on dimension of porous measures we introduce a notion of mean porosity for measures and obtain dimensional bounds for mean porous and porous measures as a corollary we generalize to all porous borel measures the estimate obtained recently by j p eckmann e jarvenpaa and m jarvenpaa for porous measures satisfying the doubling condition we also discuss various generalizations of this notion and possible applications

uniform rectifiability and harmonic measure ii poisson kernels in l p imply uniform rectifiability the first author was supported by nsf grants dms 0801079 and dms 1101244 the second author was supported by micinn grant mtm2010 16518 and by csic pie 200850i015 the third author was partially supported by grants dms 0901524 career dms 1056965 us nsf sloan research foundation and mtm2010 16232 mtm2009 14694 c02 01 spain

perturbations of elliptic operators in 1 sided chord arc domains part i small and large perturbation for symmetric operators the first and third authors acknowledge financial support from the spanish ministry of economy and competitiveness through the severo ochoa programme for centres of excellence in r d sev 2015 0554 they also acknowledge that the research leading to these results has received funding from the european research council under the european union s seventh framework programme fp7 2007 2013 erc agreement no 615112 hapdegmt the second author was supported by nsf grant dms 1664047

area and hausdorff dimension of julia sets of entire functions we show the julia set of a sin z has positive area and the action of a sin z on its julia set is not ergodic the julia set of a exp z has hausdorff dimension two but in the presence of an attracting periodic cycle its area is zero

conformal metrics and size of the boundary we establish a lower bound for the hausdorff dimension of the boundary associated with a conformal deformation of the euclidean metric on the unit ball in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the deformations we consider are motivated by quasiconformal maps our abstract approach leads to new results on the boundary behavior of these maps we prove a conjecture by hanson on the compression of sets and obtain an improved version of the so called wall theorem we also establish a riesz privalov theorem in higher dimensions

harmonic measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem it is well known that quantitative scale invariant absolute continuity more precisely the weak a infty property of harmonic measure with respect to surface measure on the boundary of an open set omega subset mathbb r n 1 with ahlfors david regular boundary is equivalent to the solvability of the dirichlet problem in omega with data in l p partial omega for some p infty in this paper we give a geometric characterization of the weak a infty property of harmonic measure and hence of solvability of the l p dirichlet problem for some finite p this characterization is obtained under background hypotheses an interior corkscrew condition along with ahlfors david regularity of the boundary that are natural and in a certain sense optimal we provide counter examples in the absence of either of them or even one of the two upper or lower ahlfors david bounds moreover the examples show that the upper and lower ahlfors david bounds are each quantitatively sharp

dimension gap under sobolev mappings abstract we prove an essentially sharp estimate in terms of generalized hausdorff measures for the images of boundaries of holder domains under continuous sobolev mappings satisfying suitable orlicz sobolev conditions this estimate marks a dimension gap which was first observed in 2 for conformal mappings

symmetry of some entire solutions to the allen cahn equation in two dimensions abstract in this paper we prove even symmetry and monotonicity of certain solutions of allen cahn equation in a half plane we also show that entire solutions with finite morse index and four ends must be evenly symmetric with respect to two orthogonal axes a classification scheme of general entire solutions with finite morse index is also presented using energy quantization

on de giorgi s conjecture in dimensions 4 and 5 in this paper we develop an approach for establishing in some important cases a conjecture made by de giorgi more than 20 years ago the problem originates in the theory of phase transition and is so closely connected to the theory of minimal hypersurfaces that it is sometimes referred to as the eversion of bernstein s problem for minimal graphs the conjecture has been completely settled in dimension 2 by the authors 15 and in dimension 3 in 2 yet the approach in this paper seems to be the first to use in an essential way the solution of the bernstein problem stating that minimal graphs in euclidean space are necessarily hyperplanes provided the dimension of the ambient space is not greater than 8 we note that the solution of bernstein s problem was also used in 18 to simplify an argument in 9 here is the conjecture as stated by de giorgi 12 conjecture 1 1 suppose that u is an entire solution of the equation

analysis of wiener functionals malliavin calculus and its applications to heat kernels on donne un expose facilement accessible du calcul de malliavin du calcul differentiel de dimension infinie sur l espace de wiener

two end solutions to the allen cahn equation in r3 abstract in this paper we study axially symmetric solutions of allen cahn equation in the three dimensional euclidean space using a sophisticated continuation method we show the existence of a complete family of axially symmetric solutions with a range of logarithmic growth rates which may be regarded as the analogue of the family of catenoids and hence called two end solutions nonexistence of two end solution with a small growth rate is also shown which differs from the theory of minimal surfaces

