dirichlet conditions at infinity for parabolic and elliptic equations we investigate existence and uniqueness of bounded solutions of parabolic equations with unbounded coefficients in rn r under specific assumptions we establish existence of solutions satisfying prescribed conditions at infinity depending on the direction along which infinity is approached moreover the large time behavior of such solutions is addressed we consider also elliptic equations in rn with similar conditions at infinity

the poisson boundary of the mapping class group a theory of random walks on the mapping class group and its non elementary subgroups is developed we prove convergence of sample paths in the thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the poisson boundary of random walks the methods are based on an analysis of the asymptotic geometry of teichmuller space and of the contraction properties of the action of the mapping class group on the thurston boundary we prove in particular that teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees using our analysis of the mapping class group action on the thurston boundary we prove that no non elementary subgroup of the mapping class group can be a lattice in a higher rank semi simple lie group

boundary value problems with measures for elliptic equations with singular potentials we study the boundary value problem with radon measures for nonnegative solutions of lvu u vu 0 in a bounded smooth domain when v is a locally bounded nonnegative function introducing some specific capacity we give sufficient conditions on a radon measure on so that the problem can be solved we study the reduced measure associated to this equation as well as the boundary trace of positive solutions in appendix a a ancona solves a question raised by m marcus and l veron concerning the vanishing set of the poisson kernel of lv for an important class of potentials v

positive harmonic functions on complete manifolds of negative curvature this paper studies the interaction between the geometry of complete riemannian manifolds of negative curvature and some aspects of function theory on these spaces the study of harmonic functions on the unit disc provides a classical and beautiful example of this interaction we recall some aspects of this below there is a well known representation of positive harmonic functions on the unit disc u due to herglotz 13 in terms of positive borel measures tt on the circle s1

the dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality we study the spatial behaviour of random walks on infinite graphs which are not necessarily invariant under some transitive group action and whose transition probabilities may have infinite range we assume that the underlying graphg satisfies a strong isoperimetric inequality and that the transition operatorp is strongly reversible uniformly irreducible and satisfies a uniform first moment condition we prove that under these hypotheses the random walk converges almost surely to a random end ofg and that the dirichlet problem forp harmonic functions is solvable with respect to the end compactification if in addition the graph as a metric space is hyperbolic in the sense of gromov then the same conclusions also hold for the hyperbolic compactification in the place of the end compactification the main tool is the exponential decay of the transition probabilities implied by the strong isoperimetric inequality finally it is shown how the same technique can be applied to brownian motion to obtain analogous results for riemannian manifolds satisfying cheeger s isoperimetric inequality in particular in this general context new and simpler proofs of well known results on the dirichlet problem for negatively curved manifolds are obtained

twisted alexander norms give lower bounds on the thurston norm we introduce twisted alexander norms of a compact connected orientable 3 manifold with first betti number greater than one generalizing norms of mcmullen and turaev we show that twisted alexander norms give lower bounds on the thurston norm of a 3 manifold using these we completely determine the thurston norm of many 3 manifolds which are not determined by norms of mcmullen and turaev

local limit theorem for symmetric random walks in gromov hyperbolic groups completing a strategy of gouezel and lalley we prove a local limit theorem for the random walk generated by any symmetric finitely supported probability measure on a non elementary gromov hyperbolic group denoting by r the inverse of the spectral radius of the random walk the probability to return to the identity at time n behaves like c r n n 3 2 an important step in the proof is to extend ancona s results on the martin boundary up to the spectral radius we show that the martin boundary for r harmonic functions coincides with the geometric boundary of the group in an appendix we explain how the symmetry assumption of the measure can be dispensed with for surface groups

martin compactification of a complete surface with negative curvature in this paper we consider the martin compactification associated with the operator mathcal l delta 1 of a complete non compact surface sigma 2 with negative curvature in particular we investigate positive eigenfunctions with eigenvalue one of the laplace operator delta of sigma 2 and prove a uniqueness result such eigenfunctions are unique up to a positive multiplicative constant if they vanish on the part of the geometric boundary s infty sigma 2 of sigma 2 where the curvature is bounded above by a negative constant and satisfy some growth estimate on the other part of s infty sigma 2 where the curvature approaches zero this uniqueness result plays an essential role in our recent paper cao and he infinitesimal rigidity of steady gradient ricci soliton in dimension three arxiv 1412 2714 2014 preprint in which we prove an infinitesimal rigidity theorem for deformations of certain three dimensional collapsed gradient steady ricci soliton with a non trivial killing vector field

