boundary behavior in the loewner nirenberg problem abstract let r n be a bounded domain of class c 2 0 1 we show that if n 3 and u is the maximal solution of equation u n n 2 u n 2 n 2 in then the hyperbolic radius v u 2 n 2 is of class c 2 up to the boundary the argument rests on a reduction to a nonlinear fuchsian elliptic pde

fractional fokker planck equation for nonlinear stochastic differential equations driven by non gaussian levy stable noises the fokker planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by gaussian noises however there are both theoretical and empirical reasons to consider similar equations driven by strongly non gaussian noises in particular they yield strongly non gaussian anomalous diffusion which seems to be relevant in different domains of physics in this paper we therefore derive a fractional fokker planck equation for the probability distribution of particles whose motion is governed by a nonlinear langevin type equation which is driven by a levy stable noise rather than a gaussian we obtain in fact a general result for a markovian forcing we also discuss the existence and uniqueness of the solution of the fractional fokker planck equation

stationary states in langevin dynamics under asymmetric levy noises properties of systems driven by white non gaussian noises can be very different from these of systems driven by the white gaussian noise we investigate stationary probability densities for systems driven by ensuremath alpha stable l evy type noises which provide natural extension to the gaussian noise having however a new property namely a possibility of being asymmetric stationary probability densities are examined for a particle moving in parabolic quartic and in generic double well potential models subjected to the action of ensuremath alpha stable noises relevant solutions are constructed by methods of stochastic dynamics in situations where analytical results are known they are compared with numerical results furthermore the problem of estimation of the parameters of stationary densities is investigated

sparre andersen theorem with spatiotemporal correlations the sparre andersen theorem is a remarkable result in one dimensional random walk theory concerning the universality of the ubiquitous first passage time distribution it states that the probability distribution n of the number of steps needed for a walker starting at the origin to land on the positive semiaxes does not depend on the details of the distribution for the jumps of the walker provided this distribution is symmetric and continuous where in particular n n 3 2 for large number of steps n on the other hand there are many physical situations in which the time spent by the walker in doing one step depends on the length of the step and the interest concentrates on the time needed for a return not on the number of steps here we modify the sparre andersen proof to deal with such cases in rather general situations in which the time variable correlates with the step variable as an example we present a natural process in 2d that shows that deviations from normal scaling are present for the first passage time distribution on a semiplane

boundary case of equality in optimal loewner type inequalities we prove certain optimal systolic inequalities for a closed riemannian manifold x g depending on a pair of parameters n and b here n is the dimension of x while b is its first betti number the proof of the inequalities involves constructing abel jacobi maps from x to its jacobi torus t b which are area decreasing on b dimensional areas with respect to suitable norms these norms are the stable norm of g the conformally invariant norm as well as other l p norms here we exploit l p minimizing differential 1 forms in cohomology classes we characterize the case of equality in our optimal inequalities in terms of the criticality of the lattice of deck transformations of t b while the abel jacobi map is a harmonic riemannian submersion that the resulting inequalities are actually nonvacuous follows from an isoperimetric inequality of federer and fleming under the assumption of the nonvanishing of the homology class of the lift of the typical fiber of the abel jacobi map to the maximal free abelian cover

levy flights on the half line we study the probability distribution function pdf of the position of a levy flight of index 0 2 in the presence of an absorbing wall at the origin the solution of the associated fractional fokker planck equation can be constructed using a perturbation scheme around the brownian solution corresponding to 2 as an expansion in e 2 we obtain an explicit analytical solution exact at the first order in e which allows us to conjecture the precise asymptotic behavior of this pdf including the first subleading corrections for any careful numerical simulations as well as an exact computation for 1 confirm our conjecture

a product estimate for ginzburg landau and corollaries abstract we prove a new inequality for the jacobian or vorticity associated to the ginzburg landau energy in any dimension it allows to retrieve existing lower bounds on the energy to extend them to the case of unbounded vorticity and to get a few other corollaries it also provides a new estimate on the time variation for time dependent families which has applications for the study of ginzburg landau dynamics

a remark on rotationally symmetric solutions to the centroaffine minkowski problem abstract in this paper we study the solvability of the rotationally symmetric centroaffine minkowski problem by delicate blow up analyses we remove a technical condition in the existence result obtained by lu and wang 30

