transience recurrence and local extinction properties of the support for supercritical finite measure valued diffusions whose log laplace equation is associated with the semilinear equation ut lu flu au2 where a 8 0 and l 1 ed j l aij d92 xxi dxj ed 1 bi d 8xi a path x is said to survive if x t 0 for all t 2 0 since 0 p x q survives 0 for all 0 w u e4 rd where rd denotes the space of finite measures on rd we define transience recurrence and local extinction for the support of the supercritical measure valued diffusion starting from a finite measure as follows the support is recurrent if p x t b 0 for some t 2 0 i x survives 1 for every 0 u ej rd and every open set b c rd for d 2 the support is transient if p x t b 0 for some t 2 0 i x survives 1 for every ut ej rd and bounded b c rd which satisfy supp u n b 0 a similar definition taking into account the topology of r1 is given for d 1 the support exhibits local extinction if for each t e 4 rd and each bounded b c rd there exists a p l almost surely finite random time b such that x t b 0 for all t 2 b criteria for transience recurrence and local extinction are developed in this paper also studied is the

the boundary condition at an impedance wall in a non uniform duct with potential mean flow the boundary condition at an impedance wall in a duct with a steady mean flow requiring the specification of the normal component of acoustic particle velocity is examined it is found that when implemented in the weak formulation of the finite element method it can be considerably simplified the boundary condition would appear to require data which includes the tangential derivative of the tangential mean flow velocity the normal derivative of the normal component of mean flow velocity and the derivatives of the mean flow density and the boundary admittance along the boundary it is shown that with suitable rearrangement the normal and tangential velocity derivatives can be eliminated as can the derivatives of the mean flow density and admittance the boundary condition becomes only slightly more complicated than the corresponding boundary condition when mean flow is absent and is no more difficult to implement requiring only local values of tangential mean flow velocity density and admittance which are already required as data for the weak formulation of the field equation

weak uniqueness for the heat equation with noise the uniqueness in law for the equation x t t 1 2 x t x t w is established for 1 2 1 the proof uses a duality technique and requires the construction of an approximating sequence of dual processes

markov snakes and superprocesses we suggest the name markov snakes for a class of path valued markov processes introduced recently by j f le gall in connection with the theory of branching measure valued processes le gall applied this class to investigate path properties of superdiffusions and to approach probabilistically partial differential equations involving a nonlinear operator v v2 we establish an isomorphism theorem which allows to translate results on continuous superprocesses into the language of markov snakes and vice versa by using this theorem we get limit theorems for discrete markov snakes

a weighted occupation time for a class of measured valued branching processes a weighted occupation time is defined for measure valued processes and a representation for it is obtained for a class of measure valued branching random motions on rd considered as a process in its own right the first and second order asymptotics are found as time t specifically the finiteness of the total weighted occupation time is determined as a function of the dimension d and when infinite a central limit type renormalization is considered yielding gaussian or asymmetric stable generalized random fields in the limit in one gaussian case the results are contrasted in high versus low dimensions

birth delay of a markov process and the stopping distributions for regular processes we consider markov processes with a fixed transition functionp r x t b and with random birth times we show that a process tilde x t tilde p can be obtained from x t p by birth delay if and only if tilde p left tilde x t in b right leqq p left x t in b right for allt andb as an application we give a new version and a new proof of the results of rost r and fitzsimmons f2 on stopping distributions of markov processes the key lemma 1 1 replaces the filling scheme used by the previous authors birth delay was considered from a different prospective in f1

ground state solutions for the electrostatic nonlinear klein gordon maxwell system abstract in this paper we study the nonlinear klein gordon equation coupled with the maxwell equation in the electrostatic case p u m 2 e 2 u f u in r 3 e e u 2 in r 3 where m e 0 benci and fortunato 2002 3 and d aprile and mugnai 2004 6 showed that for any u h 1 r 3 the second equation of problem p has a unique solution u d 1 2 r 3 the map u h 1 r 3 u d 1 2 r 3 is continuously differentiable and u e 0 furthermore we prove that max e u u u 0 where u u u 2 then we consider the ground state solution of problem p with f u u p 2 u 2 p 6

infinitely many solutions and least energy solutions for klein gordon maxwell systems with general superlinear nonlinearity abstract this paper is concerned with the following klein gordon maxwell system u v x u 2 u f x u x r 3 u 2 x r 3 where 0 is a constant v c r 3 r f c r 3 r r and f is superlinear at infinity using some weaker superlinear conditions instead of the common super cubic conditions on f we prove that the above system has 1 infinitely many solutions when v x is coercive and sign changing 2 a least energy solution when v x is positive periodic these results improve the related ones in the literature

