uniform boundary harnack principle and generalized triangle property let d be a bounded open subset in rd d 2 and let g denote the green function for d with respect to 2 0 2 d if 2 assume that d satisfies the interior corkscrew condition if 2 i e if g is the classical green function on d assume more restrictively that d is a uniform domain let g g y0 1 for some y0 d based on the uniform boundary harnack principle it is shown that g has the generalized triangle property which states that g z y g z cg x y g x when d z x d z y an intermediate step is the approximation g x y x y dg x g y g a 2 where a is an arbitrary point in a certain set b x y this is discussed in a general setting where d is a dense open subset of a compact metric space satisfying the interior corkscrew condition and g is a quasi symmetric positive numerical function on d d which has locally polynomial decay and satisfies harnack s inequality under these assumptions the uniform boundary harnack principle the approximation for g and the generalized triangle property turn out to be equivalent

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

gevrey semigroup generated by λα b in lp rn abstract for every p belongs to 1 and n dimensional vector b it is shown that b generates an analytic semigroup in l p r n if 1 2 and a gevrey semigroup of order for any 1 in l p r n if 0 1 moreover it is proved that the semigroup is not a gevrey semigroup of order with 1 if 0 1 this shows that our result is sharp

theories of causal ordering abstract this paper is a response to iwasaki and simon 14 which criticizes de kleer and brown 8 we argue that many of their criticisms particularly concerning causality modeling and stability originate from the difference of concerns between engineering and economics our notion of causality arises from considering the interconnections of components not equations when no feedback is present the ordering produced by our qualitative physics is similar to theirs however when feedback is present our qualitative physics determines a causal ordering around feedback loops as well causal ordering is a general technique not only applicable to qualitative reasoning therefore we also explore the relationship between causal ordering and propagation of constraints upon which the methods of qualitative physics are based

fractal first order partial differential equations the present paper is concerned with semi linear partial differential equations involving a particular pseudo differential operator it investigates both fractal conservation laws and non local hamilton jacobi equations the idea is to combine an integral representation of the operator and duhamel s formula to prove on the one hand the key a priori estimates for the scalar conservation law and the hamilton jacobi equation and on the other hand the smoothing effect of the operator as far as hamilton jacobi equations are concerned a non local vanishing viscosity method is used to construct a viscosity solution when existence of regular solutions fails and a rate of convergence is provided turning to conservation laws global in time existence and uniqueness are established we also show that our formula allows us to obtain entropy inequalities for the non local conservation law and thus to prove the convergence of the solution as the non local term vanishes toward the entropy solution of the pure conservation law

berkovich spaces and tubular descent we consider an algebraic variety xx together with the choice of a subvariety zz we show that any coherent sheaf on xx can be constructed out of a coherent sheaf on the formal neighborhood of zz a coherent sheaf on the complement of zz and an isomorphism between certain representative images of these two sheaves in the category of coherent sheaves on a berkovich analytic space ww which we define

estimates and structure of α harmonic functions we prove a uniform boundary harnack inequality for nonnegative harmonic functions of the fractional laplacian on arbitrary open set d this yields a unique representation of such functions as integrals against measures on dc satisfying an integrability condition the corresponding martin boundary of d is a subset of the euclidean boundary determined by an integral test

the brezis nirenberg problem for the laplacian with a singular drift in rn and sn abstract we consider the brezis nirenberg problem for the laplacian with a singular drift for a geodesic ball in both r n and s n 3 n 5 the singular drift we consider derives from a potential which is symmetric around the center of the geodesic ball here the potential is given by a parameter say times the logarithm of the distance to the center of the ball in both cases we determine the exact region in the parameter space for which positive smooth solutions of this problem exist and the exact region for which there are no solutions the parameter space is characterized by the geodesic radius of the ball and the coupling constant of the linear term of the brezis nirenberg problem

parabolic littlewood paley inequality for ϕ δ type operators and applications to stochastic integro differential equations abstract in this paper we prove a parabolic version of the littlewood paley inequality 1 4 for the operators of the type where is a bernstein function as an application we construct an l p theory for the stochastic integro differential equations of the type d u u f d t g d w t

