fractional laplacian phase transitions and boundary reactions a geometric inequality and a symmetry result we deal with symmetry properties for solutions of nonlocal equations of the type sv f v in rn where s 0 1 and the operator s is the so called fractional laplacian the study of this nonlocal equation is made via a careful analysis of the following degenerate elliptic equation div x u 0on rn 0 x ux f u on rn 0 where 1 1 y rn x 0 and u u y x this equation is related to the fractional laplacian since the dirichlet to neumann operator u r n 1 x ux r n 1 is 1 2 more generally we study the so called boundary reaction equations given by div x u g x u 0on rn 0 x ux f u on rn 0 under some natural assumptions on the diffusion coefficient and on the nonlinearities f and g we prove a geometric formula of poincare type for stable solutions from which we derive a symmetry result in the spirit of a conjecture of de giorgi

cores for feller semigroups with an invariant measure abstract let a i j 1 n a i j x d i j i 1 n b i x d i be an elliptic differential operator with unbounded coefficients on r n and assume that the associated feller semigroup t t t 0 has an invariant measure then t t t 0 extends to a strongly continuous semigroup t p t t 0 on l p l p r n for every 1 p we prove that under mild conditions on the coefficients of a the space of test functions c c r n is a core for the generator a p d p of t p t t 0 in l p for 1 p

weak stability of ricci expanders with positive curvature operator we prove the l infty stability of expanding gradient ricci solitons with positive curvature operator and quadratic curvature decay at infinity

the well posedness of the surface quasi geostrophic equations in the besov morrey spaces abstract in this paper we develop the boundedness of riesz transform in morrey spaces which generalize the corresponding result in the usual l p spaces as an application of this boundedness we establish the well posedness and blow up criterion for the surface quasi geostrophic equations in the framework of besov morrey spaces

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

some classes of non analytic markov semigroups we deal with markov semigroups tt corresponding to second order elliptic operators au u du f where f is an unbounded locally lipschitz vector field on rn we obtain new conditions on f under which tt is not analytic in cb rn in particular we prove that the one dimensional operator au u x3u with domain u c2 r u u x3u cb r is not sectorial in cb r under suitable hypotheses on the growth of f we introduce a class of non analytic markov semigroups in lp rn where is an invariant measure for tt

cores for parabolic operators with unbounded coefficients abstract let a i j 1 n q i j s x d i j i 1 n b i s x d i be a family of elliptic differential operators with unbounded coefficients defined in r n 1 in m kunze l lorenzi a lunardi nonautonomous kolmogorov parabolic equations with unbounded coefficients trans amer math soc in press under suitable assumptions it has been proved that the operator g a d s generates a semigroup of positive contractions t p t in l p r n 1 for every 1 p where is an infinitesimally invariant measure of t p t here under some additional conditions on the growth of the coefficients of a which cover also some growths with an exponential rate at we provide two different cores for the infinitesimal generator g p of t p t in l p r n 1 for p 1 and we also give a partial characterization of d g p finally we extend the results so far obtained to the case when the coefficients of the operator a are t periodic with respect to the variable s for some t 0

an asymptotic solution of the first passage problem for singular diffusion in population biology the first passage problem is studied for a singular diffusion process arising in population biology with the deterministic part having a stable equilibrium point and small diffusion the laplace transform of the first passage time density satisfies an ordinary differential equation in a finite interval with two point boundary conditions a formal asymptotic solution to this problem is constructed when the singular boundary point is absorbing and the regular boundary point is either absorbing or reflecting the solution depends upon the initial value z of the process for z near the equilibrium point the density is approximated by the ornstein uhlenbeck equation for z near the singular boundary point it is approximated by the diffusion limit of the linear birth death process and elsewhere it is approximated by the wkb solution these different solutions are then connected together using these results the process is shown to exit with probability one in the limit of small diffusion the mean and varianc

boundary harnack inequality for markov processes with jumps we prove a boundary harnack inequality for jump type markov processes on metric measure state spaces under comparability estimates of the jump kernel and urysohn type property of the domain of the generator of the process the result holds for positive harmonic functions in arbitrary open sets it applies e g to many subordinate brownian motions levy processes with and without continuous part stable like and censored stable processes jump processes on fractals and rather general schrodinger drift and jump perturbations of such processes