on the maximal displacement of subcritical branching random walks we study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin for each n in mathbb n let m n be the rightmost position reached by the branching random walk up to generation n under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function we show that there exists rho 1 such that the function begin aligned g c n rho cn p m n ge cn quad hbox for each c 0 hbox and n in mathbb n end aligned satisfies the following properties there exist 0 underline delta le overline delta infty such that if c underline delta then begin aligned 0 liminf n rightarrow infty g c n le limsup n rightarrow infty g c n le 1 end aligned while if c overline delta then begin aligned lim n rightarrow infty g c n 0 end aligned moreover if the jump distribution has a finite right range r then overline delta r if furthermore the jump distribution is nearly right continuous then there exists kappa in 0 1 such that lim n rightarrow infty g c n kappa for all c underline delta we also show that the tail distribution of m sup n ge 0 m n namely the rightmost position ever reached by the branching random walk has a similar exponential decay without the cutoff at underline delta finally by duality these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior

association and random measures our point of departure is the result due to burton and waymire that every infinitely divisible random measure has the property variously known as association positive correlations or the fkg property this leads into a study of stationary associated random measures onrd we establish simple necessary and sufficient conditions for ergodicity and mixing when second moments are present we also study the second moment condition that is usually referrent to as finite susceptibility as one consequence of these results we can easily rederive some central limit theorems of burton and waymire using association techniques we obtain a law of the iterated logarithm for infinitely divisible random measures under simple moment hypotheses finally we apply these results to a class of stationary random measures related to measure valued markov branching processes

temporal monotonicity of the solutions of some semilinear parabolic equations with fractional diffusion abstract suppose that the functions g and are nonnegative and satisfy suitable regularity conditions then we prove in this work that the parabolic semilinear problem t u t x u t x g x f u t x x t 0 x r n u 0 x x x r n has a unique positive and time monotone solution here is the fractional laplacian with 0 2 and the source term f is a convex function with f 0 0 moreover using the temporal monotonicity we show that the elliptic equation v x g x f v x x x r d with boundary condition lim x v x 0 has a positive solution we provide also sufficient conditions for the integrability of both solutions

absolute continuity results for superprocesses with some applications let x1 and x2 be instances of a measure valued dawsonwatanabe 4 super process where the underlying spatial motions are given by a borel right process x and where the branching mechanism has finite variance a necessary and sufficient condition on x0 and x2 is found for the law of xi 2 to be absolutely continuous with respect to the law of x2 the conditions are the natural absolute continuity conditions on s but some care must be taken with the set of times s t being considered the result is used to study the closed support of super brownian motion and give sufficient conditions for the existence of a nontrivial collision measure for a pair of independent superlevy processes or more generally for a super levy process and a fixed measure the collision measure gauges the extent of overlap of the two measures as a final application we give an elementary proof of the instantaneous propagation of a super levy process to all points to which the underlying levy process can jump this result is then extended to a much larger class of superprocesses using different techniques

large time behavior of interface solutions to the heat equation with fisher wright white noise the one dimensional heat equation driven by fisher wright white noise is studied from initial conditions with compact support solutions retain this compact support and die out in finite time there exist interface solutions which change from the value 1 to the value 0 in a finite region the motion of the interface location is shown to approach that of a brownian motion under rescaling solutions with a finite number of interfaces are approximated by a system of annihilating brownian motions

sample path large deviations for super brownian motion we derive two large deviation principles of freidlin wentzell type for rescaled super brownian motion for one of the appearing rate functions an integral representation is given and interpreted as kakutani hellinger energy as a tool we develop estimates for the laplace functionals of historical super brownian motion and certain maximal inequalities also it is shown that the holder norm of index 1 2 of the processt f x t possesses some finite exponential moments provided the functionf is smooth

positive ground state solutions for the critical klein gordon maxwell system with potentials abstract this paper deals with the klein gordon maxwell system when the nonlinearity exhibits critical growth we prove the existence of positive ground state solutions for this system when a periodic potential v is introduced the method combines the minimization of the corresponding euler lagrange functional on the nehari manifold with the brezis and nirenberg technique

on the construction and support properties of measure valued diffusions on d subseteq mathbb r d with spatially dependent branching in this paper we construct a measure valued diffusion on d r d whose underlying motion is a diffusion process with absorption at the boundary corresponding to an elliptic operator l 1 2 a b on d r d and whose spatially dependent branching term is of the form x z x z 2 x d where satisfies a very general condition and 0 in the case that and are bounded from above we show that the measure valued process can also be obtained as a limit of approximating branching particle systems we give criteria for extinction survival recurrence transience of the support compactness of the support compactness of the range and local extinction for the measure valued diffusion we also present a number of examples which reveal that the behavior of the measure valued diffusion may be dramatically different from that of the approximating particle systems