partial regularity for mass minimizing currents in hilbert spaces recently the theory of currents and the existence theory for plateau s problem have been extended to the case of finite dimensional currents in infinite dimensional manifolds or even metric spaces see acta math 185 2000 1 80 and also proc lond math soc 3 106 2013 1121 1142 adv calc var 7 2014 227 240 for the most recent developments in this paper in the case when the ambient space is hilbert we provide the first partial regularity result in a dense open set of the support for n dimensional integral currents which locally minimize the mass our proof follows with minor variants indiana univ math j 31 1982 415 434 implementing lipschitz approximation and harmonic approximation without indirect arguments and with estimates which depend only on the dimension n and not on codimension or dimension of the target space

lusternik schnirelmann category and systolic category of low dimensional manifolds we show that the geometry of a riemannian manifold m is sensitive to the apparently purely homotopy theoretic invariant of m known as the lusternik schnirelmann category denoted catls m here we introduce a riemannian analogue of catls m called the systolic category of m it is denoted catsys m and defined in terms of the existence of systolic inequalities satisfied by every metric as initiated by c loewner and later developed by m gromov we compare the two categories in all our examples the inequality catsysm catlsm is satisfied which typically turns out to be an equality e g in dimension 3 we show that a number of existing systolic inequalities can be reinterpreted as special cases of such equality and that both categories are sensitive to massey products the comparison with the value of catls m leads us to prove or conjecture new systolic inequalities on m 2006 wiley periodicals inc

testing hypotheses about sun climate complexity linking we reexamine observational evidence presented in support of the hypothesis of a sun climate complexity linking by n scafetta and b j west phys rev lett 90 248701 2003 which contended that the integrated solar flare index sfi and the global temperature anomaly gta both follow l evy walk statistics with the same waiting time exponent ensuremath mu ensuremath approx 2 1 however their analysis does not account for trends in the signal cannot deal correctly with infinite variance processes l evy flights and suffers from considering only the second moment our analysis shows that properly detrended the integrated sfi is well described as a l evy flight and the integrated gta as a persistent fractional brownian motion these very different stochastic properties of the solar and climate records do not support the hypothesis of a sun climate complexity linking

some fine properties of currents and applications to distributional jacobians we study fine properties of currents in the framework of geometric measure theory on metric spaces developed by ambrosio and kirchheim and we prove a rectifiability criterion for flat currents of finite mass we apply these tools to study the structure of the distributional jacobians of functions in the space bnv defined by jerrard and soner we define the subspace of special functions of bounded higher variation and we prove a closure theorem

levy flights in a steep potential well levy flights in steeper than harmonic potentials have been shown to exhibit finite variance and a critical time at which a bifurcation from an initial monomodal to a terminal bimodal distribution occurs chechkin et al phys rev e 67 010102 r 2003 in this paper we present a detailed study of levy flights in potentials of the type u x x c with c 2 apart from the bifurcation into bimodality we find the interesting result that for c 4 a trimodal transient exists due to the temporal overlap between the decay of the central peak around the initial condition and the building up of the two emerging side peaks which are characteristic for the stationary state thus for certain system parameters there exists a transient trimodal distribution of the levy flight these properties of levy flights in external potentials of the power law type can be represented by certain phase diagrams we also present details about the proof of multimodality and the numerical procedures to establish the probability distribution of the process

levy anomalous diffusion and fractional fokker planck equation we demonstrate that the fokker planck equation can be generalized into a fractional fokker planck equation i e an equation which includes fractional space differentiations in order to encompass the wide class of anomalous diffusions due to a levy stable stochastic forcing a precise determination of this equation is obtained by substituting a levy stable source to the classical gaussian one in the langevin equation this yields not only the anomalous diffusion coefficient but a non trivial fractional operator which corresponds to the possible asymmetry of the levy stable source both of them cannot be obtained by scaling arguments the mono scaling behaviors of the fractional fokker planck equation and of its solutions are analysed and a generalization of the einstein relation for the anomalous diffusion coefficient is obtained this generalization yields a straightforward physical interpretation of the parameters of levy stable distributions furthermore with the help of important examples we show the applicability of the fractional fokker planck equation in physics

anomalous diffusion resulting from strongly asymmetric random walks for leftward and rightward flights pl t t 2m and pr t t 2h with mfih we calculate the dependence of the variance exponent g s 2 t g on the pdf exponents n m and h we find thatg is determined by the two smaller of the three pdf exponents and in some cases by only the smallest a pdf with decay exponent less

boundary holder estimates for nonlinear singular elliptic equations abstract in this paper we study the dirichlet problem of a class of nonlinear elliptic equations which includes monge ampere equation k hessian equation and the usual linear elliptic equation this problem becomes singular near the boundary of the domain by carefully constructing sub solutions we obtain the boundary holder estimate for the convex solution to the problem when the domain satisfies exterior sphere condition