the boundary of a square tiling of a graph coincides with the poisson boundary answering a question of benjamini and schramm ann probab 24 3 1219 1238 1996 we show that the poisson boundary of any planar uniquely absorbing e g one ended and transient graph with bounded degrees can be realised geometrically as a circle which arises from a discrete version of riemann s mapping theorem this implies a conjecture of northshield potential anal 2 4 299 314 1993 some of our technique apply to the non planar case and might have further applications when the graph is also hyperbolic then under mild conditions we prove the equivalence of several boundary constructions

negatively curved manifolds elliptic operators and the martin boundary consider a riemannian manifold m of dimension n 2 and assume that m is complete simply connected and that its sectional curvatures are bounded between two negative constants in a recent paper 7 m anderson and r schoen have proved that the martin boundary of m with respect to the laplace beltrami operator a is naturally homeomorphic to soo m the sphere at infinity of m here we shall first establish a similar result for a wide class of elliptic operators on m giving at the same time a new proof of the theorem of anderson and schoen our proof relies heavily on several concepts of potential theory but on the other hand avoids the quite difficult calculations of 7 and perhaps makes the result more understandable roughly speaking we show that if y is a second order elliptic operator defined over m and has a uniformly nice behaviour on each ball of radius 1 in m then dq m the y martin boundary of m is s m provided that the following condition holds there is a positive y ci superharmonic function on m for some positive if instead we only assume the existence of the green function for y a minimal requirement when studying the y martin boundary then d m may be very different from so m even for a rotation invariant operator y on the unit ball of rn equipped

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

the dirichlet problem at infinity for a negatively curved manifold on considere une n variete de riemann complete m qui est connexe et simplement connexe et a des courbures sectionnelles k bornees par 2 constantes finies negatives on etudie l existence de fonctions harmoniques bornees non constantes pour ces varietes

recursive algorithms for two dimensional filters spectral transformations in this correspondence a fast algorithm for two dimensional 2 d digital filters spectral transformations is developed the whole algorithm is based on the 1 d and 2 d horner s form all the considered spectral transformations are arbitrary 2 d rational transformations the method is successfully applied in an example concerning 2 d digital filter design

positive solutions of schrodinger equations and fine regularity of boundary points given a lipschitz domain in mathbb r n and a nonnegative potential v in such that v x d x 2 is bounded we study the fine regularity of boundary points with respect to the schrodinger operator l v v in using potential theoretic methods several conditions are shown to be equivalent to the fine regularity of z in partial omega the main result is a simple explicit if is smooth necessary and sufficient condition involving the size of v for z in partial omega to be finely regular an intermediate result consists in a majorization of int a vert frac u d partial omega vert 2 dx for u positive harmonic in and a subset omega conditions for almost everywhere regularity in a subset a of are also given as well as an extension of the main results to a notion of fine mathcal l 1 vert mathcal l 0 regularity if mathcal l j mathcal l v j v 0 v 1 being two potentials with v 0 v 1 and mathcal l a second order elliptic operator

on families of optimal hardy weights for linear second order elliptic operators abstract we construct families of optimal hardy weights for a subcritical linear second order elliptic operator using a one dimensional reduction more precisely we first characterize all optimal hardy weights with respect to one dimensional subcritical sturm liouville operators on a b a b and then apply this result to obtain families of optimal hardy inequalities for general linear second order elliptic operators in higher dimensions as an application we prove a new rellich inequality