local extinction versus local exponential growth for spatial branching processes let x be either the branching diffusion corresponding to the operator lu u2 u on d rd where x 0 and 0 is bounded from above or the superprocess corresponding to the operator lu u u2 on d rd with 0 and is bounded from above but no restriction on its sign let c denote the generalized principal eigenvalue for the operator l on d we prove the following dichotomy either c 0 and x exhibits local extinction or c 0 and there is exponential growth of mass on compacts of d with rate c for superdiffusions this completes the local extinction criterion obtained by pinsky ann probab 24 1996 237 267 and a recent result on the local growth of mass under a spectral assumption given by englander and turaev ann probab 30 2002 683 722 the proofs in the above two papers are based on pde techniques however the proofs we offer are probabilistically conceptual for the most part they are based on spine decompositions or immortal particle representations along with martingale convergence and the law of large numbers further they are generic in the sense that they work for both types of processes

spatial extent of branching brownian motion we study the one dimensional branching brownian motion starting at the origin and investigate the correlation between the rightmost x max geq 0 and leftmost x min leq 0 visited sites up to time t at each time step the existing particles in the system either diffuse with diffusion constant d die with rate a or split into two particles with rate b we focus on the regime b leq a where these two extreme values x max and x min are strongly correlated we show that at large time t the joint probability distribution function pdf of the two extreme points becomes stationary p x y t to infty to p x y our exact results for p x y demonstrate that the correlation between x max and x min is nonzero even in the stationary state from this joint pdf we compute exactly the stationary pdf p zeta of the dimensionless span zeta x max x min sqrt d b which is the distance between the rightmost and leftmost visited sites this span distribution is characterized by a linear behavior p zeta sim frac 1 2 left 1 delta right zeta for small spans with delta left frac a b 1 right in the critical case delta 0 this distribution has a non trivial power law tail p zeta sim 8 pi sqrt 3 zeta 3 for large spans on the other hand in the subcritical case delta 0 we show that the span distribution decays exponentially as p zeta sim a 2 2 zeta exp left sqrt delta zeta right for large spans where a is a non trivial function of delta which we compute exactly we show that these asymptotic behaviors carry the signatures of the correlation between x max and x min finally we verify our results via direct monte carlo simulations

frequency domain prediction of turbofan noise radiation this paper describes a frequency domain numerical method for predicting noise radiation from ducted fans including acoustic treatment and non uniform background flow effects the method solves the euler equations linearized about a mean flow in the frequency domain a pseudo time derivative term is added to the frequency domain equations so that a time marching technique can be employed to drive the acoustic field to steady state explicitly this approach makes distributed parallel computing more viable for equations of this type and will allow for future use of well known convergence acceleration techniques such as multigrid to obtain the solutions efficiently simulations of the jt15d static test inlet are performed including the effects of liners and the results are compared with experimental data a generic engine geometry is used for demonstrating further the prediction capability of the code calculating the attenuation effects of different liner impedances and liner installation locations on the radiated sound fields

the support of measure valued branching processes in a random environment we consider the one dimensional catalytic branching process introduced by dawson and fleischmann which is a modification of the super brownian motion the catalysts are given by a nonnegative infinitely divisible random measure with independent increments we give sufficient conditions for the global support of the process to be compact and sufficient conditions for noncompact global support since the catalytic process is related to the heat equation compact support may be surprising on the other hand the super brownian motion has compact global support we find that all nonnegative stable random measures lead to compact global support and we give an example of a very rarified levy process which leads to noncompact global support

a phase transition for measure valued sir epidemic processes we consider measure valued processes xt that solve the following martingale prob lem for a given initial measure x0 and for all smooth compactly supported test functions xt x0 1 z t 2 ds such processes arise as scaling limits of sir epidemic models we show that there exist critical values c d 0 for dimensions d 2 3 such that if c d then the solution survives forever with positive probability but if c d then the solution dies out in finite time with probability 1 for d 1 we prove that the solution dies out almost surely for all values of we also show that in dimensions d 2 3 the process dies out locally almost surely for any value of that is for any compact set k the process xt k 0 eventually

regular transition functions and regular superprocesses the class of regular markov processes is very close to the class of right processes studied by meyer getoor and others we say that a transition function p is regular if it is the transition function of a well defined regular markov process a characterization of regular transition functions is given which implies that if p is regular then the dawson watanabe and the flemingviot supertransition functions over p belong to the same class

on the range of lattice models in high dimensions we investigate the scaling limit of the range the set of visited vertices for a general class of critical lattice models starting from a single initial particle at the origin conditions are given on the random sets and an associated ancestral relation under which conditional on longterm survival the rescaled ranges converge weakly to the range of super brownian motion as random sets these hypotheses also give precise asymptotics for the limiting behaviour of the probability of exiting a large ball that is for the extrinsic one arm probability we show that these conditions are satisfied by the voter model in dimensions d ge 2 sufficiently spread out critical oriented percolation and critical contact processes in dimensions d 4 and sufficiently spread out critical lattice trees in dimensions d 8 the latter result proves conjecture 1 6 of van der hofstad et al ann probab 45 278 376 2017 and also has important consequences for the behaviour of random walks on lattice trees in high dimensions