spectrum of ornstein uhlenbeck operators in lp spaces with respect to invariant measures abstract let a i j 1 n q ij d ij i j 1 n b ij x j d i be a possibly degenerate ornstein uhlenbeck operator in r n and assume that the associated markov semigroup has an invariant measure we compute the spectrum of a in l p for 1 p

schauder theorems for ornstein uhlenbeck equations in infinite dimension abstract we prove schauder type estimates for stationary and evolution equations driven by the classical ornstein uhlenbeck operator in a separable banach space endowed with a centered gaussian measure

asymptotic behaviour of densities of stable semigroups of measures we prove that densities of the measures in a strictly stable semigroup ht of symmetric measures on a homogeneous group if they exist have the following asymptotic behaviour mathop lim limits x to infty x q alpha cdot h 1 x k bar x where is the characteristic exponent bar x x 1 x andk is the density of the levy measure associated to the semigroup moreover if k bar x 0 a more precise description is given

global well posedness for the critical 2d dissipative quasi geostrophic equation we give an elementary proof of the global well posedness for the critical 2d dissipative quasi geostrophic equation the argument is based on a non local maximum principle involving appropriate moduli of continuity

the radiative transfer equation in the forward peaked regime in this work we study the radiative transfer equation in the forward peaked regime in free space specifically it is shown that the equation is well posed by proving instantaneous regularization of weak solutions for arbitrary initial datum in l1 classical techniques for hypo elliptic operators such as averaging lemma are used in the argument among the interesting aspects of the proof are the use of the stereographic projection and the presentation of a rigorous expression for the scattering operator given in terms of a fractional laplace beltrami operator on the sphere or equivalently a weighted fractional laplacian analog in the projected plane such representations may be used for accurate numerical simulations of the model as a bonus given by the methodology we show convergence of henyey greenstein scattering models and vanishing of the solution at time algebraic rate due to scattering diffusion

l p uniqueness for elliptic operators with unbounded coefficients in r n let a be an elliptic operator with unbounded and sufficiently smooth coefficients and let be a sub invariant measure of the operator a in this paper we give sufficient conditions guaranteeing that the closure of the operator a cc rn generates a sub markovian strongly continuous semigroup of contractions in lp rn applications are given in the case when a is a generalized schrodinger operator

an asymptotic solution to a two dimensional exit problem arising in population dynamics a study is made of a two dimensional stochastic system with small stochastic fluctuations arising in population biology at the boundary of the state space the diffusion matrix becomes singular by an asymptotic analysis expressions are derived that determine the probability of exit at each of the two boundaries and the expectation and variance of the exit time these expressions contain constants that can be computed numerically

elliptic problems on ℝ n with unbounded coefficients in classical sobolev spaces we discuss the existence and uniqueness in h1 n and the h2 n regularity of the solutions of au f when f l2 n and a is a second order linear elliptic operator whose first and zeroth order coefficients may be unbounded at infinity we also investigate whether a generates a c0 or analytic semigroup on l2 the approach in this nonweighted setting is based on a new and general method the idea consists in embedding a into a suitable one parameter family of operators as s with a0 a the properties of as when s 0 make it possible to prove that the boundary integrals arising from simple integration by parts over balls with increasing radius tend to 0 at infinity this provides the needed estimates for uniqueness and regularity

regularity theory for general stable operators abstract we establish sharp regularity estimates for solutions to l u f in r n l being the generator of any stable and symmetric levy process such nonlocal operators l depend on a finite measure on s n 1 called the spectral measure first we study the interior regularity of solutions to l u f in b 1 we prove that if f is c then u belong to c 2 s whenever 2 s is not an integer in case f l we show that the solution u is c 2 s when s 1 2 and c 2 s for all 0 when s 1 2 then we study the boundary regularity of solutions to l u f in u 0 in r n in c 1 1 domains we show that solutions u satisfy u d s c s for all 0 where d is the distance to finally we show that our results are sharp by constructing two counterexamples

gradient estimates for dirichlet parabolic problems in unbounded domains we consider autonomous parabolic dirichlet problems in a regular unbounded open set rn involving second order operator a with possibly unbounded coefficients we determine new conditions on the coefficients of a yielding global gradient estimates for the bounded classical solution

on the ornstein uhlenbeck operator in spaces of continuous functions we study the realization of the ornstein uhlenbeck operator a in the space of the uniformly continuous and bounded functions in rn we prove that it generates a semigroup which is neither strongly continuous nor analytic but enjoys nice smoothing properties such properties are used to prove a schauder type theorem for the elliptic equation u au f