random walks and induced dirichlet forms on self similar sets abstract let k be a self similar set satisfying the open set condition following kaimanovich s elegant idea 25 it has been proved that on the symbolic space x of k a natural augmented tree structure e exists it is hyperbolic and the hyperbolic boundary h x with the gromov metric is holder equivalent to k in this paper we consider certain reversible random walks with return ratio 0 1 on x e we show that the martin boundary m can be identified with h x and k with this setup and a device of silverstein 41 we obtain precise estimates of the martin kernel and the naim kernel in terms of the gromov product moreover the naim kernel turns out to be a jump kernel satisfying the estimate where is the hausdorff dimension of k and depends on for suitable the kernel defines a regular non local dirichlet form on k this extends the results of kigami 27 concerning random walks on certain trees with cantor type sets as boundaries see also 5

a new approach to the martin boundary via diffusions conditioned to hit a compact set let l generate a transient diffusion x t on rd and let d be an exterior domain let h be the smallest positive solution of lh 0 in d and h 1 on dd define xh t to be the process x t conditioned to hit dd by doob s h transform theory xh t is also a markov diffusion and its generator lh is defined by lhf l h l hf letting rd be the hitting time of dd define the harmonic measure for xh t on dd starting from x e d by i4x dy p r xh trd e dy let xn n 1 c d be a sequence satisfying limp xn i for which i4h converges weakly call two such sequences xn 1 and x 1 equivalent if lim k limn o x we call the set of equivalence classes thus generated the harmonic measure boundary at infinity for lh this boundary is independent of the particular exterior domain d we prove that the harmonic measure boundary at infinity for lh coincides with the martin boundary for l on rd the formal adjoint of the operator l on rd in the case that l generates a reversible diffusion the martin boundaries of l and l coincide and hence the harmonic measure boundary of lh coincides with the martin boundary for l on rd a similar probabilistic description of the martin boundary for l on rd can be given in the nonreversible case these results are then used to give explicit representations of the martin boundaries of l and l for several classes of diffusion processes

boundary behavior of harmonic maps on non smooth domains and complete negatively curved manifolds abstract in this paper we develop a theory for harmonic maps which is analogous to the classical theory for bounded harmonic functions in harmonic analysis we solve the dirichlet problem in bounded lipschitz domains and manifolds with negative sectional curvatures generalize the wienner criterion and prove the fatou theorem for harmonic maps into convex balls

on the existence of positive solutions for semilinear elliptic equations with singular lower order coefficients and dirichlet boundary conditions we give sufficient conditions for the existence of positive solutions to some semilinear elliptic equations in bounded domains d subset bbb r n n ge 3 with dirichlet boundary conditions we impose mild conditions on the domains and lower order nonlinear coefficients of the equations in that the bounded domains are only required to satisfy an exterior cone condition and we allow the coefficients to have singularities controlled by kato class functions our approach uses an implicit probabilistic representation schauder s fixed point theorem and new a priori estimates for solutions of the corresponding linear elliptic equations in the course of deriving these a priori estimates we show that the green functions for operators of the form 1 over 2 delta b cdot nabla on d are comparable when one modifies the drift term b on a compact subset of d this generalizes a previous result of ancona 2 obtained under an l p condition on b to a kato condition on b 2

the oshima sekiguchi and liouville theorems on heintze groups abstract let l be an elliptic operator on a riemannian manifold m a function f annihilated by l is said to be l harmonic f is said to have moderate growth if and only if f grows at most exponentially in the riemannian distance if m is a rank one symmetric space and l is the laplace beltrami operator for m the oshima sekiguchi theorem t oshima j sekiguchi eigenspaces of invariant differential operators on an affine symmetric space invent math 57 1980 1 81 states that a l harmonic function f has moderate growth if and only if f is the poisson integral of a distribution on the furstenberg boundary in this work we prove that this result generalizes to a very large class of homogeneous riemannian manifolds of negative curvature we also i prove a liouville type theorem that characterizes the polynomial like harmonic functions which vanish on the boundary in terms of their growth properties ii describe all polynomial like harmonic functions and iii give asymptotic expansions for the poisson kernel one consequence of this work is that every schwartz distribution on the boundary is the boundary value for a l harmonic function f which is uniquely determined modulo polynomial like harmonic functions

