uniform boundedness of conditional gauge and schrodinger equations we prove that for a bounded domaind rn withc2 boundary and q in k n loc n geqq 3 if e x exp int limits 0 tau d q x t dt mathop ddag limits infty ind then mathop sup limits mathop x in d limits z in partial d e z x exp int limits 0 tau d q x t dt infty xt brownian motion

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

potential theory for the schrodinger equation soit a a ij x x i x j un operateur uniformement elliptique dont les coefficients sont des fonctions bornees mesurables sur un domaine de lipschitz borne d r d on definit la classe de kato k d on demontre des resultats sur l operateur l a v en utilisant des resultats connus pour a et en etudiant la jauge conditionnelle

conditional transformation of drift formula and potential theory for 1 2 b using conditional brownian motion and the transformation of drift formula of cameron martin girsarov maruyama we give integral conditions on a vector fieldb which imply the harmonic measures and green functions for 1 2 and 1 2 b on a bounded lipschitz domaind are equivalent by equivalent we mean there exist two sided inequalities with constants depending only onb andd this enables one to conclude the potential theory for 1 2 b ond and 1 2 ond are the same

gauge theorems for resolvents with application to markov processes letv v 0 be a not necessarily sub markovian resolvent such that the kernelv for some 0 is compact and irreducible we prove the following general gauge theorem if there exists at least onev excessive function which is notv inviriant thenv0 is bounded

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

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

the gauge theorem for a class of additive functionals of zero energy in earlier works the gauge theorem was proved for additive functionals of brownian motion of the form 0 q b s ds whereq is a function in the kato class subsequently the theorem was extended to additive functionals with revuz measures in the kato class we prove that the gauge theorem holds for a large class of additive functionals of zero energy which are in general of unbounded variation these additive functionals may not be semi martingales but correspond to a collection of distributions that belong to the kato class in a suitable sense our gauge theorem generalizes the earlier versions of the gauge theorem

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

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

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

filling polygonal holes with minimal energy surfaces on powell sabin type triangulations in this paper we present two different methods for filling in a hole in an explicit 3d surface defined by a smooth function f in a part of a polygonal domain email protected r 2 we obtain the final reconstructed surface over the whole domain d we do the filling in two different ways discontinuous and continuous in the discontinuous case we fill the hole with a function in a powell sabin spline space that minimizes a linear combination of the usual seminorms in an adequate sobolev space and approximates in the least squares sense the values of f and those of its normal derivatives at an adequate set of points in the continuous case we will first replace f outside the hole by a smoothing bivariate spline s f and then we fill the hole also with a powell sabin spline minimizing a linear combination of given seminorms in both cases we obtain existence and uniqueness of solutions and we present some graphical examples and in the continuous case we also give a local convergence result

homomorphisms on infinite direct product algebras especially lie algebras abstract we study surjective homomorphisms f i a i b of not necessarily associative algebras over a commutative ring k for i a generally infinite set especially when k is a field and b is countable dimensional over k our results have the following consequences when k is an infinite field the algebras are lie algebras and b is finite dimensional if all the lie algebras a i are solvable then so is b if all the lie algebras a i are nilpotent then so is b if k is not of characteristic 2 or 3 and all the lie algebras a i are finite dimensional and are direct products of simple algebras then i so is b ii f splits and iii under a weak cardinality bound on i f is continuous in the pro discrete topology a key fact used in getting i iii is that over any such field every finite dimensional simple lie algebra l can be written l x 1 l x 2 l for some x 1 x 2 l which we prove from a recent result of j m bois the general technique of the paper involves studying conditions under which a homomorphism on i a i must factor through the direct product of finitely many ultraproducts of the a i several examples are given and open questions noted