a modica mortola approximation for branched transport and applications the m energy which is usually minimized in branched transport problems among singular one dimensional rectifiable vector measures is approximated by means of a sequence of elliptic energies defined on more regular vector fields the procedure recalls the one of modica mortola related to the approximation of the perimeter in our context the double well potential is replaced by a concave term the paper contains a proof of convergence and numerical simulations of optimal networks based on that previous result

boundary blow up and degenerate equations let r2 be a bounded domain of class c2 0 1 we show that if u is the solution of u 4exp 2u which tends to as x y then the hyperbolic radius v exp u is also of class c2 up to the boundary the proof relies on new schauder estimates for degenerate elliptic equations of fuchsian type

levy flights in confining potentials we analyze confining mechanisms for levy flights when they evolve in suitable external potentials their variance may exist and show signatures of a superdiffusive transport two classes of stochastic jump type processes are considered those driven by langevin equation with levy noise and those named topological levy processes occurring in systems with topological complexity such as folded polymers or complex networks whose langevin representation is unknown and possibly nonexistent our major finding is that both above classes of processes stay in affinity and may share common stationary probability density even if their detailed dynamical behavior look different this near equilibrium observation seems to be generic to a broad class of levy noise driven processes such as e g superdiffusion on folded polymers geophysical flows and even climatic changes

rectifiable and flat g chains in a metric space chains in a complete metric space which have coefficients in a normed abelian group g are studied an m dimensional rectifiable chain is the lipschitz push forward of a region in bbb r m equipped with a measurable g valued density flat chains are obtained by completion using a certain flat norm on polyhedral or lipschitz chain approximations numerous basic results of geometric measure theory for these chains are derived including the rectifiability of finite mass flat chains provided that g contains no nonconstant lipschitz curves the work here generalizes and uses many ideas from the 1999 paper of b white on at g chains in bbb r n and the 2000 paper of l ambrosio and b kirchheim on currents in a metric space the use of flat and rectifiable chains in geometric variational problems or in defining geometric homology theories may reveal geometric properties of spaces

some basic theorems on flat g chains abstract for a complete normed abelian group g we show that a flat g chain respectively normal flat g chain of codimension 0 in a finite dimensional banach space corresponds to a g valued functions that is h m summable respectively of bounded variation we also prove a constancy theorem for an m dimensional flat g chain t in a metric space assuming supp t is contained in the closure of a connected m dimensional lipschitz submanifold m with m supp t and h m m h m clos m

limit theorems for levy walks in d dimensions rare and bulk fluctuations we consider super diffusive levy walks in dimensions when the duration of a single step i e a ballistic motion performed by a walker is governed by a power law tailed distribution of infinite variance and finite mean we demonstrate that the probability density function pdf of the coordinate of the random walker has two different scaling limits at large times one limit describes the bulk of the pdf it is the d dimensional generalization of the one dimensional levy distribution and is the counterpart of the central limit theorem clt for random walks with finite dispersion in contrast with the one dimensional levy distribution and the clt this distribution does not have a universal shape the pdf reflects anisotropy of the single step statistics however large the time is the other scaling limit the so called infinite density describes the tail of the pdf which determines second dispersion and higher moments of the pdf this limit repeats the angular structure of the pdf of velocity in one step a typical realization of the walk consists of anomalous diffusive motion described by anisotropic d dimensional levy distribution interspersed with long ballistic flights described by infinite density the long flights are rare but due to them the coordinate increases so much that their contribution determines the dispersion we illustrate the concept by considering two types of levy walks with isotropic and anisotropic distributions of velocities furthermore we show that for isotropic but otherwise arbitrary velocity distributions the d dimensional process can be reduced to a one dimensional levy walk we briefly discuss the consequences of non universality for the d 1 dimensional fractional diffusion equation in particular the non uniqueness of the fractional laplacian

on the lower semicontinuous envelope of functionals defined on polyhedral chains abstract in this note we prove an explicit formula for the lower semicontinuous envelope of some functionals defined on real polyhedral chains more precisely denoting by h r 0 an even subadditive and lower semicontinuous function with h 0 0 and by h the functional induced by h on polyhedral m chains namely h p i 1 n h i h m i for every p i 1 n i i p m r n we prove that the lower semicontinuous envelope of h coincides on rectifiable m currents with the h mass m h r e h x d h m x for every r e r m r n