ground state solutions for generalized quasilinear schrodinger equations with variable potentials and berestycki lions nonlinearities by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature by introducing some new tricks we prove that the following generalized quasilinear schrodinger equation div g2 u u g u g u u 2 v x u f u x rn admits two classes of ground state solutions under the general berestycki lions assumptions on the nonlinearity f which are almost necessary conditions as well as some weak assumptions on the potential v moreover we also give a minimax characterization of the ground state energy our results improve and complement the previous ones in the literature

superprocesses and their linear additive functionals let x x t p be a measure valued stochastic process linear functionals of x are the elements of the minimal closed subspace l of l 2 p which contains all x t b with x t b 2 dp we represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their laplace transforms we consider markov processes with nonstationary transition functions to reveal better the principal role of the backward equations

fundamental problems with the model of uniform flow over acoustic linings abstract the walls of ducts containing flow are often acoustically lined in order to reduce sound many simple models of acoustic linings assume the lining to be linear and locally reacting examples considered here include the three parameter mass spring damper helmholtz resonator and enhanced helmholtz resonator models all of these models have been found to have stability issues with uniform mean grazing flow and there has been some confusion over the existence of hydrodynamic instability surface waves over such linings mathematically the standard proven briggs bers stability analysis is not applicable computationally the hydrodynamic modes are routinely ignored in the frequency domain and instabilities filtered out in the time domain this confusion also causes significant problems for mode matching green functions and scattering analyses in this paper it is shown that any situation not capable of being analysed using the briggs bers criterion is illposed and that this is the root cause of the confusion over hydrodynamic instabilities a large class of lining models including all those mentioned above are shown to be illposed with uniform mean flow an explanation is given of the effects of this illposedness in practice and it is argued that illposed models should not be used an alternative stability criterion to briggs bers has been occasionally used in the literature this alternative criterion involving analytic continuation to purely imaginary frequencies was recently christened the crighton leppington stability criterion however this stability criterion is incorrect and several counter examples are given here

time domain numerical simulation of a flow impedance tube an efficient method has been developed for the application of the surface acoustic impedance condition in time domain solutions of aeroacoustic problems such as the broadband frequency simulation of a flow impedance tube the basis for this method is the standard impedance condition stated in the frequency domain as the particle displacement continuity equation the development of the time domain impedance condition follows the relations among the frequency z and discrete time domains and a rational function representation of the impedance in thez domain the resultant impedance condition consists of finite infinite impulse response type digital filter operations in the time domain which is very suitable to computational aeroacoustics algorithms this paper describes the present approach and discusses the time domain numerical simulations of the nasa langley flow impedance tube with a constant depth ceramic tubular liner both single and broadband frequency simulations are performed excellent agreement is shown with experimental data at various frequencies and flow conditions

conditioned super brownian motion we investigate classes of conditioned super brownian motions namely h transformsp h with non negative finitely based space time harmonic functionsh t we prove thath h is the unique solution of a martingale problem with interaction and is a weak limit of a sequence of rescaled interacting branching brownian motions we identify the limit behaviour of h transforms with functionsh t h t 1 depending only on the total mass 1 using the palm measures of the super brownian motion we describe for an additive spacetime harmonic functionh t h t x dx theh transformp h as a conditioned super brownian motion in which an immortal particle moves like an h transform of brownian motion

a phase diagram for a stochastic reaction diffusion system in this paper a stochastic reaction diffusion system is considered which models the spread of a finite population reacting with a non renewable resource in the presence of individual based noise a two parameter phase diagram is established to describe the large time evolution distinguishing between certain death or possible life of the population

far field calculations for turbofan noise the development of variable order mapped infinite wave envelope elements for finite element modeling of acoustic radiation from turbofan nacelles is reviewed these elements can be used outside of the near field as a nonreflecting boundary condition and as a finite dimensional discretization of the infinite exterior domain for calculation of far field acoustic radiation the kirchhoff method previously has been investigated in the context of the turbofan inlet radiation problem to extend a near field solution to the far field when the results of the mapped infinite wave envelope element analysis are compared with results of the kirchhoff method it is found that the mapped element formulation efficiently provides the acoustic far field obviating the need to carry out an additional kirchhoff integral formulation relative efficiencies are considered and it is found that finite dimensional discretization of the far field using mapped infinite elements is superior to the kirchhoff formulation when radiation directivity in the entire far field is required

on the large time growth rate of the support of supercritical super brownian motion consider the supercritical super brownian motion x t on r d corresponding to the evolution equation u t d 2 u u u 2 we obtain rather tight bounds on p x s b c n 0 0 for all s 0 t and on p x t b c n 0 0 for large n where p denotes the measure corresponding to the supercritical super brownian motion starting from the finite measure b n 0 r d denotes the ball of radius n centered at the origin and b c n 0 denotes its complement in particular we show for example that if is a compactly supported finite measure on r d then formula chem and formula chem