probabilistic approach to the dirichlet problem of perturbed stable processes letx t be a symmetric stable process of index 0 2 inr d d 2 in this paper we deal with the perturbation ofx t by a multiplicative functional of the following form m t exp left sum limits s mathop leqslant limits t f x s x s right withf being a function onr d r d satisfying certain conditions first we prove the following gauge theorem ifd is a bounded open domain ofr d then the functiong x e x m d is either identically infinite ond or bounded ond where d is the first exit time fromd then we formulate the dirichlet problem associated with the perturbed symmetric stable process by using dirichlet form theory finally we apply the gauge theorem to prove the existence and uniqueness of solutions to the dirichlet problem mentioned above

relativistic schrodinger operators asymptotic behavior of the eigenfunctions nonrelativistic schrodinger operators are perturbations of the negative laplacian and the connection with stochastic processes and brownian motion in particular is well known and usually goes under the name of feynman and kac we present a similar connection between a class of relativistic schrodinger operators and a class of processes with stationary independent increments in particular we investigate the decay of the eigenfunctions of these operators and we show that not only exponential decay but also polynomial decay can occur

heat kernels and analyticity of non symmetric jump diffusion semigroups let d geqslant 1 and alpha in 0 2 consider the following non local and non symmetric levy type operator on mathbb r d begin aligned fancyscript l kappa alpha f x hbox p v int mathbb r d f x z f x frac kappa x z z d alpha mathord mathrm d z end aligned where 0 kappa 0 leqslant kappa x z leqslant kappa 1 kappa x z kappa x z and kappa x z kappa y z leqslant kappa 2 x y beta for some beta in 0 1 using levi s method we construct the fundamental solution also called heat kernel p kappa alpha t x y of fancyscript l kappa alpha and establish its sharp two sided estimates as well as its fractional derivative and gradient estimates we also show that p kappa alpha t x y is jointly holder continuous in t x the lower bound heat kernel estimate is obtained by using a probabilistic argument the fundamental solution of fancyscript l kappa alpha gives rise a feller process x mathbb p x x in mathbb r d on mathbb r d we determine the levy system of x and show that mathbb p x solves the martingale problem for fancyscript l kappa alpha c 2 b mathbb r d furthermore we show that the c 0 semigroup associated with fancyscript l kappa alpha is analytic in l p mathbb r d for every p in 1 infty a maximum principle for solutions of the parabolic equation partial t u fancyscript l kappa alpha u is also established as an application of the main result of this paper sharp two sided estimates for the transition density of the solution of mathord mathrm d x t a x t mathord mathrm d y t is derived where y is a rotationally symmetric stable process on mathbb r d and a x is a holder continuous d times d matrix valued function on mathbb r d that is uniformly elliptic and bounded

sharp estimates for the green function in lipschitz domains we give sharp global estimates for the green function and martin kernel in lipschitz domains we apply the estimates to obtain short proofs of some classical results

stable estimates for source solution of critical fractal burgers equation abstract in this paper we provide two sided estimates for the source solution of d dimensional critical fractal burgers equation u t 2 b u u q 0 q 1 d 1 2 b r d by the density function of the isotropic stable process

schauder estimates for drifted fractional operators in the supercritical case abstract we consider a non local operator l which is the sum of a fractional laplacian 2 0 1 plus a first order term which is measurable in the time variable and locally holder continuous in the space variables importantly the fractional laplacian 2 does not dominate the first order term we show that global parabolic schauder estimates hold even in this case under the natural condition 1 thus the constant appearing in the schauder estimates is in fact independent of the l norm of the first order term in our approach we do not use the so called extension property and we can replace 2 with other operators of stable type which are somehow close including the relativistic stable operator moreover when 1 2 1 we can prove schauder estimates for more general stable type operators like the singular cylindrical one i e when 2 is replaced by a sum of one dimensional fractional laplacians k 1 d x k x k 2 2

