density and tails of unimodal convolution semigroups abstract we give sharp bounds for the isotropic unimodal probability convolution semigroups when their levy khintchine exponent has matuszewska indices strictly between 0 and 2

weighted poincare inequality and heat kernel estimates for finite range jump processes it is well known that there is a deep interplay between analysis and probability theory for example for a markovian infinitesimal generator mathcal l the transition density function p t x y of the markov process associated with mathcal l if it exists is the fundamental solution or heat kernel of mathcal l a fundamental problem in analysis and in probability theory is to obtain sharp estimates of p t x y in this paper we consider a class of non local integro differential operators mathcal l on mathbb r d of the form mathcal l u x lim limits varepsilon downarrow 0 int limits y in mathbb r d y x varepsilon u y u x j x y dy where displaystyle j x y frac c x y x y d alpha bf 1 x y leq kappa for some constant kappa 0 and a measurable symmetric function c x y that is bounded between two positive constants associated with such a non local operator mathcal l is an mathbb r d valued symmetric jump process of finite range with jumping kernel j x y we establish sharp two sided heat kernel estimate and derive parabolic harnack principle for them along the way some new heat kernel estimates are obtained for more general finite range jump processes that were studied in barlow et al in trans am math soc 2008 one of our key tools is a new form of weighted poincare inequality of fractional order which corresponds to the one established by jerison in duke math j 53 2 503 523 1986 for differential operators using meyer s construction of adding new jumps we also obtain various a priori estimates such as holder continuity estimates for parabolic functions of jump processes not necessarily of finite range where only a very mild integrability condition is assumed for large jumps to establish these results we employ methods from both probability theory and analysis extensively

the hidden subgroup problem and mktp abstract we show that the hidden subgroup problem for group families where products and inverses can be computed efficiently is in bpp mktp where mktp is the minimum kt problem using the techniques of allender et al 2018 1 we also show that the problem is in zpp mktp provided that there is a pac overestimator computable in zpp mktp for the logarithm of the order of the input group this last result implies that for permutation groups the dihedral group and many types of matrix groups the problem is in zpp mktp lastly we also show that two decision versions of the problem admit statistical zero knowledge proofs these results help classify the relative difficulty of the hidden subgroup problem

on lp estimates for a class of non local elliptic equations abstract we consider non local elliptic operators with kernel k y a y y d where 0 2 is a constant and a is a bounded measurable function by using a purely analytic method we prove the continuity of the non local operator l from the bessel potential space h p to l p and the unique strong solvability of the corresponding non local elliptic equations in l p spaces as a byproduct we also obtain interior l p estimates the novelty of our results is that the function a is not necessarily to be homogeneous regular or symmetric an application of our result is the uniqueness for the martingale problem associated to the operator l

an lp theory for a class of non local elliptic equations related to nonsymmetric measurable kernels abstract we study the integro differential operators l with kernels k y a y j y where j y is rotationally invariant and j y d y is a levy measure on r d i e r d 1 y 2 j y d y and a y is an only measurable function with positive lower and upper bounds under few additional conditions on j y we prove the unique solvability of the equation l u u f in l p spaces and present some l p estimates of the solutions

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

potential theory of truncated stable processes for any alpha in 0 2 a truncated symmetric stable process is a symmetric levy process in mathbb r d with a levy density given by c x d alpha 1 x 1 for some constant c in this paper we study the potential theory of truncated symmetric stable processes in detail we prove a harnack inequality for nonnegative harmonic functions of these processes we also establish a boundary harnack principle for nonnegative functions which are harmonic with respect to these processes in bounded convex domains we give an example of a non convex domain for which the boundary harnack principle fails

heat kernel upper bounds for jump processes and the first exit time for all t 0 x m and f l2 m the function pt x y can be considered as the transition density of the associated markov processx xt t 0 and the question of estimating of pt x y which is the main subject of this paper is closely related to the properties of x the function pt x y is also referred to as a heat kernel and this terminology goes back to the classical gauss weierstrass heat kernel associated with the heat semigroup e t 0 in r n whose markov process is brownian motion a somewhat more general but still well treated case is when m d is a riemannian metric measure space that is m is a riemannian manifold

transition probabilities for symmetric jump processes we consider symmetric markov chains on the integer lattice in d dimensions where a 0 2 and the conductance between x and y is comparable to x y d we establish upper and lower bounds for the transition probabilities that are sharp up to constants