ground state sign changing solutions for kirchhoff type problems in bounded domains abstract in the present paper we consider the existence of ground state sign changing solutions for a class of kirchhoff type problems 0 1 a b u 2 d x u f u x u 0 x where r n is a bounded domain with a smooth boundary n 1 2 3 a 0 b 0 and f c r r under some weak assumptions on f with the aid of some new analytical skills and non nehari manifold method we prove that 0 1 possesses one ground state sign changing solution u b and its energy is strictly larger than twice that of the ground state solutions of nehari type furthermore we establish the convergence property of u b as the parameter b 0 our results improve and generalize some results obtained by w shuai 2015 34

super brownian motion path properties and hitting probabilities sample path properties of super brownian motion including a one sided modulus of continuity and exact hausdorff measure function of the range and closed support are obtained analytic estimates for the probability of hitting balls lead to upper bounds on the hausdorff measure of the set of k multiple points and a sufficient condition for a set to be polar

path processes and historical superprocesses a superprocessx over a markov process can be obtained by a passage to the limit from a branching particle system for which describes the motion of individual particles the historical process hat xi for is the process whose state at timet is the path of over time interval 0 t the superprocess hat x over hat xi the historical superprocess over reflects not only the particle distribution at any fixed time but also the structure of family trees the principal property of a historical process hat xi is that hat xi s is a function of hat xi t for alls t every process with this property is calleda path process we develop a theory of superprocesses over path processes whose core is the integration with respect to measure functionals by applying this theory to historical superprocesses we construct the first hitting distributions and prove a special markov property for superprocesses

a three dimensional hybrid finite element spectral analysis of noise radiation from turbofan inlets this paper describes a new three dimensional 3d analysis of tonal noise radiated from non axisymmetric turbofan inlets the novelty of the method is in combining a standard finite element discretisation of the acoustic field in the axial and radial coordinates with a fourier spectral representation in the circumferential direction the boundary conditions at the farfield fan face and acoustic liners are treated using the same spectral representation the resulting set of discrete acoustic equations are solved employing the well established bicgstab or qmr iterative algorithms and a very effective specialised preconditioner based on the axisymmetric mean geometry and flow field numerical examples demonstrate the suitability of the new method to engine configurations with realistic 3d features such as relatively large degrees of asymmetry and spliced acoustic liners the examples also illustrate the two advantages of the new method over a traditional 3d finite element approach the new method requires a significantly smaller number of unknowns as relatively few circumferential fourier modes in the spectral solution ensure an accurate field representation also due to the effective preconditioner the spectral linear solver benefits from stable iterations at a high rate of convergence

a super brownian motion with a locally infinite catalytic mass a super brownian motion x in bbb r with hyperbolic branching rate varrho 2 b 1 b 2 b in bbb r is constructed which symbolically could be described by the formal stochastic equation with a space time white noise rm d w starting at x 0 delta a a ne 0 this superprocess x will never hit the catalytic center there is an increasing sequence of brownian stopping times tau n strictly smaller than the hitting time of 0 such that with probability one dynkin s stopped measures x tau n vanish except for finitely many n

existence of multiple solutions of kirchhoff type equation with sign changing potential abstract in this paper we study the existence of infinitely many nontrivial solutions for a class of nonlinear schrodinger kirchhoff type equation a b r n u 2 dx u v x u f x u in r n u x 0 as x where constants a 0 b 0 and the potential v x is allowed to be sign changing under general superlinear assumption on nonlinearity f x u we establish the existence of infinitely many solutions via variational methods which unifies and improves the recent results of wu 2011 9

effect of noise on front propagation in reaction diffusion equations of kpp type we consider reaction diffusion equations of kpp type in one spatial dimension perturbed by a fisher wright white noise under the assumption of uniqueness in distribution examples include the randomly perturbed fisher kpp equations partial tu partial x 2u u 1 u epsilon sqrt u 1 u dot w and partial tu partial x 2u u 1 u epsilon sqrt u dot w where dot w dot w t x is a space time white noise we prove the brunet derrida conjecture that the speed of traveling fronts for small e is 2 pi 2 log epsilon 2 2 o log log epsilon log epsilon 3

persistence of critical multitype particle and measure branching processes we consider a class of systems of particles ofk types inr d undergoing spatial diffusion and critical multitype branching where the diffusions the particle lifetimes and the branching laws depend on the types we prove persistence criteria for such systems and for their corresponding high density limits known as multitype dawson watanabe processes the main tool is a representation of the palm distributions for a general class of inhomogeneous critical branching particle systems constructed by means of a backward tree

a probabilistic approach to one class of nonlinear differential equations we establish connections between positive solutions of one class of nonlinear partial differential equations and hitting probabilities and additive functionals of superdiffusion processes as an application we improve results on superprocesses by using the recent progress in the theory of removable singularities for differential equations

a note on superprocesses subject to a mild restriction ona generator of the one particle motion we show thea fleming viot superprocess can be obtained from thea dawson watanabe superprocess by conditioning the latter to have constant total mass