vanishing cycles for formal schemes let k be a non archimedean field and let x be a formal scheme locally finitely presented over the ring of integers k see 1 in this work we construct and study the vanishing cycles functor from the category of etale sheaves on the generic fibre x of x which is a k analytic space to the category of etale sheaves on the closed fibre xs of x which is a scheme over the residue field of k we prove that if x is the formal completion x of a scheme x finitely presented over k along the closed fibre then the vanishing cycles sheaves of x are canonically isomorphic to those of x as defined in sga7 exp xiii in particular the vanishing cycles sheaves of x depend only on x and any morphism x induces a homomorphism from the pullback of the vanishing cycles sheaves of x under s ys xs to those of y furthermore we prove that for each x one can find a nontrivial ideal of k such that if two morphisms x coincide modulo this ideal then the homomorphisms between the vanishing cycles sheaves induced by and coincide these facts were conjectured by p deligne in 1 we associate with a formal scheme x locally finitely presented over k a k analytic space x in the sense of ber1 and ber2 in 2 we find that the morphism y x which is induced by an etale morphism of formal schemes y x possesses a certain property morphisms of k analytic spaces with this property are called quasi etale and they give rise to a quasi etale site xqet of a k analytic space x there is a canonical morphism of sites xqet xet where xet is the etale site introduced in ber2 we show that the inverse image functor identifies the category of etale sheaves x et with a full subcategory of

global qualitative description of a class of nonlinear dynamical systems in this paper we propose a methodology to derive a qualitative description of the behavior of a system from an incompletely known nonlinear dynamical model the model is written as an algebraic structure with unknown parameters and or functions under some hypotheses we obtain a graph describing the possible transitions between regions defined by the trends of the state variables and their relative positions a qualitative simulation of the model can be compared with on line data for fault detection purpose we give the example of a nonlinear biological model in dimension three for the growth of cells in a bioreactor

upper bounds for fundamental solutions to non local diffusion equations with divergence free drift we consider some non local diffusion equations in the presence of a divergence free drift term where the diffusion operators are related to certain dirichlet forms of jump type we derive pointwise upper bounds for fundamental solutions of the equations under weak assumptions for the velocity of the drift term our class of the velocity includes functions with the scale critical regularity and some growing functions at spatial infinity

dirichlet heat kernel estimates for fractional laplacian with gradient perturbation suppose that d 2 and 1 2 let d be a bounded c1 1 open set in rd and b an rd valued function on rd whose components are in a certain kato class of the rotationally symmetric stable process in this paper we derive sharp two sided heat kernel estimates for lb 2 b in d with zero exterior condition we also obtain the boundary harnack principle for lb in d with explicit decay rate

pointwise estimates for solutions of fractal burgers equation abstract in this paper we provide two sided estimates and uniform asymptotics for the solution of d dimensional critical fractal burgers equation u t 2 u b u u q 0 1 2 b r d for q 1 d and u 0 l 1 r d we consider also q 1 d under additional condition u 0 l r d in both cases we assume u 0 0 which implies that the solution is non negative the estimates are given in the terms of the function p t u 0 where p t denotes the semigroup for the operator t 2

asymptotic estimates of multi dimensional stable densities and their applications the relation between the upper and lower asymptotic estimates of the density and the fractal dimensions on the sphere of the spectral measure for a multivariate stable distribution is discussed in particular the problem and the conjecture on the asymptotic estimates of multivariate stable densities in the work of pruitt and taylor in 1969 are solved the proper asymptotic orders of the stable densities in the case where the spectral measure is absolutely continuous on the sphere or discrete with the support being a finite set or a mixture of such cases are obtained those results are applied to the moment of the last exit time from a ball and the spitzer type limit theorem involving capacity for a multi dimensional transient stable process

comparison and regularity results for the fractional laplacian via symmetrization methods abstract in this paper we establish a comparison result through symmetrization for solutions to some boundary value problems involving the fractional laplacian this allows to get sharp estimates for the solutions obtained by comparing them with solutions of suitable radial problems furthermore we use such result to prove a priori estimates for solutions in terms of the data providing several regularity results which extend the well known ones for the classical laplacian

on harnack inequality for α stable ornstein uhlenbeck processes we consider the stable ornstein uhlenbeck process in mathbb r d with the generator l delta alpha 2 lambda x cdot nabla x we show that if 2 1 or 1 d the harnack inequality holds for 1 d we construct a counterexample that shows that the harnack inequality does not hold

estimates of heat kernel of fractional laplacian perturbed by gradient operators we construct a continuous transition density of the semigroup generated by delta alpha 2 b x cdot nabla for 1 alpha 2 d ge 1 and b in the kato class mathcal k d alpha 1 on mathbb r d for small time the transition density is comparable with that of the fractional laplacian