global well posedness in the super critical dissipative quasi geostrophic equations we consider the quasi geostrophic equation with the dissipation term in the case constantin cordoba wu 6 proved the global existence of strong solution in h 1 and h 2 under the assumption of small l norm of initial data in this paper we prove the global existence in the scale invariant besov space b 2 2 2 1 for initial data small in the b 2 2 2 1 norm we also prove a global stability result in b 1 2 1

heat kernels for time dependent non symmetric stable like operators abstract when studying non symmetric nonlocal operators on r d l f x r d f x z f x f x z 1 z 1 x z z d d z where 0 2 d 1 and x z is a function on r d r d that is bounded between two positive constants it is customary to assume that x z is symmetric in z in this paper we study heat kernel of l and derive its two sided sharp bounds without the symmetric assumption x z x z in fact we allow the kernel to be time dependent and x t x z to be only locally holder continuous with holder constant possibly growing at a polynomial rate in z we also derive gradient estimate when 0 1 1 as well as fractional derivative estimate of order 0 2 for the heat kernel moreover when 1 2 drift perturbation of the time dependent non local operator l t with drift in kato s class is also studied in this paper as an application when x z z does not depend on x we show the boundedness of nonlocal riesz s transformation for any p 2 d d l 1 2 f p f 1 2 p where f 1 2 l f 2 f l f is the carre du champ operator associated with l and l 1 2 is the square root operator of l defined by using bochner s subordination here means that both sides are comparable up to a constant multiple

an exact algorithm for the reliability redundancy allocation problem abstract the redundancy allocation problem is the problem of finding an optimal allocation of redundant components subject to a set of resource constraints the problem studied in this paper refers to a series parallel system configuration and allows for component mixing we propose a new modeling solution approach in which the problem is transformed into a multiple choice knapsack problem and solved to optimality via a branch and cut algorithm the algorithm is tested on well known sets of benchmark instances all instances have been solved to optimality in milliseconds or very few seconds on a normal workstation

a pde approach to fractional diffusion a space fractional wave equation we study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers of order s in 0 1 of symmetric coercive linear elliptic second order operators in bounded domains varomega we realize fractional diffusion as the dirichlet to neumann map for a nonuniformly elliptic problem posed on the semi infinite cylinder mathcal c varomega times 0 infty we thus rewrite our evolution problem as a quasi stationary elliptic problem with a dynamic boundary condition and derive space time and space time regularity estimates for its solution the latter problem exhibits an exponential decay in the extended dimension and thus suggests a truncation that is suitable for numerical approximation we propose and analyze two fully discrete schemes the discretization in time is based on finite difference discretization techniques the trapezoidal and leapfrog schemes the discretization in space relies on the tensorization of a first degree fem in varomega with a suitable hp fem in the extended variable for both schemes we derive stability and error estimates we consider a first degree fem in varomega with mesh refinement near corners and the aforementioned hp fem in the extended variable and extend the a priori error analysis of the trapezoidal scheme for open bounded polytopal but not necessarily convex domains varomega subset mathbb r 2 we discuss implementation details and report several numerical examples

regularity for a local nonlocal transmission problem we formulate and study an elliptic transmission like problem combining local and nonlocal elements let mathbb r n be separated into two components by a smooth hypersurface on one side of a function satisfies a local second order elliptic equation on the other it satisfies a nonlocal one of lower order in addition a nonlocal transmission condition is imposed across the interface which has a natural variational interpretation we deduce the existence of solutions to the corresponding dirichlet problem and show that under mild assumptions they are holder continuous following the method of de giorgi the principal difficulty stems from the lack of scale invariance near which we circumvent by deducing a special energy estimate which is uniform in the scaling parameter we then turn to the question of optimal regularity and qualitative properties of solutions and show that in the case of constant coefficients and flat they satisfy a kind of transmission condition where the ratio of fractional conormal derivatives on the two sides of the interface is fixed a perturbative argument is then given to show how to obtain regularity for solutions to an equation with variable coefficients and smooth interface throughout we pay special attention to a nonlinear version of the problem with a drift given by a singular integral operator of the solution which has an interpretation in the context of quasigeostrophic dynamics

