boundary condition and an application to continuous n particle systems with singular interactions abstract assuming that r n n 2 is an open relatively compact set with boundary of lebesgue measure zero we prove strong feller properties for a class of distorted brownian motions in with reflecting boundary condition dirichlet form techniques give the existence of a weak solution to the corresponding stochastic differential equation for quasi all starting points in the sense of the associated martingale problem combining this result with the strong feller properties we can construct a weak solution for specified starting points if has c 2 boundary the construction works for all starting points where the drift term is not singular even on the boundary but also for a certain class of sets with less smooth boundary our approach works for all points in where the drift term is not singular and at least some points from our techniques allow very singular drift terms this enables us to construct continuous n particle gradient stochastic dynamics in cuboids r d d n with reflecting boundary condition and singular interactions for d n 2 we can start the stochastic dynamics in all initial configurations having at most one particle in provided is locally smooth there

on the existence of a three point boundary value problem at resonance in rn abstract we apply a coincidence degree theorem due to mawhin to obtain a sufficient condition for the existence of at least one solution for a class of nonlinear second order differential equations in r n associated with three point boundary conditions

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

approximate solution of three point boundary value problems for second order ordinary differential equations with variable coefficients three point boundary value problem is investigated and solved numerically a novel simple and effective numerical method is proposed the convergence and error estimate of approximate solution are given numerical results are carried out to verify the proposed methods some three point boundary value problems for a second order ordinary differential equation with variable coefficients are investigated in the present paper by using the integration method the second order three point boundary value problems are transformed into a fredholm integral equation of the second kind the solutions and green s functions for some special cases of the second order three point boundary value problems can be determined easily the existence and uniqueness of the solutions of the given fredholm integral equations are considered by using the fixed point theorem in banach spaces a new numerical method is further proposed to solve the second kind fredholm integral equation and an approximate solution is made the convergence and error estimate of the obtained approximate solution are further analyzed numerical results are carried out to verify the feasibility and novelty of the proposed solution procedures

boundary trace of reflecting brownian motions we establish a uniform dimensional result for normally reflected brownian motion rbm in a large class of non smooth domains hausdorff dimensions for the boundary occupation time and the boundary trace of rbm are determined extensions to stable like jump processes and to symmetric reflecting diffusions are also given

normal derivative for bounded domains with general boundary let d be a general bounded domain in the euclidean space rn a brownian motion which enters from and returns to the boundary symmetrically is used to define the normal derivative as a functional for f with f vf and a f all in l2 on d the corresponding neumann condition normal derivative 0 is an honest boundary condition for the l2 generator of reflected brownian notion on d a conditioning argument shows that for d and f sufficiently smooth this general definition of the normal derivative agrees with the usual one introduction bt t 0 is standard brownian motion on the euclidean space rn the case when the dimension n 1 is well understood and so we assume always that n 2 d is a bounded open connected subset of rn an important role will be played by a inf t 0 bt is not in d the hitting time for the complement rn d the absorbed process xt t 0 is xt bt for 0 a with a a dead point adjoined to d as an isolated point all functions are understood to vanish at 9 the absorbed transition operators po t 0 with action ptf x exf xt form a strongly continuous symmetric semigroup on l2 the hilbert space of real valued functions on d that are square integrable with respect to lebesgue measure dx the strong generator ao is selfadjoint and negative definite by the spectral theorem as formulated and proved in chapter ix of segal and kunze 1968 the operator a0 1 2 is a well defined positive definite selfadjoint operator the absorbed dirichlet space is the pair fo e where 0 1 f0 domain ao 2 e f g dx a 1 2f x a0 1 2g x f g in f received by the editors june 7 1987 1980 mathematics subject classification 1985 revision primary 60b85 secondary 35a85

some notes on graphs whose index is close to 2 abstract we consider two classes of graphs i trees of order n and diameter d n 3 and ii unicyclic graphs of order n and girth g n 2 assuming that each graph within these classes has two vertices of degree 3 at distance k we order by the index i e spectral radius the graphs from i for any fixed k 1 k d 2 and the graphs from ii independently of k

three positive solutions for second order m point boundary value problems for the second order m point boundary value problemu t f t u 0 0 0 i 1 2 m 2 0 x 1 x 2 x m 2 1 growth conditions are imposed on f which yield the existence of at least three positive solutions by using the leggett williams fixed point theorem the associated green s function for the m point boundary value problem is also given