terminal valuations and the nash problem let x be an algebraic variety of characteristic zero terminal valuations are defined in the sense of the minimal model program as those valuations given by the exceptional divisors on a minimal model over x we prove that every terminal valuation over x is in the image of the nash map and thus it corresponds to a maximal family of arcs through the singular locus of x in dimension two this result gives a new proof of the theorem of fernandez de bobadilla and pe pereira stating that for surfaces the nash map is a bijection

existence of solutions for an inhomogeneous fractional semilinear heat equation abstract we obtain necessary conditions and sufficient conditions on the existence of solutions to the cauchy problem for a fractional semilinear heat equation with an inhomogeneous term we identify the strongest spatial singularity of the inhomogeneous term for the solvability of the cauchy problem

heat kernel estimates for the fractional laplacian with dirichlet conditions we give sharp estimates for the heat kernel of the fractional laplacian with dirichlet condition for a general class of domains including lipschitz domains

estimates of transition densities and their derivatives for jump levy processes abstract we give upper and lower estimates of densities of convolution semigroups of probability measures under explicit assumptions on the corresponding levy measure and the levy khinchin exponent we obtain also estimates of derivatives of densities

stability of dirichlet heat kernel estimates for non local operators under feynman kac perturbation in this paper we show that dirichlet heat kernel estimates for a class of not necessarily symmetric markov processes are stable under non local feynman kac perturbations this class of processes includes among others reflected symmetric stable like processes in closed d sets in r d killed symmetric stable processes censored stable processes in c 1 1 open sets as well as stable processes with drifts in bounded c 1 1 open sets these two sided estimates are explicit involving distance functions to the boundary

bmo and morrey campanato estimates for stochastic convolutions and schauder estimates for stochastic parabolic equations abstract in this paper we are aiming to prove several regularity results for the following stochastic fractional heat equations with additive noises d u t x 2 u t x d t g t x d t u 0 0 t 0 t x g for a random field u t x 0 t g u t x u t x r where 2 2 0 2 is the fractional laplacian t 0 is arbitrarily fixed g r d is a bounded domain g 0 t g r is a joint measurable coefficient and t t 0 is either a brownian motion or a levy process on a given filtered probability space f p f t t 0 t to this end we derive the bmo estimates and morrey campanato estimates respectively for stochastic singular integral operators arising from the equations concerned then by utilizing the embedding theory between the campanato space and the holder space we establish the controllability of the norm of the space c 2 d where 0 d 0 t g with all these in hand we are able to show that the q th order bmo quasi norm of the q 0 order derivative of the solution u is controlled by the norm of g under the condition that t is a levy process finally we derive the schauder estimate for the p moments of the solution of the above stochastic fractional heat equations driven by levy noise

heat kernels for non symmetric diffusion operators with jumps abstract for d 2 we prove the existence and uniqueness of heat kernels to the following time dependent second order diffusion operator with jumps l t 1 2 i j 1 d a i j t x i j 2 i 1 d b i t x i l t where a a i j is a uniformly bounded elliptic and holder continuous matrix valued function b belongs to some suitable kato s class and l t is a non local stable type operator with bounded kernel moreover we establish sharp two sided estimates gradient estimate and fractional derivative estimate for the heat kernel under some mild conditions

estimates of gradient perturbation series abstract we give upper and lower bounds of perturbation series for transition densities corresponding to additive gradient perturbations satisfying certain space time integrability conditions