quantization and motion law for ginzburg landau vortices we study the vortex trajectories for the two dimensional complex parabolic ginzburg landau equation without a well preparedness assumption we prove that the trajectory set is rectifiable and satisfies a weak motion law in the case of degree 1 vortices the motion law is satisfied in the classical sense moreover dissipation occurs only at a finite number of times away from these times possible collisions and splittings of vortices are constrained by algebraic equations involving their topological degrees

improved partial regularity for manifold constrained minimisers of subquadratic energies we consider minimising p harmonic maps from three dimensional domains to the real projective plane for 1 p 2 these maps arise as least energy configurations in variational models for nematic liquid crystals we show that the singular set of such a map decomposes into a 1 dimensional set which can be physically interpreted as a non orientable line defect and a locally finite set i e a collection of point defects

global smoothness for a singular monge ampere equation abstract the global c n 2 regularity for the graph of solutions to a singular monge ampere equation was established in 7 this equation arises in the study of hyperbolic affine spheres and is singular at the boundary a compatibility condition for higher regularity was found in 7 in this paper we verify the compatibility condition and obtain the global c regularity in the even dimensional case

optimal boundary regularity for nonlinear singular elliptic equations abstract in this paper we prove the optimal boundary regularity under natural structural conditions for a large class of nonlinear elliptic equations with singular terms near the boundary by a careful construction of super and sub solutions we obtain precise growth estimates for solutions at the boundary and reduce the boundary regularity to the interior one by a rescaling argument

existence and regularity of the solutions of some singular monge ampere equations abstract in this paper we investigate the following singular monge ampere equation 0 1 det d 2 u 1 h u n k 2 u k in r n u 0 on where k 0 h 0 are constants and u x u x u x is the legendre transformation of u equation 0 1 is related to proper affine hyperspheres we will show the existence of solutions of 0 1 u c c via regularization method using the technique in 10 12 we also obtain the optimal graph regularity of the solution of 0 1

approximation of rectifiable 1 currents and weak relaxation of the h mass abstract based on smirnov s decomposition theorem we prove that every rectifiable 1 current t with finite mass m t and finite mass m t of its boundary t can be approximated in mass by a sequence of rectifiable 1 currents t n with polyhedral boundary t n and m t n no larger than m t using this result we can compute the relaxation of the h mass for polyhedral 1 currents with respect to the joint weak convergence of currents and their boundaries we obtain that this relaxation coincides with the usual h mass for normal currents this shows that the concepts of so called generalized branched transport and the h mass are equivalent

a generalized rotationally symmetric case of the centroaffine minkowski problem abstract in this paper the centroaffine minkowski problem a critical case of the l p minkowski problem in the n 1 dimensional euclidean space is studied by its variational structure and the method of blow up analyses we obtain two sufficient conditions for the existence of solutions for a generalized rotationally symmetric case of the problem

gradient bounds for elliptic problems singular at the boundary let be a bounded smooth domain in r n n geqq 2 and let us denote by d x the distance function d x we study a class of singular hamilton jacobi equations arising from stochastic control problems whose simplest model is alpha delta u u frac nabla u cdot b x d x c x nabla u 2 f x quad rm in omega where f belongs to w 1 infty rm loc omega and is possibly singular at partial omega c in w 1 infty omega with no sign condition and the field b in w 1 infty omega n has an outward direction and satisfies b cdot nu geqq alpha at is the outward normal despite the singularity in the equation we prove gradient bounds up to the boundary and the existence of a globally lipschitz solution we show that in some cases this is the unique bounded solution we also discuss the stability of such estimates with respect to as vanishes obtaining lipschitz solutions for first order problems with similar features the main tool is a refined weighted version of the classical bernstein method to get gradient bounds the key role is played here by the orthogonal transport component of the hamiltonian

rectifiability of flat chains we prove without using federer s structure theorem that a finite mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0 dimensional slices are rectifiable this implies that every flat chain of finite mass and finite size is rectifiable it also leads to a simple necessary and sufficient condition on the coefficient group in order for every finite mass flat

bifurcation bimodality and finite variance in confined levy flights we investigate the statistical behavior of levy flights confined in a symmetric quartic potential well u x proportional variant x 4 at stationarity the probability density function features a distinct bimodal shape and decays with power law tails which are steep enough to give rise to a finite variance in contrast to free levy flights from a delta initial condition a bifurcation of the unimodal state is observed at t c 0 the nonlinear oscillator with potential u x ax 2 2 bx 4 4 a b 0 shows a crossover from unimodal to bimodal behavior at stationarity depending on the ratio a b