on neumann eigenfunctions in lip domains atar s research partially supported by the fund for the promotion of research at the technion burdzy gratefull acknowledges the hospitality and financial support of technion israel and institut mittag leffler sweden this research was partially supported by nsf grant dms 0071486 and isf grant 12 98

existence of multiple positive solutions for nonlinear m point boundary value problems abstract in this paper we afford some sufficient conditions to guarantee the existence of multiple positive solutions for the nonlinear m point boundary value problem for the one dimensional p laplacian p u a t f t u 0 t 0 1 u 0 0 u 1 i 1 m 2 a i u i

transition densities for brownian motion on the sierpinski carpet upper and lower bounds are obtained for the transition densitiesp t x y of brownian motion on the sierpinski carpet these are of the same form as those which hold for the sierpinski gasket in addition the joint continuity ofp t x y is proved the existence of the spectral dimension is established and the einstein relation connecting the spectral dimension the hausdorff dimension and the resistance exponent is shown to hold

the heat equation with nonlinear generalized robin boundary conditions abstract let r n be a bounded domain and let be an admissible measure on we show in the first part that if has the h 1 extension property then a realization of the laplace operator with generalized nonlinear robin boundary conditions formally given by u d x u d 0 on generates a strongly continuous nonlinear submarkovian semigroup s b s b t t 0 on l 2 we also obtain that this semigroup is ultracontractive in the sense that for every u v l p p 2 and every t 0 one has s b t u s b t v c 1 e c 2 t t n 2 p u v p for some constants c 1 c 2 0 in the second part we prove that if is a bounded lipschitz domain one can also define a realization of the laplacian with nonlinear robin boundary conditions on c and this operator generates a strongly continuous and contractive nonlinear semigroup

on high spots of the fundamental sloshing eigenfunctions in axially symmetric domains we investigate the classical eigenvalue problem that arises in hydrodynamics and is referred to as the sloshing problem it describes free liquid oscillations in a liquid container w r we study the case when w is an axially symmetric convex bounded domain satisfying the john condition the cartesian coordinates x y z are chosen so that the mean free surface of the liquid lies in x z plane and y axis is directed upwards y axis is the axis of symmetry our first result states that the fundamental eigenvalue has multiplicity 2 and for each fundamental eigenfunction there is a change of x z coordinates by a rotation around y axis so that is odd in x variable the second result of the paper gives the following monotonicity property of the fundamental eigenfunction if is odd in x variable then it is strictly monotonic in x variable this property has the following hydrodynamical meaning if liquid oscillates freely with fundamental frequency according to then the free surface elevation of liquid is increasing along each line parallel to x axis during one period of time and decreasing during the other half period the proof of the second result is based on the method developed by d jerison and n nadirashvili for the hot spots problem for neumann laplacian

uniqueness and regularity for a system of interacting bessel processes via the muckenhoupt condition we study the regularity of a diusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with coulomb interaction and reflection between nearest neighbors as our main result we establish the strong feller property for the process in both cases of repulsion and attraction in particular the system can be started from any initial state including multiple point configurations moreover we show that the process is a euclidean semi martingale if and only if the interaction is repulsive hence contrary to classical results about reflecting brownian motion in smooth domains in the attractive regime a construction via a system of skorokhod sdes is impossible finally we establish exponential heat kernel gradient estimates in the repulsive regime the main proof for the attractive case is based on potential theory in sobolev spaces with muckenhoupt weights

positive solutions of nonlinear three point boundary value problems abstract let a c 0 1 b c 0 1 0 let 1 t be the unique solution of the linear boundary value problem u t a t u t b t u t 0 t 0 1 u 0 0 u 1 1 we study the existence of positive solutions to the nonlinear boundary value problem u t a t u t b t u t h t f u 0 t 0 1 u 0 0 u u 1 where 0 0 and f c 0 0 we show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem in cones

systems of interacting diffusions with partial annihilation through membranes we introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface ii this system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition a related interacting random walk model with discrete state spaces has recently been introduced and studied in chen and fan 2014 in this paper we establish the functional law of large numbers for this new system thereby extending the hydrodynamic limit in chen and fan 2014 to reflected diffusions in domains with mixed type boundary conditions which include absorption harvest of electric charges we employ a new and direct approach that avoids going through the delicate bbgky hierarchy