absorption semigroups their generators and schrodinger semigroups on presente des resultats de perturbations pour des c 0 semi groupes on considere le probleme aux valeurs initiales pour l equation de la chaleur avec absorption avec une solution dans l p r v avec une mauvaise fonction v r r on considere le meme probleme dans l 2 r pour l equation de schrodinger

regularity for anisotropic fully nonlinear integro differential equations we consider fully nonlinear integro differential equations governed by kernels that have different homogeneities in different directions we prove a nonlocal version of the abp estimate a harnack inequality and the interior c 1 gamma regularity extending the results of caffarelli and silvestre comm pure appl math 62 597 638 2009 to the anisotropic case

enhanced multi level block ilu preconditioning strategies for general sparse linear systems abstract this paper introduces several strategies to deal with pivot blocks in multi level block incomplete lu factorization bilum preconditioning techniques these techniques are aimed at increasing the robustness and controlling the amount of fill ins of bilum for solving large sparse linear systems when large size blocks are used to form block independent set techniques proposed in this paper include double dropping strategies approximate singular value decomposition variable size blocks and use of an arrowhead block submatrix we point out the advantages and disadvantages of these strategies and discuss their efficient implementations numerical experiments are conducted to show the usefulness of the new techniques in dealing with hard to solve problems arising from computational fluid dynamics in addition we discuss the relation between multi level ilu preconditioning methods and algebraic multi level methods

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

lower tail probability estimates for subordinators and nondecreasing random walks f xn ex 00 oo and a is the infimum of the support of the distribution of x1 for a xn lk we obtain the asymptotic behavior of log p sn nxn as n oo under the additional assumption of stochastic compactness a stronger result is obtained which gives the asymptotic behavior of p sn nxn itself analogues of these results are given for subordinators when t o oo or t o 0

aleksandrov bakelman pucci type estimates for integro differential equations in this work we provide an aleksandrov bakelman pucci type estimate for a certain class of fully nonlinear elliptic integro differential equations the proof of which relies on an appropriate generalization of the convex envelope to a nonlocal fractional order setting and on the use of riesz potentials to interpret second derivatives as fractional order operators this result applies to a family of equations involving some nondegenerate kernels and as a consequence provides some new regularity results for previously untreated equations furthermore this result also gives a new comparison theorem for viscosity solutions of such equations which depends only on the l and ln norms of the right hand side in contrast to previous comparison results which utilize the continuity of the right hand side for their conclusions these results appear to be new even for the linear case of the relevant equations

meyers inequality and strong stability for stable like operators let 0 2 let e u u rd rd u y u x 2a x y x y d dydx be the dirichlet form for a stable like operator let u x rd u y u x 2a x y x y d dy 1 2 let l be the associated infinitesimal generator and suppose a x y is jointly measurable symmetric bounded and bounded below by a positive constant we prove that if u is the weak solution to lu h then u lp for some p 2 this is the analogue of an inequality of meyers for solutions to divergence form elliptic equations as an application we prove strong stability results for stable like operators if a is perturbed slightly we give explicit bounds on how much the semigroup and fundamental solution are perturbed

two sided eigenvalue estimates for subordinate processes in domains let x xt t 0 be a symmetric markov process in a state space e and d an open set of e denote by x d the subprocess of x killed upon leaving d let s st t 0 be a subordinator with laplace exponent that is independent of x the processes x xst t 0 and x d x dt t 0 are called the subordinate processes of x and x d respectively under some mild conditions we show that if n n 1 and n n 1 denote the eigenvalues of the generators of the subprocess of x killed upon leaving d and of the process x d respectively then

graph isomorphism is in spp we show that graph isomorphism is in the complexity class spp and hence it is in p in fact in modkp for each k 2 these inclusions for graph isomorphism were not known prior to membership in spp we derive this result as a corollary of a more general result we show that a generic problem find group has an fpspp algorithm this general result has other consequences for example it follows that the hidden subgroup problem for permutation groups studied in the context of quantum algorithms has an fpspp algorithm also some other algorithmic problems over permutation groups known to be at least as hard as graph isomorphism e g coset intersection are in spp and thus in modkp for each k 2