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

stable processes with drift on the line the stable processes on the line having a drift are investigated except for the symmetric cauchy processes with drift these are all transient and points are nonpolar sets explicit information about the potential kernel is obtained and this is used to obtain specific results about hitting times and places for various sets 1 statement of results let be a process with stationary independent increments having log characteristic function 0 l iasgn 0 where 0 0 there is an enormous literature on the behavior of drift free stable processes surprisingly little has been written on stable processes with drift in view of the extensive results known for the potential theory 6 and the path behavior 3 4 for infinitely divisible processes in general the interest today in special processes such as stable processes with drift lies in the fact that for such processes rather explicit results may be obtained the incorporation of a drift term changes the behavior of the process as compared to a drift free process however the drift acts substantially different for processes with index a 1 in all cases except for a 1 and s 0 i e the symmetric cauchy process the processes with drift are transient and hit points with positive probability this fact follows from the general theory 3 and 6 this is in contrast to the drift free processes for drift free processes if a 1 the processes are transient but do not hit points received by the editors july 31 1986 and in revised form february 1 1988 1980 mathematics subject classification 1985 revision primary 60j30 secondary 60j45

bounds on short cylinders and uniqueness in cauchy problem for degenerate kolmogorov equations abstract we consider the cauchy problem for degenerate kolmogorov equations in the form t u i j 1 m a i j x t x i x j u j 1 m a j x t x j u i j 1 n b i j x i x j u x t r n 0 t 1 m n as well as in its divergence form we prove that if u x t m exp a t x 2 for some positive constants a m and 0 1 and u 0 0 then u 0 the proof of the main result is based on some previous uniqueness result and on the application of some estimates in short cylinders previously used by ferretti in the study of uniformly parabolic operators

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

radial symmetry results for fractional laplacian systems abstract in this paper we generalize the direct method of moving planes for the fractional laplacian to the system case considering a coupled nonlinear system with fractional laplacian we first establish a decay at infinity principle and a narrow region principle using these principles we then obtain two radial symmetry results for the decaying solutions of the fractional laplacian systems finally we apply our method to fractional schrodinger systems and fractional henon systems

diffusion across characteristic boundaries with critical points we consider the problems of the effect of small white noise perturbations on a deterministic dynamical system in the plane with i an asymptotically stable equilibrium point or limit cycle and ii an equilibrium point surrounded by closed trajectories the mean exit time and the distribution of exit points for each problem is determined by solving singularly perturbed elliptic boundary value problems in domains with closed characteristic boundaries with critical points uniformly valid asymptotic solutions are constructed for each of the problems for the asymptotically stable equilibrium point the method of matched asymptotic expansions with the integral condition of matkowsky and schuss is employed a method of averaging combined with boundary layer analysis is used for the problem of an equilibrium point surrounded by closed trajectories the influence on the solutions of the critical points on the boundary is exhibited and explained an application to the physical pendulum is given finally our r

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

reliability redundancy allocation an improved realization for nonconvex nonlinear programming problems the redundancy allocation problem which seeks to find the optimal allocation of redundancy that maximizes system reliability is one of the representative problems in reliability optimization the problem can be formulated as a nonconvex integer nonlinear programming problem this paper presents an efficient branch and bound approach to solve this problem where the considered system is coherent that is the objective and constraint functions have monotonic increasing properties the proposed method is based primarily on a search space elimination of disjoint sets in a solution space that does not require any relaxation of branched subproblems the main advantage of the proposed method is flexibility i e it does not rely on any assumptions of linearity separability single constraint or convexity which make the method adaptive to various applications numerical experiments demonstrate that the method is superior to the existing exact algorithms for redundancy allocation problems in terms of computation time

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

reliability stochastic optimization for a series system with interval component reliability via genetic algorithm this paper deals with chance constraints based reliability stochastic optimization problem in the series system this problem can be formulated as a nonlinear integer programming problem of maximizing the overall system reliability under chance constraints due to resources the assumption of traditional reliability optimization problem is that the reliability of a component is known as a fixed quantity which lies in the open interval 0 1 however in real life situations the reliability of an individual component may vary due to some realistic factors and it is sensible to treat this as a positive imprecise number and this imprecise number is represented by an interval valued number in this work we have formulated the reliability optimization problem as a chance constraints based reliability stochastic optimization problem with interval valued reliabilities of components then the chance constraints of the problem are converted into the equivalent deterministic form the transformed problem has been formulated as an unconstrained integer programming problem with interval coefficients by big m penalty technique then to solve this problem we have developed a real coded genetic algorithm ga for integer variables with tournament selection uniform crossover and one neighborhood mutation to illustrate the model two numerical examples have been solved by our developed ga finally to study the stability of our developed ga with respect to the different ga parameters sensitivity analyses have been done graphically