entropy and drift in word hyperbolic groups the fundamental inequality of guivarc h relates the entropy and the drift of random walks on groups it is strict if and only if the random walk does not behave like the uniform measure on balls we prove that in any nonelementary hyperbolic group which is not virtually free endowed with a word distance the fundamental inequality is strict for symmetric measures with finite support uniformly for measures with a given support this answers a conjecture of s lalley for admissible measures this is proved using previous results of ancona and blachere haissinsky mathieu for non admissible measures this follows from a counting result interesting in its own right we show that in any infinite index subgroup the number of non distorted points is exponentially small the uniformity is obtained by studying the behavior of measures that degenerate towards a measure supported on an elementary subgroup

brownian motion and dirichlet problems at infinity we discuss angular convergence of riemannian brownian motion on a cartan hadamard manifold and show that the dirichlet problem at infinity for such a manifold is uniquely solvable under the curvature conditions ce 2 ar x km x a2 vspace 0 5pt 0 and cr x 2 km x 1 r x 2 2 2 respectively

integral representations of nonnegative solutions for parabolic equations and elliptic martin boundaries abstract we consider nonnegative solutions of a parabolic equation in a cylinder d 0 t where d is a noncompact domain of a riemannian manifold under the assumption iu i e the associated heat kernel is intrinsically ultracontractive we establish an integral representation theorem any nonnegative solution is represented uniquely by an integral on d 0 m d 0 t where m d is the martin boundary of d for the associated elliptic operator we apply it in a unified way to several concrete examples to explicitly represent nonnegative solutions we also show that iu implies the condition sp i e the constant function 1 is a small perturbation of the elliptic operator on d

schrodinger equations with singular potentials linear and nonlinear boundary value problems let omega subset mathbb r n n ge 3 be a c 2 bounded domain and f subset partial omega be a c 2 submanifold with dimension 0 le k le n 2 denote delta f mathrm dist cdot f v delta f 2 and c h v the hardy constant relative to v in omega we study positive solutions of equations le l gamma v u 0 and ne l gamma v u f u 0 in omega where l gamma v delta gamma v gamma c h v and f in c mathbb r is an odd monotone increasing function we extend the notion of normalized boundary trace introduced in marcus and nguyen ann inst h poincare c non linear anal 34 69 88 2015 and employ it to investigate the linear equation le using these results we obtain properties of moderate solutions of ne finally we determine a criterion for subcriticality of points on partial omega relative to f and study b v p for ne in particular we establish existence and stability results when the data is concentrated on the set of subcritical points

semilinear elliptic equations with subcritical absorption in lipschitz domains abstract let be a bounded lipschitz domain in r n consider the equation u g x u 0 g c r and g x positive and increasing on r x we say that g is subcritical at y if has a solution u k y with boundary trace k y k 0 for a large family of functions g we establish existence and stability results for boundary value problems for with data given by measures concentrated on the set of subcritical points in addition we describe the precise asymptotic behavior of u k y at y some related results have been obtained in marcus and veron 2011 when g t t q in the case that g satisfies the keller osserman condition we prove if u is a positive solution with strong singularity at y i e u is not dominated by a harmonic function then u lim k u k y finally we extend estimates related to the keller osserman condition that are well known in the case of c 2 domains when g is independent of the space variable to lipschitz domains and a large class of functions g including cases where g x t 0 or g x t as x

the poisson equation on complete manifolds with positive spectrum and applications abstract in this paper we investigate the existence of a solution to the poisson equation on complete manifolds with positive spectrum and ricci curvature bounded from below we show that if a function f has decay f o r 1 e for some e 0 where r is the distance function to a fixed point then the poisson equation u f has a solution u with at most exponential growth we apply this result on the poisson equation to study the existence of harmonic maps between complete manifolds and also existence of hermitian einstein metrics on holomorphic vector bundles over complete manifolds thus extending some results of li tam and ni assuming moreover that the manifold is simply connected and of ricci curvature between two negative constants we can prove that in fact the poisson equation has a bounded solution and we apply this result to the ricci flow on complete surfaces

the dirichlet problem at infinity for manifolds of negative curvature on considere l existence de fonctions harmoniques bornees sur des varietes simplement connexes n n de courbure negative