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

pointwise eigenfunction estimates and intrinsic ultracontractivity type properties of feynman kac semigroups for a class of levy processes we introduce a class of l evy processes subject to specific regularity conditions and consider their feynman kac semigroups given under a kato class potential using new techniques first we analyze the rate of decay of eigenfunctions at infinity we prove bounds on subaveraging functions from which we derive two sided sharp pointwise estimates on the ground state and obtain upper bounds on all other eigenfunctions next by using these results we analyze intrinsic ultracontractivity and related properties of the semigroup refining them by the concept of ground state domination and asymptotic versions we establish the relationships of these properties derive sharp necessary and sufficient conditions for their validity in terms of the behavior of the l evy density and the potential at infinity define the concept of borderline potential for the asymptotic properties and give probabilistic and variational characterizations these results are amply illustrated by key examples

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

solvable black box group problems are low for pp in this paper we show that several algorithmic problems for abelian black box groups and solvable black box groups are low for pp in particular we prove that membership testing group intersection order verification and group isomorphism for abelian groups are in spp and group factorization coset intersection and double coset membership for abelian groups are in lwpp similarly we show that membership testing order verification and group isomorphism for solvable groups are in zppspp we also show that group intersection group factorization coset intersection and double coset membership for solvable groups are low for pp

aperiodic fractional obstacle problems abstract we determine the asymptotic behavior of bilateral obstacle problems for fractional energies in rather general aperiodic settings via convergence arguments as further developments we consider obstacles with random sizes and shapes located on points of standard lattices and the case of random homothetics obstacles centered on random delone sets of points obstacle problems for non local energies occur in several physical phenomena for which our results provide a description of the zeroth order asymptotic behavior

harnack inequalities for non local operators of variable order we consider harmonic functions with respect to the operator formula math under suitable conditions on n x h we establish a harnack inequality for functions that are nonnegative and harmonic in a domain the operator l is allowed to be anisotropic and of variable order

harnack inequality for some classes of markov processes in this paper we establish a harnack inequality for nonnegative harmonic functions of some classes of markov processes with jumps

non local dirichlet forms and symmetric jump processes we consider the non local symmetric dirichlet form e f given by with f the closure with respect to e 1 of the set of c 1 functions on r d with compact support where e 1 f f e f f f rd f x 2 dx and where the jump kernel j satisfies for 0 2 x y 1 this assumption allows the corresponding jump process to have jump intensities whose sizes depend on the position of the process and the direction of the jump we prove upper and lower estimates on the heat kernel we construct a strong markov process corresponding to e f we prove a parabolic harnack inequality for non negative functions that solve the heat equation with respect to e finally we construct an example where the corresponding harmonic functions need not be continuous

on stochastic completeness of jump processes we prove the following sufficient condition for stochastic completeness of symmetric jump processes on metric measure spaces if the volume of the metric balls grows at most exponentially with radius and if the distance function is adapted in a certain sense to the jump kernel then the process is stochastically complete we use this theorem to prove the following criterion for stochastic completeness of a continuous time random walk on a graph with a counting measure if the volume growth with respect to the graph distance is at most cubic then the random walk is stochastically complete where the cubic volume growth is sharp

perturbation of dirichlet forms lower semiboundedness closability and form cores we study perturbations e e qu of dirichlet forms e on some l2 space l2 m given by quadratic forms q g g d with a signed borel measure whose positive and negative parts are smooth measures with respect to the given dirichlet form we give necessary and sufficient conditions for e to be a lower semibounded closed quadratic form with a unique associated continuous semigroup pt on l2 m we also study the associated resolvent semigroup as well as the relation between e and the largest closed quadratic form which is smaller than e in particular we exhibit form cores

developments and perspectives in nonlinear potential theory abstract nonlinear potential theory aims at replicating the classical linear potential theory when nonlinear equations are considered in recent years there has been a substantial development of this subject mostly linked to the possibility of proving pointwise estimates for solutions to nonlinear equations via linear and nonlinear potentials here we give a brief account of such developments and outline the connections with different fields

heat kernel estimates for jump processes of mixed types on metric measure spaces in this paper we investigate symmetric jump type processes on a class of metric measure spaces with jumping intensities comparable to radially symmetric functions on the spaces the class of metric measure spaces includes the alfors d regular sets which is a class of fractal sets that contains geometrically self similar sets a typical example of our jump type processes is the symmetric jump process with jumping intensity e c 0 x y x y int alpha 1 alpha 2 frac c alpha x y x y d alpha nu d alpha where is a probability measure on alpha 1 alpha 2 subset 0 2 c x y is a jointly measurable function that is symmetric in x y and is bounded between two positive constants and c 0 x y is a jointly measurable function that is symmetric in x y and is bounded between 1 and 2 where either 2 1 0 or 1 2 0 this example contains mixed symmetric stable processes on mathbb r n as well as mixed relativistic symmetric stable processes on mathbb r n we establish parabolic harnack principle and derive sharp two sided heat kernel estimate for such jump type processes