on critical klein gordon maxwell systems with super linear nonlinearities abstract this paper is concerned with two classes of critical klein gordon maxwell systems as follows u v x u 2 u f u u 5 x r 3 u 2 x r 3 and e 2 u v x u 2 u g x u k x u 5 x r 3 u 2 x r 3 where 0 e 0 v k c r 3 r f c r r and g c r 3 r r are super linear at infinity instead of the expression sufficiently large in the existing works we give a certain range 0 under which the first system admits a ground state solution when v is positive and periodic under some mild conditions on v k and g we show that the second system has at least one nontrivial solution provided e 0 e 0 where the bound e 0 0 is expressed explicitly in terms of v k and g

a phase transition for a stochastic pde related to the contact process we consider the one dimensional heat equation with a semilinear term and with a nonlinear white noise term r durrett conjectured that this equation arises as a weak limit of the contact process with longrange interactions we show that our equation possesses a phase transition to be more precise we assume that the initial function is nonnegative with bounded total mass if a certain parameter in the equation is small enough then the solution dies out to 0 in finite time with probability 1 if this parameter is large enough then the solution has a positive probability of never dying out to 0 this result answers a question of durett

on the supports of measure valued critical branching brownian motion on etudie la taille globale et la localisation des supports des etats du processus de markov a valeur de mesure qui est un analogue du mouvement brownien ramifie critique classique sur l espace euclidien a d dimensions

on the topological boundary of the range of super brownian motion we show that if partial mathcal r is the boundary of the range of super brownian motion and dim denotes hausdorff dimension then with probability one for any open set u u cap partial mathcal r neq varnothing implies operatorname dim u cap partial mathcal r begin cases 4 2 sqrt 2 approx1 17 quad text if d 2 frac 9 sqrt 17 2 approx2 44 quad text if d 3 end cases this improves recent results of the last two authors by working with the actual topological boundary rather than the boundary of the zero set of the local time and establishing a local result for the dimension

conditioned limit theorems of stopped critical branching bessel processes we consider critical branching bessel processes initially at r 1 and stopped at r 1 let n be the number of descendants hitting r 1 we give the norming constant k r and prove convergence as r of n k r conditioned on n 0 the distribution of conditioned limit laws is also investigated a feature of this study is an interplay between probabilistic insights and analytic techniques for emden fowler s equation

natural linear additive functionals of superprocesses we investigate natural linear additive nla functionals of a general critical xi k psi superprocess x we prove that all of them have only fixed discontinuities all homogeneous nla functionals of time homogeneous superprocesses are continuous this was known before only in the case of quadratic branching we introduce an operator mathscr e u defined in terms of xi k psi and we prove that the potential h and the log potential u of a nla functional a are connected by the equation u mathscr e u h the potential is always an exit rule for xi and the condition h mathscr e h infty a e is sufficient for an exit rule h to be a potential in an accompanying paper these results are applied to boundary value problems for partial differential equations involving nonlinear operator lu u alpha where l is a second order elliptic differential operator and 1 alpha leq 2

infinitely many solutions for a nonlinear klein gordon maxwell system abstract in this paper a nonlinear klein gordon maxwell system with solitary wave solution is considered using critical point theory we establish sufficient conditions for the existence of infinitely many solitary wave solutions results obtained complement and improve the existing ones

tree based models for random distribution of mass a mathematical model for distribution of mass ind dimensional space based upon randomly embedding random trees into space is introduced and studied the model is a variant of thesuper brownian motion process studied by mathematicians we present calculations relating to i the distribution of position of a typical mass element ii moments of the center of mass iii large deviation behavior and iv a recursive self similarity property

existence of multiple solutions for modified schrdingerkirchhoffpoisson type systems via perturbation method with sign changing potential in this paper we prove the existence of positive solutions and negative solutions for the following modified schrdingerkirchhoffpoisson type systems a br3u2 u v x u u12u u2 f x u inr3 u2 inr3 where a 0 b0 v and f are continuous functions and v x is allowed to be sign changing under some certain assumptions on v and f we prove the existence of nontrivial nonnegative solutions nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method

large deviations for the three dimensional super brownian motion let t dx denote a three dimensional super brownian motion with deterministic initial state 0 dx dx the lebesgue measure let v r 3 r be holder continuous with compact support not identically zero and such that r 3v x dx 0 we show that log p 0 t r 3v x s dx ds bt 3 4 is of order t 1 2 as t for b 0 this should be compared with the known result for the case r 3v x dx 0 in that case the normalization bt 3 4 b 0 must be replaced by bt b r 3v x dx in order that the same statement hold true while this result only captures the logarithmic order the method of proof enables us to obtain complete results for the corresponding moderate deviations and central limit theorems

