on the invariant mass conjecture in general relativity an asymptotic symmetries theorem is proved under certain hypotheses on the behaviour of the metric at spatial infinity this implies that the einstein von freud adm mass can be invariantly assigned to an asymptotically flat four dimensional end of an asymptotically empty solution of einstein equations if the metric is a no radiation metric or if the end is defined in terms of a collection of boost type domains

fuzzy nilpotent algebra abstract we define the concept of fuzzy nilpotent algebra prove that the homomorphic inverse image of a fuzzy nilpotent algebra is also nilpotent and study the intersection and union of fuzzy nilpotent algebras

on witten s proof of the positive energy theorem this paper gives a mathematically rigorous proof of the positive energy theorem using spinors this completes and simplifies the original argument presented by edward witten we clarify the geometric aspects of this argument and prove the necessary analytic theorems concerning the relevant dirac operator

on the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes let x t be a diffusion process on r d with generator l 1 2 nabla cdot a nabla b nabla and let p x x in r d be the corresponding measures on paths pick 0 s which are always satisfied if a 1 b is a gradient function we show that y t s is an inhomogeneous diffusion process and that as t rightarrow infty y t s 0 leq s leq t converges to a limiting homogeneous positive recurrent diffusion y s 0 leq s leq t with state space g since t is arbitrary we actually obtain a limiting process y s 0 leq s infty the generator of the limiting process may be written in the form l g 1 2 nabla cdot a nabla b nabla a nabla g 0 g 0 nabla a nabla h g 0 nabla where g 0 is the square root of the density of a measure mu 0 which minimizes the i function for the process over all gamma in mathscr p bar g the set of probability measures on bar g the function h g 0 appears in the explicit calculation of i mu 0 and solves a certain variational equation the invariant measure for the process is mu 0

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

positivity of the bondi energy we give a new proof that the bondi momentum is time like and future directed this is done by extending a theorem showing the existence of solutions of witten s equation on a complete asymptotically flat surface to the case where the surface is asyptotically hyperbolic and spans a cut s of i by choosing appropriate boundary conditions the positivity of the bondi energy is also shown for the case where the surface is no longer free of inner boundaries but is internally bounded by the union of a finite number of marginally trapped surfaces this argument also applies to the case of asymptotically flat surfaces

a strictly positive mass theorem we show that the adm 4 momentum of an isolated gravitational system spatially asymptotically flat spacetime satisfying the dominant energy condition cannot be null like unless it is flat together with the positive mass theorem this implies that the adm 4 momentum of an isolated gravitational system must be strictly time like

quasi stationary distributions for structured birth and death processes with mutations we study the probabilistic evolution of a birth and death continuous time measure valued process with mutations and ecological interactions the individuals are characterized by phenotypic traits that take values in a compact metric space each individual can die or generate a new individual the birth and death rates may depend on the environment through the action of the whole population the offspring can have the same trait or can mutate to a randomly distributed trait we assume that the population will be extinct almost surely our goal is the study in this infinite dimensional framework of quasi stationary distributions when the process is conditioned on non extinction we firstly show in this general setting the existence of quasi stationary distributions this result is based on an abstract theorem proving the existence of finite eigenmeasures for some positive operators we then consider a population with constant birth and death rates per individual and prove that there exists a unique quasi stationary distribution with maximal exponential decay rate the proof of uniqueness is based on an absolute continuity property with respect to a reference measure

non intersecting paths random tilings and random matrices we investigate certain measures induced by families of non intersecting paths in domino tilings of the aztec diamond rhombus tilings of an abc hexagon a dimer model on a cylindrical brick lattice and a growth model the measures obtained e g the krawtchouk and hahn ensembles have the same structure as the eigenvalue measures in random matrix theory like gue which can in fact can be obtained from non intersecting brownian motions the derivations of the measures are based on the karlin mcgregor or lindstrom gessel viennot method we use the measures to show some asymptotic results for the models