regularity results for stable like operators for 1 2 we consider operators of the form lf x rd f x h f x 1 h 1 f x h a x h h d dh and for 0 1 we consider the same operator but where the f term is omitted we prove under appropriate conditions on a x h that any solution u to lu f will be in c if f c

global heat kernel estimates for symmetric jump processes in this paper we study sharp heat kernel estimates for a large class of symmetric jump type processes in r d for all t 0 a prototype of the processes under consideration are symmetric jump processes on r d with jumping intensity

two sided green function estimates for killed subordinate brownian motions a subordinate brownian motion is a levy process that can be obtained by replacing the time of the brownian motion by an independent subordinator the infinitesimal generator of a subordinate brownian motion is where is the laplace exponent of the subordinator in this paper we consider a large class of subordinate brownian motions without diffusion component and with comparable to a regularly varying function at infinity this class of processes includes symmetric stable processes relativistic stable processes sums of independent symmetric stable processes sums of independent relativistic stable processes and much more we give sharp two sided estimates on the green functions of these subordinate brownian motions in any bounded fat open set d w hend is a bounded c 1 1 open set we establish an explicit form of the estimates in terms of the distance to the boundary as a consequence of such sharp green function estimates we obtain a boundary harnack principle in c 1 1 open sets with explicit rate of decay

the liouville property and a conjecture of de giorgi we consider bounded entire solutions of the nonlinear pde u u u 3 0i n r d and prove that under certain monotonicity conditions these solutions must be constant on hyperplanes the proof uses a liouville theorem for harmonic functions associated with a nonuniformly elliptic divergence form operator c 2000 john wiley sons inc

incomplete lu preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems we consider the preconditioned iterative solution of large dense linear systems where the coefficient matrix is a complex valued matrix arising from discretizing the integral equation of electromagnetic scattering for some scattering structures this matrix can be poorly conditioned the main purpose of this study is to evaluate the efficiency of a class of incomplete lu ilu factorization preconditioners for solving this type of matrices we solve the electromagnetic wave equations using the bicg method with an ilu preconditioner in the context of a multilevel fast multipole algorithm mlfma the novelty of this work is that the ilu preconditioner as constructed using the near part block diagonal submatrices generated from the mlfma experimental results show that the ilu preconditioner reduces the number of bicg iterations substantially compared to the block diagonal preconditioner the preconditioned iteration scheme also maintains the computational complexity of the mlfma and consequently reduces the total cpu time

regularity results for fully nonlinear parabolic integro differential operators in this paper we consider the regularity theory for fully nonlinear parabolic integro differential equations with symmetric kernels we are able to find parabolic versions of alexandrov backelman pucci estimate with 0 sigma 2 and we show a harnack inequality holder regularity and c 1 alpha regularity of the solutions by obtaining decay estimates of their level sets

holder estimates for non local parabolic equations with critical drift abstract in this paper we extend previous results on the regularity of solutions of integro differential parabolic equations the kernels are non necessarily symmetric which could be interpreted as a non local drift with the same order as the diffusion we provide a growth lemma and a harnack inequality which can be used to prove higher regularity estimates

a de giorgi nash type theorem for time fractional diffusion equations we study the regularity of weak solutions to linear time fractional diffusion equations in divergence form of arbitrary time order alpha in 0 1 the coefficients are merely assumed to be bounded and measurable and they satisfy a uniform parabolicity condition our main result is a de giorgi nash type theorem which gives an interior holder estimate for bounded weak solutions in terms of the data and the l infty bound of the solution the proof relies on new a priori estimates for time fractional problems and uses de giorgi s technique and the method of non local growth lemmas which has been introduced recently in the context of nonlocal elliptic equations involving operators like the fractional laplacian