a new proof of the gerritzen grauert theorem the gerritzen grauert theorem gg bgr 7 3 5 1 is one of the most important foundational results of rigid analytic geometry it describes so called locally closed immersions between affinoid varieties and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains in its turn the latter fact allowed one to extend tate s theorem see tate bgr 8 2 1 1 on acyclicity of the cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains the same fact also plays an important role in foundations of non archimedean analytic geometry developed by v berkovich in ber1 and ber2 recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry the latter were called in ber1 strictly affinoid and besides the valuation on the ground field is not assumed to be nontrivial in the recent papers by a ducros duc 2 4 and the author tem 3 5 the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces but its proof was based on the case of strictly affinoid ones i e affinoid varieties the original proof of the gerritzen grauert theorem is not easy and since then the only different proof was found by m raynaud in the framework of his approach to rigid analytic geometry see ray bl although that proof is more conceptual it is based on a complicated algebraic technics the purpose of this paper is to give a new proof of the gerritzen grauert theorem which uses basic properties of affinoid algebras in a standard way the only novelty is in using the whole spectrum m a of an affinoid algebra a introduced in ber1 instead of the maximal spectrum max a considered in rigid analytic geometry the use of the whole spectrum allows one to apply additional but standard compactness arguments in 1 2 we work in the setting of rigid analytic geometry i e the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered in 1 we recall basic definitions of an affinoid algebra an affinoid domain all notions necessary for the formulation of the gerritzen grauert theorem and formulate it theorem 1 1 the only new fact is proposition 1 2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

on fundamental solutions for non local parabolic equations with divergence free drift abstract we are concerned with non local parabolic equations in the presence of a divergence free drift term by using the classical nash approach we show the existence of fundamental solutions together with continuity estimates under weak regularity assumptions on the kernel of the non local term and the velocity of the drift term as an application we give an alternative proof of global regularity for the two dimensional dissipative quasi geostrophic equations in the critical case

large global in time solutions to a nonlocal model of chemotaxis abstract we consider the parabolic elliptic model for the chemotaxis with fractional anomalous diffusion global in time solutions are constructed under nearly optimal assumptions on the size of radial initial data moreover criteria for blowup of radial solutions in terms of suitable morrey spaces norms are derived

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

fractional semi linear parabolic equations with unbounded data this paper is devoted to the study of semi linear parabolic equations whose principal term is fractional i e is integral and eventually singular a typical example is the fractional laplace operator this work sheds light on the fact that if the initial datum is not bounded assumptions on the non linearity are closely related to its behavior at infinity the sub linear and super linear cases are first treated by classical techniques we next present a third original case if the associated first order hamilton jacobi equation is such that perturbations propagate at finite speed then the semi linear parabolic equation somehow keeps memory of this property by using such a result locally bounded initial data that are merely integrable at infinity can be handled next regularity of the solution is proved eventually strong convergence of gradients as the fractional term disappears is proved for strictly convex non linearity

a dynamic systems perspective on qualitative simulation abstract this paper presents a simple geometric interpretation of qualitative simulation qs based on the phase space representation of dynamic systems theory qs consists of two steps transition analysis determines the sequence of qualitative states that a system traverses and global interpretation derives its long term behavior i recast transition analysis as a search problem in phase space and replace the assorted transition rules with two algebraic conditions the first condition determines transitions between arbitrarily shaped regions in phase space as opposed to qs which only handles n dimensional rectangles it also provides more accurate results by considering only the boundaries between regions the second condition determines whether nearby trajectories approach a fixed point asymptotically it obtains better results than qs by exploiting local stability properties i recast global interpretation of dissipative systems as a search for attractors in phase space and present a global interpretation strategy for the subset of these systems whose local behavior determines global behavior uniquely although limited in scope this strategy handles many systems that appear in the qs literature

a trace formula for rigid varieties and motivic weil generating series for formal schemes we establish a trace formula for rigid varieties x over a complete discretely valued field which relates the set of unramified points on x to the galois action on its etale cohomology next we show that the analytic milnor fiber of a morphism f at a point x completely determines the formal germ of f at x we develop a theory of motivic integration for formal schemes of pseudo finite type over a complete discrete valuation ring r and we introduce the weil generating series of a regular formal r scheme mathfrak x of pseudo finite type via the construction of a gelfand leray form on its generic fiber when mathfrak x is the formal completion of a morphism f from a smooth irreducible variety to the affine line then its weil generating series coincides modulo normalization with the motivic zeta function of f when mathfrak x is the formal completion of f at a closed point x of the special fiber f 1 0 we obtain the local motivic zeta function of f at x

causality in device behavior abstract this paper shows how formal characterizations of causality and of the method of comparative statics long used in econometrics thermodynamics and other domains can be applied to clarify and make rigorous the qualitative causal calculus recently proposed by de kleer and brown 2 the formalization shows exactly what assumptions are required to carry out causal analysis of a system of interdependent variables in equilibrium and to propagate disturbances through such a system the intuitive concepts of causality captured by de kleer and brown provide a rough approximation to the standard analytic techniques that are used in the treatment of simultaneous algebraic and differential equations