existence and non existence of solitary waves for the critical klein gordon equation coupled with maxwell s equations abstract in this paper we investigate existence and non existence of solutions for a nonlinear elliptic system arising from the coupling of the nonlinear klein gordon equation with the maxwell equations when the nonlinearity exhibits critical growth we prove a non existence result by a poho aev type argument and then adding a suitable perturbation we recover existence of at least a radially symmetric solution we overcome the lack of compactness relying on the brezis nirenberg method

some local approximations of dawson watanabe superprocesses let be a dawson watanabe superprocess in rd such that t is a s locally finite for every t 0 then for d 2 and fixed t 0 the singular ran dom measure t can be a s approximated by suitably normalized restrictions of lebesgue measure to the neighborhoods of suppff when d 3 the local distributions of near a hitting point can be approximated in total vari ation by those of a stationary and self similar pseudo random measure f by contrast the corresponding distributions for d 2 are locally invariant fur ther results include improvements of some classical extinction criteria and some limiting properties of hitting probabilities our main proofs are based on a detailed analysis of the historical structure of f

infinitely many standing wave solutions for the nonlinear klein gordon maxwell system with sign changing potential in this paper we study the infinitely many solutions for the nonlinear klein gordon maxwell system email protected v x u email protected email protected fu f x u email protected r 3 email protected email protected u 2 email protected r 3 where w 0 is a constant u f r 3 r the potential v x is allowed to be sign changing and the primitive of the nonlinearity f is of super linear growth near infinity our super linear conditions are weaker than the usual super linear conditions

occupation time large deviations for critical branching brownian motion super brownian motion and related processes we derive a large deviation principle for the occupation time functional acting on functions with zero lebesgue integral for both super brownian motion and critical branching brownian motion in three dimensions our technique based on a moment formula of dynkin allows us to compute the exact rate functions which differ for the two processes obtaining the exact rate function for the super brownian motion solves a conjecture of lee and remillard we also show the corresponding clt and obtain similar results for the superprocesses and critical branching process built over the symmetric stable process of index in r d with d 2 2 d

regularity and irregularity of bolds 1 beta stable super brownian motion this paper establishes the continuity of the density of 1 stable super brownian motion 0 1 for fixed times in d 1 and local unboundedness of the density in all higher dimensions where it exists we also prove local unboundedness of the density in time for a fixed spatial parameter in any dimension where the density exists and local unboundedness of the occupation density the local time in the spatial parameter for dimensions d 2 where the local time exists

the dimension of the boundary of super brownian motion we show that the hausdorff dimension of the boundary of d dimensional super brownian motion is 0 if d 1 4 2 sqrt 2 if d 2 and 9 sqrt 17 2 if d 3

decay of mass for the equation ut δu a x up u q abstract consider the quasilinear cauchy problem u t u a x u p u q x r d t 0 u x 0 x 0 x r d where a 0 p and q satisfy p 0 and q 1 or p 1 and q 0 and 0 l 1 r d c 3 b r d this paper proves that the above equation possesses a unique positive classical solution and then investigates whether or not limt rd u x t dx 0 in particular it is shown that if a is on the order x m for large x then 0 if dp d 1 q d 2 m under the assumption that for compactly supported u e t ec t 1 for some 0 and for e 0 1 where ue denotes the solution to the above equation with initial condition e it is shown that 0 if dp 2 q d 2 max m d for a certain range of the parameters d p q m it is proved that holds with d 1 2 and for many other parameter values it is proved that holds with d 2 note that if d 1 2 then the above condition for 0 becomes dp d 1 q d 2 max m d which in light of the parameter restrictions is equivalent to dp d 1 q d 2 m

polylinear additive functionals of superprocesses abstract letxbe a superdiffusion in a domaineof r d a polylinear additive functional ofxcorresponding to a positive borel function is given by the formula a b b dt 1 dt k e k t 1 z 1 t k z k x t 1 dz 1 x t k dz k by a passage to the limit we extend this definition to a certain class of generalized functions more precisely we associate an additive functionala of xt p with every measure subject to the condition dt 1 dz 1 dt k dz k dt 1 dz 1 dt k dz k emsp q t 1 z 1 t k dz k t 1 z 1 t k dz k for a certain kernelq we prove that if t ti s 0 fori 1 kand for alls then measurea has a s the same property this result is applicable in particular to self intersection local times ofx

on regularity of superprocesses three theorems on regularity of measure valued processesx with branching property are established which improve earlier results of fitzsimmons f1 and the author d5 the main difference is that we treatx as a family of random measures associated with finely open setsq in time space heuristically x describes an evolution of a cloud of infinitesimal particles to everyq there corresponds a random measurex which arises if each particle is observed at its first exit time fromq the statex t at a fixed timet is a particular case we consider a monotone increasing familyq t of finely open sets and we establish regularity properties of bar x t x tau t as a function oft the results are used in d6 d7 and d10 for investigating the relations between superprocesses and non linear partial differential equations basic definitions on markov processes and superprocesses are introduced in sect 1 the next three sections are devoted to proving the regularity theorems they are applied in sect 5 to study parts of superprocess the relation to the previous work is discussed in more detail in the concluding section it may be helpful to look briefly through this section before reading sects 2 5