two renewal theorems for general random walks tending to infinity necessary and sufficient conditions for the existence of moments of the first passage time of a random walk s n into x for fixed x 0 and the last exit time of the walk from x are given under the condition that s n a s the methods which are quite different from those applied in the previously studied case of a positive mean for the increments of s n are further developed to obtain the order of magnitude as x of the moments of the first passage and last exit times when these are finite a number of other conditions of interest in renewal theory are also discussed and some results for the first time for which the random walk remains above the level x on k consecutive occasions which has applications in option pricing are given

gradient estimates for diffusion semigroups with singular coefficients abstract uniform gradient estimates are derived for diffusion semigroups possibly with potential generated by second order elliptic operators having irregular and unbounded coefficients we first consider the r d case by using the coupling method due to the singularity of the coefficients the coupling process we construct is not strongly markovian so that additional difficulties arise in the study then more generally we treat the case of a possibly unbounded smooth domain of r d with dirichlet boundary conditions we stress that the resulting estimates are new even in the r d case and that the coefficients can be holder continuous our results also imply a new liouville theorem for space time bounded harmonic functions with respect to the underlying diffusion semigroup

rational approximation schemes for bi continuous semigroups abstract this paper extends the hille phillips functional calculus and rational approximations results due to r hersh t kato p brenner and v thomee to generators of bi continuous semigroups the method yields error estimates for rational time discretization schemes for such semigroups in particular for dual semigroups feller semigroups such as the ornstein uhlenbeck semigroup the heat semigroup semigroups induced by nonlinear flows implemented semigroups and evolution semigroups furthermore the results provide error estimates for a new class of inversion formulas for the laplace transform

matched uniform approximations for a singular boundary point and an interior turning point the singular perturbation problem for a second order ode with singular boundary point and interior turning point is studied two uniform asymptotic approximations are separately constructed around singular point and turning point the restricted uniform approximation about the turning point corresponds to o malley s uniform reduction while that about the singular point is found by a similar method and is entirely new higher order matching of both dominant and recessive terms of the two approximations leads to a complete asymptotic approximation of the general solution a subsequent paper applies these results to higher order resonance criteria

nonsymmetric ornstein uhlenbeck semigroups in banach spaces letebe a separable real banach space and letq l e e be positive and symmetric lets s t t 0be ac0 semigroup onewe study the relations between the reproducing kernel hilbert spaces associated with the operatorsqt t0 s s qs s ds under the assumption that these operators are the covariances of centered gaussian measures tone we also study equivalence t sfor different values of s and t and we calculate their radon nikodym derivatives

global well posedness for the euler boussinesq system with axisymmetric data abstract in this paper we prove the global well posedness for the three dimensional euler boussinesq system with axisymmetric initial data without swirl this system couples the euler equation with a transport diffusion equation governing the temperature

mild solutions of semilinear elliptic equations in hilbert spaces abstract this paper extends the theory of regular solutions c 1 in a suitable sense for a class of semilinear elliptic equations in hilbert spaces the notion of regularity is based on the concept of g derivative which is introduced and discussed a result of existence and uniqueness of solutions is stated and proved under the assumption that the transition semigroup associated to the linear part of the equation has a smoothing property that is it maps continuous functions into g differentiable ones the validity of this smoothing assumption is fully discussed for the case of the ornstein uhlenbeck transition semigroup and for the case of invertible diffusion coefficient covering cases not previously addressed by the literature it is shown that the results apply to hamilton jacobi bellman hjb equations associated to infinite horizon optimal stochastic control problems in infinite dimension and that in particular they cover examples of optimal boundary control of the heat equation that were not treatable with the approaches developed in the literature up to now

a maximum principle applied to quasi geostrophic equations we study the initial value problem for dissipative 2d quasi geostrophic equations proving local existence global results for small initial data in the super critical case decay of lp norms and asymptotic behavior of viscosity solution in the critical case our proofs are based on a maximum principle valid for more general flows