infinity laplacian equation with strong absorptions abstract we study regularity properties of solutions to reaction diffusion equations ruled by the infinity laplacian operator we focus our analysis in models presenting plateaus i e regions where a non negative solution vanishes identically we obtain sharp geometric regularity estimates for solutions along the boundary of plateaus sets in particular we show that the n hausdorff measure of the plateaus boundary is finite for a universal number 0

closed form expansion conditional expectation and option valuation enlightened by the theory of watanabe watanabe s 1987 analysis of wiener functionals malliavin calculus and its applications to heat kernels ann probab 15 1 39 for analyzing generalized random variables and its further development in yoshida yoshida n 1992a asymptotic expansions for statistics related to small diffusions j japan statist soc 22 139 159 takahashi takahashi a 1995 essays on the valuation problems of contingent claims ph d thesis haas school of business university of california berkeley takahashi a 1999 an asymptotic expansion approach to pricing contingent claims asia pacific financial markets 6 115 151 as well as kunitomo and takahashi kunitomo n takahashi a 2001 the asymptotic expansion approach to the valuation of interest rate contingent claims math finance 11 1 117 151 kunitomo n takahashi a 2003 on validity of the asymptotic expansion approach in contingent claim analysis ann appl probab 13 3 914 952 etc we focus on a wide range of mult

dahlberg s theorem in higher co dimension abstract in 1977 the celebrated theorem of b dahlberg established that the harmonic measure is absolutely continuous with respect to the hausdorff measure on a lipschitz graph of dimension n 1 in r n and later this result has been extended to more general non tangentially accessible domains and beyond in the present paper we prove the first analogue of dahlberg s theorem in higher co dimension on a lipschitz graph of dimension d in r n d n 1 with a small lipschitz constant we construct a linear degenerate elliptic operator l such that the corresponding harmonic measure l is absolutely continuous with respect to the hausdorff measure on more generally we provide sufficient conditions on the matrix of coefficients of l which guarantee the mutual absolute continuity of l and the hausdorff measure

σ finiteness of elliptic measures for quasilinear elliptic pde in space abstract in this paper we study the hausdorff dimension of a elliptic measure f in space associated to a positive weak solution to a certain quasilinear elliptic pde in an open subset and vanishing on a portion of the boundary of that open set we show that this measure is concentrated on a set of finite n 1 dimensional hausdorff measure for p n and the same result holds for p n with an assumption on the boundary we also construct an example of a domain in space for which the corresponding measure has hausdorff dimension n 1 for p n for some which depends on various constants including p the first result generalizes the authors previous work in 3 when the pde is the p laplacian and the second result generalizes the well known theorem of wolff in 24 when p 2 and n 2

holder domains and poincare domains a domain d c rd of finite volume is said to be a p poincare domain if there is a constant mp d so that fu udii dx d then d is a p poincare domain this answers a question of axler and shields regarding the image of the unit disk under a holder continuous conformal mapping we also consider geometric conditions which imply that the imbedding of the sobolev space w p d lp d is compact and prove that this is the case for a holder domain d

rectifiability and elliptic measures on 1 sided nta domains with ahlfors david regular boundaries let omega subset mathbb r n 1 n geq 2 be 1 sided nta domain aka uniform domain i e a domain which satisfies interior corkscrew and harnack chain conditions and assume that partial omega is n dimensional ahlfors david regular we characterize the rectifiability of partial omega in terms of the absolute continuity of surface measure with respect to harmonic measure we also show that these are equivalent to the fact that partial omega can be covered mathcal h n a e by a countable union of portions of boundaries of bounded chord arc subdomains of omega and to the fact that partial omega possesses exterior corkscrew points in a qualitative way mathcal h n a e our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally lipschitz coefficients whose derivatives satisfy a natural qualitative carleson condition

asymptotic expansions of maximum likelihood estimators for small diffusions via the theory of malliavin watanabe the asymptotic expansions of the probability distributions of statistics for the small diffusion are derived by means of the malliavin calculus from this the second order efficiency of the maximum likelihood estimator is proved

capacity and principal eigenvalues the method of enlargement of obstacles revisited we describe a coarse graining method which provides lower bounds on the principal dirichlet eigenvalue of the laplacian in regions receiving small obstacles and sharpens the previous method of enlargement of obstacles based on a quantitative wiener criterion one replaces the actual obstacles by obstacles of a much larger size controls on the shift of principal eigenvalues and capacity estimates on the locus where the wiener criterion breaks down are derived the results are written in a self contained fashion