interior a priori estimates for solutions of fully non linear equations on etend une theorie de perturbations aux solutions d equations uniformement elliptiques d ordre 2 totalement non lineaires

multidimensional levy walk and its scaling limits in this paper we obtain the scaling limit of a multidimensional levy walk and describe the detailed structure of the limiting process the scaling limit is a subordinated stable levy motion with the parent process and subordinator being strongly dependent processes the corresponding langevin picture is derived we also introduce a useful method of simulating levy walks with a predefined spectral measure which controls the direction of each jump our approach can be applied in the analysis of real life data we are able to recover the spectral measure from the data and obtain the full characterization of a levy walk we also give examples of some useful spectral measures which cover a large class of possible scenarios in the modeling of real life phenomena

fractional fokker planck equation for levy flights in nonhomogeneous environments the fractional fokker planck equation which contains a variable diffusion coefficient is discussed and solved it corresponds to the l evy flights in a nonhomogeneous medium for the case with the linear drift the solution is stationary in the long time limit and it represents the l evy process with a simple scaling the solution for the drift term in the form ensuremath lambda text text sgn x possesses two different scales which correspond to the l evy indexes ensuremath mu and ensuremath mu 1 ensuremath mu l1 the former component of the solution prevails at large distances but it diminishes with time for a given x the fractional moments as a function of time are calculated they rise with time and the rate of this growth increases with ensuremath lambda

stationary states in two dimensional systems driven by bivariate levy noises systems driven by ensuremath alpha stable noises could be very different from their gaussian counterparts stationary states in single well potentials can be multimodal moreover a potential well needs to be steep enough in order to produce stationary states here it is demonstrated that two dimensional 2d systems driven by bivariate ensuremath alpha stable noises are even more surprising than their 1d analogs in 2d systems intriguing properties of stationary states originate not only due to heavy tails of noise pulses which are distributed according to ensuremath alpha stable densities but also because of properties of spectral measures consequently 2d systems are described by a whole family of langevin and fractional diffusion equations solutions of these equations bear some common properties but also can be very different it is demonstrated that also for 2d systems potential wells need to be steep enough in order to produce bounded states moreover stationary states can have local minima at the origin the shape of stationary states reflects symmetries of the underlying noise i e its spectral measure finally marginal densities in power law potentials also have power law asymptotics

on sequences of maps into a manifold with equibounded w1 2 energies abstract we characterize weak limits of sequences of smooth functions from b n into suitable manifolds y with equibounded w 1 2 energies the relaxed w 1 2 energy and we prove strong density of smooth maps we then obtain the weak sequential density of smooth maps in w 1 2 b n y and a criterion for strong density of smooth maps

nonlinear stochastic equations with multiplicative levy noise the langevin equation with a multiplicative levy white noise is solved the noise amplitude and the drift coefficient have a power law form a validity of ordinary rules of the calculus for the stratonovich interpretation is discussed the solution has the algebraic asymptotic form and the variance may assume a finite value for the case of the stratonovich interpretation the problem of escaping from a potential well is analyzed numerically predictions of different interpretations of the stochastic integral are compared

gamma limit of the nonself dual chern simons higgs energy abstract we consider the gamma limit of the abelian chern simons higgs energy g csh 1 2 u a e u e 2 2 4 curl a e h ex 2 u e 2 1 e 2 u e 2 1 u e 2 2 d x on a bounded simply connected two dimensional domain under the e 0 limit as a first step we study the gamma limit of e csh 1 2 u u e 2 1 e 2 u e 2 1 u e 2 2 d x under two different scalings e csh log e and e csh log e 2 we apply the log e 2 scaling result to the full chern simons higgs energy g csh and as a consequence we are able to compute the first critical field h 1 h 1 u for the nucleation of a vortex the method entails estimating in certain weak topologies the jacobian j u e det u e in terms of the chern simons higgs energy e csh

rectifiability of flat chains in banach spaces with coefficients in zp in this paper we study the behaviour of a class of currents with coefficients in zp in metric spaces obtained by taking a suitable quotient of currents with coefficients in z we prove under fairly general assumption on the ambient space the rectifiability of the so called slice mass and under more restrictive assumptions the rectifiability of the mass measure

gradient estimates for singular fully nonlinear elliptic equations abstract we study a class of singular fully nonlinear elliptic equations under suitable natural conditions with the model equation being u b d u d u c d u 2 f for lipschitz functions b 0 c and f where d denotes distance to the boundary of the domain and is a suitable extension of the interior unit normal we show that such equations have a unique globally c 1 solution without any a priori prescription of boundary data