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

on the loss of continuity for super critical drift diffusion equations we show that there exist solutions of drift diffusion equations in two dimensions with divergence free super critical drifts that become discontinuous in finite time we consider classical as well as fractional diffusion however in the case of classical diffusion and time independent drifts we prove that solutions satisfy a modulus of continuity depending only on the local l 1 norm of the drift which is a super critical quantity

nonlinear fractional laplacian problems with nonlocal gradient terms let n 2 be a smooth bounded domain for s 1 2 1 we consider a problem of the form where 0 is a real parameter f belongs to a suitable lebesgue space and is a nonlocal gradient square term given by depending on the real parameter 0 we derive existence and non existence results the proof of our existence result relies on sharp calderon zygmund type regularity results for the fractional poisson equation with low integrability data we also obtain existence results for related problems involving different nonlocal diffusion terms

commutator estimate and its application to regularity criteria of the dissipative quasi geostrophic equation abstract a new commutator estimate with respect to a nonlinear convection upper bounded by a single partial derivative component in hilbert spaces is obtained as an application regularity criteria on the supercritical quasi geostrophic equation are obtained provided that solution growth conditions are assumed to involve a single partial derivative component

drift diffusion equations with fractional diffusion and the quasi geostrophic equation motivated by the critical dissipative quasi geostrophic equation we prove that drift diffusion equations with l 2 initial data and minimal assumptions on the drift are locally holder continuous as an application we show that solutions of the quasi geostrophic equation with initial l 2 data and critical diffusion 1 2 are locally smooth for any space dimension

a new proof of doobʼs theorem abstract we prove that every bounded positive irreducible stochastically continuous semigroup on the space of bounded measurable functions which is strong feller consists of kernel operators and possesses an invariant measure converges pointwise this differs from doob s theorem in that we do not require the semigroup to be markovian and request a fairly weak kind of irreducibility in addition we elaborate on the various notions of kernel operators in this context show the stronger result that the adjoint semigroup converges strongly and discuss as an example diffusion equations on rough domains the proofs are based on the theory of positive semigroups and do not use probability theory

on allocation of redundant components for systems with dependent components in this paper we consider the problem of optimal allocation of a redundant component for series parallel and k out of n systems of more than two components when all the components are dependent we show that for this problem is naturally to consider multivariate extensions of the joint bivariates stochastic orders however these extensions have not been defined or explicitly studied in the literature except the joint likelihood ratio order which was introduced by shanthikumar and yao 1991 therefore we provide first multivariate extensions of the joint stochastic hazard rate reversed hazard rate order and next we provide sufficient conditions based on these multivariate extensions to select which component performs the redundancy

note on the well posedness of a slightly supercritical surface quasi geostrophic equation abstract for the following slightly supercritical surface quasi geostrophic equation t u d 0 u d 2 m d 0 1 where m c r 2 0 is a radial non decreasing positive function which roughly has a logarithmic growth near infinity we apply the method of nonlocal maximum principle to show the global well posedness of smooth solutions

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

kolmogorov equation associated to a stochastic navier stokes equation abstract a direct solution of the kolmogorov equation associated to a stochastic navier stokes equation is given with restriction to two space dimensions and periodic boundary conditions the existence of a variational solution is proved using a special property of the nonlinear operator

global l p estimates for degenerate ornstein uhlenbeck operators we consider a class of degenerate ornstein uhlenbeck operators in mathbb r n of the kind mathcal a equiv sum i j 1 p 0 a ij partial x i x j 2 sum i j 1 n b ij x i partial x j where aij bij are constant matrices aij is symmetric positive definite on mathbb r p 0 p0 n and bij is such that mathcal a is hypoelliptic for this class of operators we prove global lp estimates 1 p of the kind left vert partial x i x j 2 u right vert l p left mathbb r n right leq c left left vert mathcal a u right vert l p left mathbb r n right left vert u right vert l p left mathbb r n right right rm for i j 1 2 ldots p 0 and corresponding weak type 1 1 estimates this result seems to be the first case of global estimates in lebesgue lp spaces for complete hormander s operators sum x i 2 x 0 proved in absence of a structure of homogeneous group we obtain the previous estimates as a byproduct of the following one which is of interest in its own left vert partial x i x j 2 u right vert l p left s right leq c left vert lu right vert l p left s right for any u in c 0 infty left s right where s is the strip mathbb r n times left 1 1 right and l is the kolmogorov fokker planck operator mathcal a partial t to get this estimate we use in a crucial way the left invariance of l with respect to a lie group structure in mathbb r n 1 and some results on singular integrals on nonhomogeneous spaces recently proved in bramanti revista matematica iberoamericana 2009 in press