malliavin calculus and asymptotic expansion for martingales we present an asymptotic expansion of the distribution of a random variable which admits a stochastic expansion around a continuous martingale the emphasis is put on the use of the malliavin calculus the uniform nondegeneracy of the malliavin covariance under certain truncation plays an essential role as the cramer condition did in the case of independent observations applications to statistics are presented

on the hausdorff dimension of harmonic measure in higher dimension assume a i n e a domain in f n where e is a compact set denote o a a x the harmonic measure for a of a evaluated at x r d according to 0ksendal s theorem o o e o a x is singular with respect to d dimensional lebesgue measure for d 2 and general domains this result seemed to be so far the only known localization property recently for d 2 it has been shown by p jones and t wolff j w that s o9 has dimension at most 1 this result completes previous work due to n g makarov and l carleson see i m and c2 let g be the green s function of a r with pole at some point of c r o9 in both the carleson and jones wolff arguments the integral g g nn log nn as boundary

moduli space theory for the allen cahn equation in the plane in this paper we study entire solutions of the allen cahn equation delta u f u 0 where f is an even bistable function we are particularly interested in the description of the moduli space of solutions which have some special structure at infinity the solutions we are interested in have their zero set asymptotic to 2k k 2 oriented affine half lines at infinity and along each of these affine half lines the solutions are asymptotic to the one dimensional heteroclinic solution such solutions are called multiple end solutions and their set is denoted by m 2k the main result of our paper states that if u in m 2k is nondegenerate then locally near u the set of solutions is a smooth manifold of dimension 2k this paper is part of a program whose aim is to classify all 2k ended solutions of the allen cahn equation in dimension 2 for k 2

a new delta expansion for multivariate diffusions via the ito taylor expansion abstract in this paper we develop a new delta expansion approach to deriving analytical approximation to the transition densities of multivariate diffusions using the ito taylor expansion of the conditional expectation of the dirac delta function our approach yields an explicit recursive formulas for the expansion coefficients and is universally applicable for a wide spectrum of models particularly the time inhomogeneous non affine irreducible multivariate diffusions we show that this new approach can be viewed as an extension of ait sahalia 2002 and lee et al 2014 to the case of multivariate models the derived expansions are proved to converge to the true probability density as the observational time interval shrinks the obtained approximations can thereby be used to carry out the maximum likelihood estimation for the diffusions with discretely observed data extensive numerical experiments demonstrate the accuracy and effectiveness of our approach

hamiltonian identities for elliptic partial differential equations abstract new identities for elliptic partial differential equations are obtained several applications are discussed in particular young s law for the contact angles in triple junction formation is proven rigorously structure of level curves of saddle solutions to allen cahn equation are also carefully analyzed

on error estimates for asymptotic expansions with malliavin weights application to stochastic volatility model this paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its greeks sensitivities under a stochastic volatility model more generally we also derive an error estimate for an asymptotic expansion around a general partially elliptic diffusion and a more general wiener functional which is applicable to various important valuation and risk management tasks in the financial business such as the ones for multidimensional diffusion and nondiffusion models in particular we take the malliavin calculus approach and estimate the error bounds for the malliavin weights of both the coefficient and the residual terms in the expansions by effectively applying the properties of kusuoka stroock functions introduced by kusuoka kusuoka s 2003 malliavin calculus revisited j math sci univ tokyo 10 261 277 functions moreover a numerical experiment under the heston type model confirms the effectiveness of our method

regularity of flat level sets in phase transitions we consider local minimizers of the ginzburg landau energy functional 1 2 u 2 1 4 1 u 2 2 dx and prove that if the 0 level set is included in a flat cylinder then in the interior it is included in a flatter cylinder as a consequence we prove a conjecture of de giorgi which states that level sets of global solutions of u u 3 u such that u 1 n u 0 lim u x x n 1 x n are hyperplanes in dimension n 8

packing dimension and ahlfors regularity of porous sets in metric spaces let x be a metric measure space with an s regular measure we prove that if a subset x is varrho porous then rm dim p a le s c varrho s where dimp is the packing dimension and c is a positive constant which depends on s and the structure constants of this is an analogue of a well known asymptotically sharp result in euclidean spaces we illustrate by an example that the corresponding result is not valid if is a doubling measure however in the doubling case we find a fixed n subset x with n 0 such that rm dim p a le rm dim p x c log tfrac1 varrho 1 varrho t for all varrho porous sets a subset x setminus n here c and t are constants which depend on the structure constant of finally we characterize uniformly porous sets in complete s regular metric spaces in terms of regular sets by verifying that a is uniformly porous if and only if there is t s and a t regular set f such that a subset f