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

scaling invariant harnack inequalities in a general setting abstract in a setting where only exit measures are given as they are associated with an arbitrary right continuous strong markov process on a separable metric space we provide simple criteria for the validity of harnack inequalities for positive harmonic functions these inequalities are scaling invariant with respect to a metric on the state space which having an associated green function may be adapted to the special situation in many cases this also implies continuity of harmonic functions and holder continuity of bounded harmonic functions the results apply to large classes of levy and similar processes

nonlocal operators of order near zero abstract we study dirichlet forms defined by nonintegrable levy kernels whose singularity at the origin can be weaker than that of any fractional laplacian we show some properties of the associated sobolev type spaces in a bounded domain such as symmetrization estimates hardy inequalities compact inclusion in l 2 or the inclusion in some lorentz space we then apply those properties to study the associated nonlocal operator l and the dirichlet and neumann problems related to the equations l u f x and l u f u in

uniqueness in law for pure jump markov processes let a be the operator defined on c2 functions by af left x right smallint left f left x h right f left x right f left x right h 1 1 1 left h right right v left x dh right sufficient conditions are given for existence and uniqueness for the martingale problem associated with a in the case of stable like processes where v x dh is equal to the levy measure for the stable symmetric process of index x for each x the conditions reduce to x continuous for existence and x dini continuous for uniqueness

automatic basis function regularization for integral equations in the presence of ill shaped mesh elements bad meshes are a common problem in using computational electromagnetics codes often much of an engineer s or analyst s time ids wasted improving a mesh until the solver finally converges we investigate the problem as it comes up in integral equation methods and observe that its effects can be detected by analyzing the singular values of the electric field integral equation matrix armed with this information we automatically remove the undesirable degrees of freedom caused by the ill shaped mesh elements so that the iterative solver behaves as if they were not present all without any extra effort by the user an a posteriori error estimator is developed which verifies the improved accuracy for test problems and also suggests potential future applications in adaptive mesh refinement for integral equation methods further numerical examples demonstrate the improved solution convergence

perturbation theory for convolution semigroups abstract we deal with convolution semigroups not necessarily symmetric in l p r n and provide a general perturbation theory of their generators by indefinite singular potentials such semigroups arise in the theory of levy processes and cover many examples such as gaussian semigroups stable semigroups relativistic schrodinger semigroups etc we give new generation theorems and feynman kac formulas in particular by using weak compactness methods in l 1 we enlarge the extended kato class potentials used in the theory of markov processes in l 2 setting dirichlet form perturbation theory is finely related to l 1 theory and the extended kato class measures is also enlarged finally various perturbation problems for subordinate semigroups are considered

boundary regularity for nonlocal operators with kernels of variable orders abstract we study the boundary regularity of solutions of the dirichlet problem for the nonlocal operator with a kernel of variable orders since the order of differentiability of the kernel is not represented by a single number we consider the generalized holder space we prove that there exists a unique viscosity solution of l u f in d u 0 in r n d where d is a bounded c 1 1 open set and that the solution u satisfies u c v d and u v d d c d with the uniform estimates where v is the renewal function and d d x dist x d

heat kernel estimates for subordinate brownian motions in this article we study transition probabilities of a class of subordinate brownian motions under mild assumptions on the laplace exponent of the corresponding subordinator sharp two sided estimates of the transition probability are established this approach in particular covers subordinators with laplace exponents that vary regularly at infinity with index one e g log 1 1 or log 1 2 0 2 that correspond to subordinate brownian motions with scaling order that is not necessarily stricty between 0 and 2 these estimates are applied to estimate green function potential of subordinate brownian motion we also prove the equivalence of the lower scaling condition of the laplace exponent and the near diagonal upper estimate of the transition estimate

a new proof of moser s parabolic harnack inequality using the old ideas of nash in 1958 nash published his fundamental work on the local holder continuity of solutions of second order parabolic equations with coefficients which need not be smooth 8 the primary purpose of that work was to study the properties of the fundamental solution corresponding to the parabolic operator and from these properties to derive regularity for a general solution though the work is often cited in the literature about weak solutions of elliptic and parabolic equations one feels that nash s ideas were never fully understood and maybe still are not and that because of this the more understandable and seemingly more fruitful ideas of de giorgi 4 and moser 6 7 were subsequently adopted

regularity of solutions to anisotropic nonlocal equations we study harmonic functions associated to systems of stochastic differential equations of the form dx t i a i1 x t dz t 1 cdots a id x t dz t d i in 1 dots d where z t j are independent one dimensional symmetric stable processes of order alpha j in 0 2 j in 1 dots d in this article we prove holder regularity of bounded harmonic functions with respect to solutions to such systems

potential theory of geometric stable processes in this paper we study the potential theory of symmetric geometric stable processes by realizing them as subordinate brownian motions with geometric stable subordinators more precisely we establish the asymptotic behaviors of the green function and the levy density of symmetric geometric stable processes the asymptotics of these functions near zero exhibit features that are very different from the ones for stable processes the green function behaves near zero as 1 x d log 2 x while the levy density behaves like 1 x d we also study the asymptotic behaviors of the green function and levy density of subordinate brownian motions with iterated geometric stable subordinators as an application we establish estimates on the capacity of small balls for these processes as well as mean exit time estimates from small balls and a harnack inequality for these processes