global comparison of perturbed green functions given a green function g e g with respect to 2 0 2 on a region x where g has a generalized triangle property and given a g bounded signed measure on x necessary and sufficient conditions are given for the existence of a perturbed green function which is comparable with g this is done in the general setting of measurable spaces applications to c1 1 regions in rd and for the classical case 2 to finitely connected regions in 2 are given

generalized 3g theorem and application to relativistic stable process on non smooth open sets let g x y and gd x y be the green functions of rotationally invariant symmetric stable process in rd and in an open set d respectively where 0 2 the inequality gd x y gd y z gd x z c g x y g y z is a very useful tool in studying local schrodinger operators when the above inequality is true with c c d 0 then we say that the 3g theorem holds in d in this paper we establish a generalized version of 3g theorem when d is a bounded fat open set which includes a bounded john domain the 3g we consider is of the form gd x y gd z w gd x w where y may be different from z when y z we recover the usual 3g the 3g form gd x y gd z w gd x w appears in non local schrodinger operator theory using our generalized 3g theorem we give a concrete class of functions belonging to the non local kato class introduced by chen and song on fat open sets as an application we discuss relativistic stable processes relativistic hamiltonian when 1 in fat open sets we identify the martin boundary and the minimal martin boundary with the euclidean boundary for relativistic stable processes in fat open sets furthermore we show that relative fatou type theorem is true for relativistic stable processes in fat open sets the main results of this paper hold for a large class of symmetric markov processes as are illustrated in the last section of this paper we also discuss the generalized 3g theorem for a large class of symmetric stable levy processes

automatic analysis of one parameter planar ordinary differential equations by intelligent numeric simulation abstract this paper describes research on automating the analysis of physical systems modeled by ordinary differential equations it describes a program called poincare that analyzes one parameter autonomous planar systems at the level of experts through a combination of theoretical dynamics numerical simulation and geometric reasoning the input is the system a bounding box for the system state a bounding interval for the parameter and error tolerances poincare partitions the parameter interval into open subintervals of equivalent behavior bounded by bifurcation points classifies the bifurcation points and constructs representative phase diagrams for the subintervals it detects local and global generic one parameter bifurcations it constructs the phase diagrams by identifying fixed points saddle manifolds and limit cycles and partitioning the remaining trajectories into open regions of uniform asymptotic behavior it obtains the region boundaries by numeric simulation guided by theoretical knowledge it determines the behavior near fixed points from the jacobian of the system scans outward to find basins of attractors and limit cycles and stops at the bounding box

symmetric stable laws and stable like jump diffusions asymptotic expansions are obtained for finite dimensional symmetric stable distributions they are used to prove the existence of continuous transition probability densities of stable and stable like jump diffusions and to construct local multiplicative asymptotics and global two sided estimates for these densities as a consequence the distribution of the first passage times for stable jump diffusions is estimated and the integral test for the limsup behaviour of their sample paths as t to 0 is provided 1991 mathematics subject classification 60e07 60g17 60j35 47d07

automatic qualitative analysis of dynamic systems using piecewise linear approximations abstract this paper explores automating the qualitative analysis of physical systems it describes a program called plr for piecewise linear reasoner that formalizes an analysis strategy employed by experts plr takes parameterized ordinary differential equations as input and produces a qualitative description of the solutions for all initial values it approximates intractable nonlinear systems with piecewise linear ones analyzes the approximations and draws conclusions about the original systems it chooses approximations that are accurate enough to reproduce the essential properties of their nonlinear prototypes yet simple enough to be analyzed completely and efficiently it derives additional properties such as boundedness or periodicity by theoretical methods plr s analysis depends on abstract properties of systems rather than on specific numeric values this makes its conclusions more robust and enables it to handle parameterized equations transparently i demonstrate plr on several common nonlinear systems and on published examples from mechanical engineering