on variational characterization of four end solutions of the allen cahn equation in the plane abstract in this paper a novel variational method is developed to construct four end solutions in r 2 for the allen cahn equation four end solutions have been constructed by del pino kowalczyk pacard and wei when the angle of the ends is close to 2 or 0 using the lyapunov schmid reduction method and later by kowalczyk liu and pacard using a continuation method for general 0 2 by a special mountain pass argument in a restricted space namely a set of monotone paths of monotone functions a family of solutions in bounded domain is constructed with very good control of their nodal sets which then leads to a four end solution after sending the domains to r 2 in this way not only a family of four end solutions is constructed for any angle 0 2 the morse index of such a four end solution is also shown to be one

partial mixing and edgeworth expansion introducing a conditional mixing property gotze and hipp s theory is generalized to a continuous time conditional markov process satisfying this property the malliavin calculus for jump processes applies to random coefficient stochastic differential equations with jumps with the aid of the support theorem to verify the non degeneracy condition i e a conditional type cramer condition

entire solutions of semilinear elliptic equations in r 3 and a conjecture of de giorgi this paper is concerned with the study of bounded solutions of semilinear elliptic equations u f u in the whole space r under the assumption that u is monotone in one direction say nu in r n the goal is to establish the one dimensional character or symmetry of u namely that u only depends on one variable or equivalently that the level sets of u are hyperplanes this type of symmetry question was raised by de giorgi in who made the following conjecture we quote literally page of dg

a implies nta for a class of variable coefficient elliptic operators abstract we consider a certain class of second order variable coefficient divergence form elliptic operators in a uniform domain with ahlfors regular boundary and we show that the a property of the elliptic measure associated to any such operator and its transpose imply that the domain is in fact nta and hence chord arc the converse was already known and follows from work of kenig and pipher 30

on the structure of partial balayage abstract we define partial balayage construct it with classical potential theoretic methods and establish the so called structure formula in full generality finally we give a stochastic construction of partial balayage

absolute continuity of harmonic measure for domains with lower regular boundaries abstract we study absolute continuity of harmonic measure with respect to surface measure on domains that have large complements we show that if r d 1 is ahlfors d regular and splits r d 1 into two nta domains then h d on this result is a natural generalization of a result of wu in 49 we also prove that almost every point in is a cone point if is a lipschitz graph combining these results and a result from 8 we characterize sets of absolute continuity with finite h d measure if d 1 for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary even in the plane this extends the results of mcmillan in 38 and pommerenke in 43 which were only known for simply connected planar domains finally we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix

average homogeneity and dimensions of measures we introduce the concept of average homogeneity of a measure by comparing the measure to the uniform distribution in a relatively simple way this leads to a very general notion which may be regarded as an inverse of porosity in this paper the emphasis is given to relations between homogeneity and dimensions of measures first we consider the effect of homogeneity on dimensions by proving an upper bound to the hausdorff dimension as a function of homogeneity and its order the opposite question of how dimensions effect homogeneity is solved by giving an upper bound to homogeneity in terms of upper packing dimension we also illustrate by examples that all our results are the best possible ones

the harmonic measure of balls in random trees we study properties of the harmonic measure of balls in typical large discrete trees for a ball of radius nn centered at the root we prove that although the size of the boundary is of order nn most of the harmonic measure is supported on a boundary set of size approximately equal to n n where 0 78 0 78 is a universal constant to derive such results we interpret harmonic measure as the exit distribution of the ball by simple random walk on the tree and we first deal with the case of critical galton watson trees conditioned to have height greater than nn an important ingredient of our approach is the analogous continuous model related to aldous continuum random tree where the dimension of harmonic measure of a level set of the tree is equal to whereas the dimension of the level set itself is equal to 11 the constant is expressed in terms of the asymptotic distribution of the conductance of large critical galton watson trees

exceptional sets for the definition of quasiconformality we show that one can allow for an exceptional set in the definition of quasiconformality even when limsup is replaced with liminf