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

on inversion of the bessel and gelfand transforms we construct the plancherel measure corresponding to the bessel model on the split special orthogonal group of odd degree 5 and the whit taker model on a connected split reductive group in general as an application we prove the inversion formula which expresses the related integral transform in terms of the satake transform

linnik levy process and some extensions abstract in the literature the linnik mittag leffler laplace and asymmetric laplace distributions are the most known examples of geometric stable distributions the geometric stable distributions are especially useful in the modeling of leptokurtic data with heavy tailed behavior they have found many interesting applications in the modeling of several physical phenomena and financial time series in this paper we define the linnik levy process llp through the subordination of symmetric stable levy motion with gamma process we discuss main properties of llp like probability density function levy measure and asymptotic forms of marginal densities we also consider the governing fractional type fokker planck equation to show practical applications we simulate the sample paths of the introduced process moreover we give a step by step procedure of the parameters estimation and calibrate the parameters of the llp with the arconic inc equity data taken from yahoo finance further some extensions of the introduced process are also discussed

symmetric markov chains on ℤ with unbounded range we consider symmetric markov chains on z d where we do not assume that the conductance between two points must be zero if the points are far apart under a uniform second moment condition on the conductances we obtain upper bounds on the transition probabilities estimates for exit time probabilities and certain lower bounds on the transition probabilities we show that a uniform harnack inequality holds if an additional assumption is made but that without this assumption such an inequality need not hold we establish a central limit theorem giving conditions for a sequence of normalized symmetric markov chains to converge to a diffusion on d corresponding to an elliptic operator in divergence form

an analytic approach to purely nonlocal bellman equations arising in models of stochastic control abstract given a bounded domain r d and two integro differential operators l 1 l 2 of the form l j u x p v u x u y k j x y x y d y we study the fully nonlinear bellman equation 0 1 max j 1 2 l j u x a j x u x f j x 0 in with dirichlet boundary conditions here a j f j r are non negative functions we prove the existence of a non negative function u r which satisfies 0 1 almost everywhere the main difficulty arises through the nonlocality of l j and the absence of regularity near the boundary

stabilizing block diagonal preconditioners for complex dense matrices in electromagnetics abstract preconditioning techniques are widely used to speed up the convergence of iterative methods for solving large linear systems with sparse or dense coefficient matrices for certain application problems however the standard block diagonal preconditioner makes the krylov iterative methods converge more slowly or even diverge to handle this problem we apply diagonal shifting and stabilized singular value decomposition svd to each diagonal block which is generated from the multilevel fast multiple algorithm mlfma to improve the stability and efficiency of the block diagonal preconditioner our experimental results show that the improved block diagonal preconditioner maintains the computational complexity of mlfma converges faster and also reduces the cpu cost

rapid solution of integral equations of scattering theory in two dimensions abstract the present paper describes an algorithm for rapid solution of boundary value problems for the helmholtz equation in two dimensions based on iteratively solving integral equations of scattering theory cpu time requirements of previously published algorithms of this type are of the order n 2 where n is the number of nodes in the discretization of the boundary of the scatterer the cpu time requirements of the algorithm of the present paper are n 4 3 and can be further reduced making it considerably more practical for large scale problems

an extension problem for sums of fractional laplacians and 1 d symmetry of phase transitions we study nonlinear elliptic equations for operators corresponding to non stable levy diffusions we include a sum of fractional laplacians of different orders such operators are infinitesimal generators of non stable i e non self similar levy processes we establish the regularity of solutions as well as sharp energy estimates as a consequence we prove a 1 d symmetry result for monotone solutions to allen cahn type equations with a non stable levy diffusion these operators may still be realized as local operators using a system of pdes in the spirit of the extension problem of caffarelli and silvestre

random conductance models with stable like jumps heat kernel estimates and harnack inequalities abstract we establish two sided heat kernel estimates for random conductance models with non uniformly elliptic possibly degenerate stable like jumps on graphs these are long range counterparts of the well known two sided gaussian heat kernel estimates by m t barlow for nearest neighbor short range random walks on the supercritical percolation cluster unlike the cases of nearest neighbor conductance models we cannot use parabolic harnack inequalities since even elliptic harnack inequalities do not hold in the present setting as an application we establish the local limit theorem for the models