exponential convergence to quasi stationary distribution and q process for general almost surely absorbed markov processes we obtain necessary and sufficient conditions for exponential convergence to a unique quasi stationary distribution in the total variation norm these conditions also ensure the existence and exponential ergodicity of the q process the process conditioned to never be absorbed we apply these results to one dimensional birth and death processes with catastrophes multi dimensional birth and death processes infinite dimensional population models with brownian mutations and neutron transport dynamics absorbed at the boundary of a bounded domain

quasistationary distributions for one dimensional diffusions with killing we extend some results on the convergence of one dimensional diffusions killed at the boundary conditioned on extended survival to the case of general killing on the interior we show under fairly general conditions that a diffusion conditioned on long survival either runs off to infinity almost surely or almost surely converges to a quasistationary distribution given by the lowest eigenfunction of the generator in the absence of internal killing only a sufficiently strong inward drift can keep the process close to the origin to allow convergence in distribution an alternative that arises when general killing is allowed is that the conditioned process is held near the origin by a high rate of killing near oo we also extend to the case of general killing the standard result on convergence to a quasistationary distribution of a diffusion on a compact interval

variational processes and stochastic versions of mechanics the dynamical structure of any reasonable stochastic version of classical mechanics is investigated including the version created by nelson e nelson quantum fluctuations princeton u p princeton nj 1985 phys rev 150 1079 1966 for the description of quantum phenomena two different theories result from this common structure one of them is the imaginary time version of nelson s theory whose existence was unknown and yields a radically new probabilistic interpretation of the heat equation the existence and uniqueness of all the involved stochastic processes is shown under conditions suggested by the variational approach of yasue k yasue j math phys 22 1010 1981

existence of quasi stationary distributions a renewal dynamical approach we consider markov processes on the positive integers for which the origin is an absorbing state quasi stationary distributions qsd s are described as fixed points of a transformation phi in the space of probability measures under the assumption that the absorption time at the origin r of the process starting from state x goes to infinity in probability as x rightarrow infty we show that the existence of a operatorname qsd is equivalent to e xe lambda r infty for some positive lambda and x we also prove that a subsequence of phi n delta x converges to a minimal operatorname qsd for a birth and death process we prove that phi n delta x converges along the full sequence to the minimal operatorname qsd the method is based on the study of the renewal process with interarrival times distributed as the absorption time of the markov process with a given initial measure mu the key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of phi mu

effect of selection on ancestry an exactly soluble case and its phenomenological generalization we consider a family of models describing the evolution under selection of a population whose dynamics can be related to the propagation of noisy traveling waves for one particular model that we shall call the exponential model the properties of the traveling wave front can be calculated exactly as well as the statistics of the genealogy of the population one striking result is that for this particular model the genealogical trees have the same statistics as the trees of replicas in the parisi mean field theory of spin glasses we also find that in the exponential model the coalescence times along these trees grow like the logarithm of the population size a phenomenological picture of the propagation of wave fronts that we introduced in a previous work as well as our numerical data suggest that these statistics remain valid for a larger class of models while the coalescence times grow like the cube of the logarithm of the population size

existence theorem for solutions of witten s equation and nonnegativity of total mass we prove that given an asymptotically flat in a very weak sense initial data set there always exists a spinor field that satisfies witten s equation and that becomes constant at infinity thus we fill a gap in witten s arguments on the nonnegativity of the total mass of an isolated system when measured at spatial infinity we also include a review of witten s argument

on ruelle s probability cascades and an abstract cavity method we construct in this work a markov process which describes a clustering mechanism through which equivalence classes on are progressively lumped together this clustering process gives a new description of ruelle s continuous probability cascades it also enables to introduce an abstract cavity method which mimicks certain features of the cavity method developed by physicists in the context of the sherrington kirkpatrick model

cauchy noise and affiliated stochastic processes by departing from the previous attempt phys rev e 51 4114 1995 we give a detailed construction of conditional and perturbed markov processes under the assumption that the cauchy law of probability replaces the gaussian law appropriate for the wiener process as the model of primordial noise all considered processes are regarded as probabilistic solutions of the so called schrodinger interpolation problem whose validity is thus extended to the jump type processes and their step process approximants