the classification of singly periodic minimal surfaces with genus zero and scherk type ends given an integer k 2 let s k be the space of complete embedded singly periodic minimal surfaces in r 3 which in the quotient have genus zero and 2k scherk type ends surfaces in s k can be proven to be proper a condition under which the asymptotic geometry of the surfaces is well known it is also known that s 2 consists of the 1 parameter family of singly periodic scherk minimal surfaces we prove that for each k 3 there exists a natural one to one correspondence between s k and the space of convex unitary nonspecial polygons through the map which assigns to each m s k the polygon whose edges are the flux vectors at the ends of m a special polygon is a parallelogram with two sides of length 1 and two sides of length k 1 as consequence s k reduces to the saddle towers constructed by karcher

dissipation in laplacian fields across irregular boundaries the entropy production associated to a laplacian field distributed across irregular boundaries is studied in the context of the active zone approximation an explicit expression is given for the entropy production in terms of geometry whose relation to the variational formulation is discussed it is shown that the entropy production diminishes for successive prefractal generations of the same fractal generator so that the final fractal object is expected to dissipate less than all previous ones the relevance of this result in the abundance of fractal surfaces or interfaces observed in nature is discussed

hausdorff dimension and σ finiteness of p harmonic measures in space when p n abstract in this paper we study a measure associated with a positive p harmonic function u defined in an open set o r n and vanishing on a portion of o if p n we show is concentrated on a set of finite h n 1 measure while if p n the same conclusion holds provided is uniformly fat in the sense of n capacity our work nearly answers in the affirmative a conjecture in lewis 2015 and also appears to be the natural extension of jones and wolff 1988 wolff 1993 to higher dimensions

nonsymmetric conical upper density and k porosity we study how the hausdorff measure is distributed in nonsymmetric narrow cones in n as an application we find an upper bound close to n k for the hausdorff dimension of sets with large k porosity with k porous sets we mean sets which have holes in k different directions on every small scale

symmetry of traveling wave solutions to the allen cahn equation in mathbb r 2 in this paper we prove even symmetry of monotone traveling wave solutions to the balanced allen cahn equation in the entire plane related results for the unbalanced allen cahn equation are also discussed

dirichlet forms quasiregular functions and brownian motion given a quasiregular function on an open setu in n it is shown that there exists a diffusionx t inu such that mapsx t inton dimensional brownian motion the process is constructed from a dirichlet form which can be described explicitly this enables us to apply stochastic methods in the investigation of quasiregular mappings some examples of applications are given including boundary behaviour and value distribution

on de giorgi s conjecture in dimension n 9 a celebrated conjecture due to de giorgi states that any bounded solution of the equation u 1 u 2 u 0 in r n with ynu 0 must be such that its level setsfu g are all hyperplanes at least for dimension n 8 a counterexample for n 9 has long been believed to exist starting from a minimal graph which is not a hyperplane found by bombieri de giorgi and giusti in r n n 9 we prove that for any small 0 there is a bounded solution u y with yn u 0 which resembles tanh a t p 2 a where t t y denotes a choice of signed distance to the blown up minimal graph 1 this solution is a counterexample to de giorgi s conjecture for n 9

multiple end solutions to the allen cahn equation in r2 we construct a new class of entire solutions for the allen cahn equation u 1 u2 u 0 in r2 c given k 1 we find a family of solutions whose zero level sets are away from a compact set asymptotic to 2k straight lines which we call the ends these solutions have the property that there exist 0 1 2k 0 2 such that limr u rei 1 j uniformly in on compact subsets of j j 1 for j 0 2k 1

a gradient bound and a liouville theorem for nonlinear poisson equations soit f c 2 r une fonction non negative et u c 3 r n une solution dans tout r n de l equation u f u ou f f est la derivee 1ere de f si u est borne sur r n et s il existe x 0 r n tel que f u x 0 0 alors u est constante

regularity for the fully nonlinear dead core problem we establish new geometric regularity estimates for reaction diffusion equations with strong absorption terms the model is given by a fully nonlinear elliptic equation with measurable coefficients and a mu holder continuous convection term f x d 2u f u the lack of lipschitz regularity of the map u mapsto f u allows the existence of plateaus i e nonnegative solutions may vanish identically within an a priori unknown region the dead core of the solution we prove that at any touching ground point z in partial u 0 solutions are varkappa mu differentiable for a sharp value varkappa mu ge 2 and in fact varkappa 1 infty the proof is based on a new flatness improvement method we apply this new regularity estimate to establish a liouville type theorem for entire solutions to dead core problems and also to obtain measure estimates on the touching ground boundary the results obtained in this article are new even for dead core problems ruled by linear equations