on the cauchy problem for integro differential operators in sobolev classes and the martingale problem abstract the existence and uniqueness in sobolev spaces of solutions of the cauchy problem to parabolic integro differential equation with variable coefficients of the order 0 2 is investigated the principal part of the operator has kernel m t x y y d with a bounded nondegenerate m holder in x and measurable in y the lower order part has bounded and measurable coefficients the result is applied to prove the existence and uniqueness of the corresponding martingale problem

a ga based penalty function technique for solving constrained redundancy allocation problem of series system with interval valued reliability of components in this paper we have considered the problem of constrained redundancy allocation of series system with interval valued reliability of components for maximizing the overall system reliability under limited resource constraints the problem is formulated as an unconstrained integer programming problem with interval coefficients by penalty function technique and solved by an advanced ga for integer variables with interval fitness function tournament selection uniform crossover uniform mutation and elitism as a special case considering the lower and upper bounds of the interval valued reliabilities of the components to be the same the corresponding problem has been solved the model has been illustrated with some numerical examples and the results of the series redundancy allocation problem with fixed value of reliability of the components have been compared with the existing results available in the literature finally sensitivity analyses have been shown graphically to study the stability of our developed ga with respect to the different ga parameters

a frequency localized maximum principle applied to the 2d quasi geostrophic equation in this paper we prove a maximum principle for a frequency localized transport diffusion equation as an application we prove the local well posedness of the supercritical quasi geostrophic equation in the critical besov spaces mathring b 1 alpha infty q and global well posedness of the critical quasi geostrophic equation in mathring b 0 infty q for all 1 q here mathring b s infty q is the closure of the schwartz functions in the norm of b s infty q

sur une conjecture de m kac we consider the following heat conduction problem let k be a compact set in euclidean space 3 suppose that k is held at the temperature 1 while the surrounding medium is at the temperature 0 at time 0 following spitzer we investigate the asymptotic behaviour of the integral e k t which represents the total energy flow in time t from the set k to the surrounding medium 3 k an asymptotic expansion is given for e k t which refines a theorem due to spitzer this expansion also verifies and improves a formal calculation of kac similar results are proved in higher dimensions up to the constant m k the quantity e k t can be interpreted as the expected value of the volume of the wiener sausage associated with k and a d dimensional brownian motion this point of view both plays a major role in the proofs and leads to a probabilistic interpretation of the different terms of the expansion

long time behavior of heat kernels of operators with unbounded drift terms abstract given a second order elliptic operator on r d with bounded diffusion coefficients and unbounded drift which is the generator of a strongly continuous semigroup on l 2 r d represented by an integral we study the time behavior of the integral kernel and prove estimates on its decay at infinity if the diffusion coefficients are symmetric a local lower estimate is also proved

trotter kato theorems for bi continuous semigroups and applications to feller semigroups abstract we give results on the convergence of bi continuous semigroups introduced and studied by kuhnemund as a consequence we obtain a lie trotter product formula and apply it to feller semigroups generated by second order elliptic differential operators with unbounded coefficients in c b r n

analytic semigroups and degenerate elliptic operators with unbounded coefficients a probabilistic approach abstract by using techniques derived from the theory of stochastic differential equations we prove that a class of second order degenerate elliptic operators having unbounded coefficients generates analytic semigroups in c b r d the space of uniformly continuous and bounded functions from r d into r

global optimization of mixed integer nonlinear programs a theoretical and computational study this work addresses the development of an efficient solution strategy for obtaining global optima of continuous integer and mixed integer nonlinear programs towards this end we develop novel relaxation schemes range reduction tests and branching strategies which we incorporate into the prototypical branch and bound algorithm in the theoretical algorithmic part of the paper we begin by developing novel strategies for constructing linear relaxations of mixed integer nonlinear programs and prove that these relaxations enjoy quadratic convergence properties we then use lagrangian linear programming duality to develop a unifying theory of domain reduction strategies as a consequence of which we derive many range reduction strategies currently used in nonlinear programming and integer linear programming this theory leads to new range reduction schemes including a learning heuristic that improves initial branching decisions by relaying data across siblings in a branch and bound tree finally we incorporate these relaxation and reduction strategies in a branch and bound algorithm that incorporates branching strategies that guarantee finiteness for certain classes of continuous global optimization problems in the computational part of the paper we describe our implementation discussing wherever appropriate the use of suitable data structures and associated algorithms we present computational experience with benchmark separable concave quadratic programs fractional 0 1 programs and mixed integer nonlinear programs from applications in synthesis of chemical processes engineering design just in time manufacturing and molecular design

growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids the growth rates of scalar gradients are studied numerically and analytically in a family of two dimensional 2d incompressible fluid equations related to the surface quasigeostrophic sqg equation the active scalar is related to the stream function by 2 0 2 a notable difference is observed in a comparison of the instantaneous growth rates in l p and in l norms depending on the stage of the time evolution the crux is the phase shift effect of singular integral operators which displaces the peak location of the scalar gradient from that of the strain rate on this basis a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow up moreover it is found in the long time evolution that a solution of the sqg equation whose regularity is not known is less singular than that of the 2d euler equations known to be regular on the time interval covered by this computation this consistently expands an earlier observation by majda and tabak physica d 98 515 1996 in some detail a 1d model problem is discussed to illustrate the present method and extensions to the 3d case are also are briefly discussed

an effective immune based two phase approach for the optimal reliability redundancy allocation problem abstract highly reliable systems can reduce loss of money and time in practice system reliability can be enhanced by i increasing component reliabilities and or ii providing redundancy at the component level a trade off between these two options is necessary for nonlinear constrained reliability optimization the problem of maximizing system reliability through component reliability choices and component redundancy is called as reliability redundancy allocation problem and it is a difficult but realistic nonlinear mixed integer optimization problem in this paper under nonlinear constraints of weight cost and volume we propose a new immune based two phase approach to solve the reliability redundancy allocation problem in the first phase an immune based algorithm ia is developed to solve the allocation problem and in the second phase we present a new procedure to improve the solutions by ia numerical results of four benchmark problems are reported and compared as shown the solutions by the new proposed approach are all superior to those best solutions by typical approaches in the literature

an existence theorem for the navier stokes flow in the exterior of a rotating obstacle we consider the three dimensional navier stokes initial value problem in the exterior of a rotating obstacle it is proved that a unique solution exists locally in time if the initial velocity possesses the regularity l 1 2 this regularity assumption is the same as that in the famous paper of fujita kato an essential step for the proof is the deduction of a certain smoothing property together with estimates near t 0 of the semigroup which is not an analytic one generated by the operator in the space l 2 where stands for the angular velocity of the rotating obstacle and p denotes the projection associated with the helmholtz decomposition

power variation for a class of stationary increments levy driven moving averages in this paper we present some new limit theorems for power variation of kkth order increments of stationary increments levy driven moving averages in the infill asymptotic setting where the sampling frequency converges to zero while the time span remains fixed the asymptotic theory gives novel results which partially have no counterpart in the theory of discrete moving averages more specifically we show that the first order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k 1k 1 the considered power p 0p 0 the blumenthal getoor index 0 2 0 2 of the driving pure jump levy process ll and the behaviour of the kernel function gg at 00 determined by the power first order asymptotic theory essentially comprises three cases stable convergence towards a certain infinitely divisible distribution an ergodic type limit theorem and convergence in probability towards an integrated random process we also prove a second order limit theorem connected to the ergodic type result when the driving levy process ll is a symmetric stable process we obtain two different limits a central limit theorem and convergence in distribution towards a k k stable totally right skewed random variable

asymptotic expansion of solutions to the drift diffusion equation with fractional dissipation abstract the initial value problem for the drift diffusion equation arising from the model of semiconductor device simulations is studied the dissipation on this equation is given by the fractional laplacian 2 large time behavior of solutions to the drift diffusion equation with 0 1 is discussed when 1 large time behavior of solutions is known however when 0 1 the perturbation methods used in the preceding works would not work in this paper the asymptotic expansion of solutions with high order is derived

regularity theory for general stable operators parabolic equations abstract we establish sharp interior and boundary regularity estimates for solutions to t u l u f t x in i with i r and r n the operators l we consider are infinitesimal generators of stable levy processes these are linear nonlocal operators with kernels that may be very singular on the one hand we establish interior estimates obtaining that u is c 2 s in x and c 1 2 s in t whenever f is c in x and c 2 s in t in the case f l we prove that u is c 2 s in x and c 1 in t for any 0 on the other hand we study the boundary regularity of solutions in c 1 1 domains we prove that for solutions u to the dirichlet problem the quotient u d s is holder continuous in space and time up to the boundary where d is the distance to this is new even when l is the fractional laplacian