extension of kirchhoff s formula to radiation from moving surfaces kirchhoff s formula for radiation from a closed surface has been used recently for prediction of the noise of high speed rotors and propellers because the closed surface on which the boundary data are prescribed in these cases is in motion an extension of kirchhoff s formula to this condition is required in this paper such a formula obtained originally by morgans for the interior problem is derived for regions exterior to surfaces moving at speeds below the wave propagation speed by making use of some results of generalized function theory it is shown that the usual kirchhoff formula is a special case of the main result of the paper the general result applieds to a deformable surface however the special form it assumes for a rigid surface in motion is also noted some possible areas of application of the formula to problems of current interest in aeroacoustics are discussed

two representations of a conditioned superprocess we consider a class of measure valued markov processes constructed by taking a superprocess over some underlying markov process and conditioning it to stay alive forever we obtain two representations of such a process the first representation is in terms of an immortal particle that moves around according to the underlying markov process and throws off pieces of mass which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess as a consequence of this representation we show that the tail field of the conditioned superprocess is trivial if the tail field of the underlying process is trivial the second representation is analogous to one obtained by legall in the unconditioned case it represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process this representation is useful for studying the transience and recurrence properties of the closed support process

nonlinear parabolic p d e and additive functionals of superdiffusions suppose that e is an arbitrary domain in r d l is a second order elliptic differential operator in s r x e and s e is the extremal part of the martin boundary for the corresponding diffusion let 1 2 we investigate a boundary value problem u r lu u in s u v on s e u 0 on x e involving two measures and for the existence of a solution we give sufficient conditions in terms of a martin capacity and necessary conditions in terms of hitting probabilities for an l superdiffusion x if a solution exists then it can be expressed by an explicit formula through an additive functional a of x an l superdiffusion is a branching measure valued process a natural linear additive nla functional a of x is determined uniquely by its potential h defined by the formula p a 0 h r x dr dx for all m the determining set of a every potential h is an exit rule for and it has a unique decomposition into extremal exit rules if and are measures which appear in this decomposition then can be replaced by an integral equation u r x p r x t dy u t y ds h r x where p r x t dy is the transition function of we prove that h is the potential of a nla functional if and only if has a solution u moreover u r x log p r x e a 0 by applying these results to homogeneous functionals of time homogeneous superdiffusions we get a stronger version of theorems proved in an earlier publication the foundation for our present investigation is laid by a general theory developed in the accompanying paper

a new approach to the single point catalytic super brownian motion a new approach is provided to the super brownian motionx with a single point catalyst c as branching rate we start from a superprocessu with constant branching rate and spatial motion given by the 1 2 stable subordinator we prove that the occupation density measure c ofx at the catalystc is distributed as the total occupation time measure ofu furthermore we show thatxt is determined from c by an explicit representation formula heuristically a mass c ds of particles leaves the catalyst at times and then evolves according to ito s brownian excursion measure as a consequence of our representation formula the density fieldx ofx satisfies the heat equation outside ofc with a noisy boundary condition atc given by the singularly continuous random measure c in particular x isc outside the catalyst we also provide a new derivation of the singularity of the measure c

numerical prediction of aft radiation of turbofan tones through exhaust jets this paper describes a numerical methodology for calculating tonal noise propagation and radiation through turbomachinery exhaust ducts including non uniform background jet flows the numerical method is based on solution of the linearized euler equations directly in the frequency domain employing a direct sparse matrix solver in parallel acoustic sources are introduced into the computational domain via the perfectly matched layer equations various test cases including propagation through infinite ducts and propagation and radiation through semi infinite ducts with and without liners are solved and results are compared with analytical solutions it is demonstrated that solutions even with thin shear layers separating the exterior and jet streams at reasonably high mach numbers can be obtained quite successfully the method is also validated by simulating the radiated sound waves from two actual engine exhaust cowl geometries and the results are compared with experimental data

ground state solutions of schrodinger poisson systems with variable potential and convolution nonlinearity abstract in the present paper we consider the following nonlinear schrodinger poisson system with convolution nonlinearity u v x u u i f u f u x r 3 u 2 x r 3 where 0 3 i r 3 r is the riesz potential v c r 3 0 f c r r and f t 0 t f s d s satisfies lim t f t t with min 2 6 4 by using some new analytic techniques and new inequalities we prove the above system admits a ground state solution under mild assumptions on v and f in particular our results cover and improve the existing ones for the schrodinger poisson system which can be considered as the limited problem when 0

a space time property of a class of measure valued branching diffusions si d on montre que la diffusion ramifiee d dimensionnelle d indice etudiee par dawson et d autres distribue sa masse sur un support aleatoire de facon uniforme relativement a la mesure de hausdorff