a widder s type theorem for the heat equation with nonlocal diffusion the main goal of this work is to prove that every non negative strong solutionu x t to the problem u t delta alpha 2 u 0 rm for x t in mathbb r n times 0 t 0 alpha 2 can be written as u x t int mathbb r n p t x y u y 0 dy where p t x frac 1 t n alpha p left frac x t 1 alpha right and p x int mathbb r n e i x cdot xi xi alpha d xi this result shows uniqueness in the setting of non negative solutions and extends some classical results for the heat equation by widder in 15 to the nonlocal diffusion framework

the nash problem for surfaces we prove that nash mapping is bijective for any surface dened over an algebraically closed eld of characteristic 0

existence and regularity of a linear nonlocal fokker planck equation with growing drift abstract the nonlocal fokker planck equations for a class of stochastic differential equations with non gaussian stable levy motion in euclidean space are studied the existence and uniqueness of weak solution are obtained with rough drift the solution is shown to be smooth on spatial variable if all derivatives of the drift are bounded moreover the solution is jointly smooth on spatial and time variable if we assume further that the drift grows like a power of logarithm function at infinity

parabolic littlewood paley inequality for a class of time dependent pseudo differential operators of arbitrary order and applications to high order stochastic pde in this paper we prove a parabolic version of the littlewood paley inequality for a class of time dependent local and non local operators of arbitrary order and as an application we show that this inequality gives a fundamental estimate for an lp theory of high order stochastic partial differential equations

topology of nonarchimedean analytic spaces and relations to complex algebraic geometry this note surveys basic topological properties of nonarchimedean analytic spaces in the sense of berkovich including the recent tameness results of hrushovski and loeser we also discuss interactions between the topology of nonarchimedean analytic spaces and classical algebraic geometry

conditional gaugeability and subcriticality of generalized schrodinger operators abstract we obtain a necessary and sufficient condition for conditional gaugeability and show the equivalence between conditional gaugeability and subcriticality of generalized schrodinger type operators we apply the condition to concrete examples

fractal hamilton jacobi kpz equations nonlinear and nonlocal evolution equations of the form u t lu u q where c is a pseudodifferential operator representing the infinitesimal generator of a levy stochastic process have been derived as models for growing interfaces in the case when the continuous brownian diffusion surface transport is augmented by a random hopping mechanism the goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the diffusive linear and hyperbolic nonlinear terms posed in the whole space r n and supplemented with nonnegative bounded and sufficiently regular initial conditions

toute forme moderement ramifiee d un polydisque ouvert est triviale let k be a complete non archimedean field and let x be a k analytic space assume that there exists a finite tamely ramified extension l of k such that x l is isomorphic to an open polydisc over l we prove that x is itself isomorphic to an open polydisc over k the proof consists in using the graded reduction a notion which is due to temkin of the algebra of functions on x together with some graded counterparts of classical commutative algebra results nakayama s lemma going up theorem basic notions about etale algebras etc

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

holder continuous of the solutions to stochastic nonlocal heat equations abstract in this paper we aim to obtain the holder continuous of solutions to stochastic nonlocal heat equations by using campanato estimates and sobolev embedding theorem we first prove the holder continuous of the mild solution of stochastic nonlocal diffusion equations locally in r d in the sense that the solution u belongs to the space c d t l p then by using tail estimates we obtain the estimates of the mild solution in l p c d t lastly we prove the holder continuous of the mild solution on bounded domain

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

non existence of positive solutions to nonlocal lane emden equations abstract this paper is concerned with non existence of positive solutions to nonlocal lane emden equations firstly we establish the fujita index for nonlocal reaction diffusion equations which is new in the blowup theory then the non existence of positive solutions to nonlocal lane emden equations is obtained because they are the limit stationary equations to the corresponding nonlocal reaction diffusion equations

optimal existence and uniqueness theory for the fractional heat equation abstract we construct a theory of existence uniqueness and regularity of solutions for the fractional heat equation t u s u 0 0 s 1 posed in the whole space r n with data in a class of locally bounded radon measures that are allowed to grow at infinity with an optimal growth rate we consider a class of nonnegative weak solutions and prove that there is an equivalence between nonnegative data and solutions which is given in one direction by the representation formula in the other one by the initial trace we review many of the typical properties of the solutions in particular we prove optimal pointwise estimates and new harnack inequalities

a generalization of the littlewood paley inequality for the fractional laplacian δ α 2 abstract we prove a parabolic version of the littlewood paley inequality for the fractional laplacian 2 where 0 2