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

convergence in distribution of random metric measure spaces λ coalescent measure trees we consider the space of complete and separable metric spaces which are equipped with a probability measure a notion of convergence is given based on the philosophy that a sequence of metric measure spaces converges if and only if all finite subspaces sampled from these spaces converge this topology is metrized following gromov s idea of embedding two metric spaces isometrically into a common metric space combined with the prohorov metric between probability measures on a fixed metric space we show that for this topology convergence in distribution follows provided the sequence is tight from convergence of all randomly sampled finite subspaces we give a characterization of tightness based on quantities which are reasonably easy to calculate subspaces of particular interest are the space of real trees and of ultra metric spaces equipped with a probability measure as an example we characterize convergence in distribution for the ultra metric measure spaces given by the random genealogies of the coalescents we show that the coalescent defines an infinite random metric measure space if and only if the so called dust free property holds

estimating input parameters from intracellular recordings in the feller neuronal model we study the estimation of the input parameters in a feller neuronal model from a trajectory of the membrane potential sampled at discrete times these input parameters are identified with the drift and the infinitesimal variance of the underlying stochastic diffusion process with multiplicative noise the state space of the process is restricted from below by an inaccessible boundary further the model is characterized by the presence of an absorbing threshold the first hitting of which determines the length of each trajectory and which constrains the state space from above we compare both in the presence and in the absence of the absorbing threshold the efficiency of different known estimators in addition we propose an estimator for the drift term which is proved to be more efficient than the others at least in the explored range of the parameters the presence of the threshold makes the estimates of the drift term biased and two methods to correct it are proposed

mean instantaneous firing frequency is always higher than the firing rate frequency coding is considered one of the most common coding strategies employed by neural systems this fact leads in experiments as well as in theoretical studies to construction of so called transfer functions where the output firing frequency is plotted against the input intensity the term firing frequency can be understood differently in different contexts basically it means that the number of spikes over an interval of preselected length is counted and then divided by the length of the interval but due to the obvious limitations the length of observation cannot be arbitrarily long then firing frequency is defined as reciprocal to the mean interspike interval in parallel an instantaneous firing frequency can be defined as reciprocal to the length of current interspike interval and by taking a mean of these the definition can be extended to introduce the mean instantaneous firing frequency all of these definitions of firing frequency are compared in an effort to contribute to a better understanding of the input output properties of a neuron

ultracontractivity and the heat kernel for schrodinger operators and dirichlet laplacians abstract abstract connections between integral kernels of positivity preserving semigroups and suitable lp contractivity properties are established then these questions are studied for the semigroups generated by v and h the dirichlet laplacian for an open connected region as an application under a suitable hypothesis sobolev estimates are proved valid up to of the form x c 0 x h k 2 where 0 is the unique positive l2 eigenfunction of h

diffusion processes with singular drift fields a class of stochastic differential equations with highly singular drift fields is considered using a purely probabilistic approach we can show the unattainability of the nodal set moreover a global existence and uniqueness theorem for diffusion processes with singular drift fields is established the finite action condition of carlen and zheng can be modified we relate our results to the diffusions which describe the time evolution of quantum systems in stochastic mechanics

killed diffusions and their conditioning let x xt t 0be a diffusion determined by an elliptic differential operator l in r n n 1 for any bounded c 1 1 domain d we define the conditional killed diffusion x on d by the semigroup t t phi f x phi 0 x 1 e x left f x t phi 0 x t tau d right left t right e lambda 0 t t 0 where 0 and 0 are the principle eigenvalue and eigenfunction respectively of l on d with the dirichlet boundary condition in this paper we prove that x is a strong feller process on d and that t is strongly continuous on c d for any t 0 we consider the conditioned process x ti e the process x in d conditioned on d t and prove that x tconverges weakly to x as t without any additional hypotheses

on evaluating the donsker varadhan i function let x t be a feller process on a complete separable metric space a and consider the occupation measure l t omega cdot int t 0 chi cdot x s ds the i function is defined for mu in mathscr p a the set of probability measures on a by i mu inf u in mathscr d int a lu u d mu where l mathscr d is the generator of the process and mathscr d subset mathscr d consists of the strictly positive functions in mathscr d the i function determines the asymptotic rate of decay of p 1 t l t omega cdot in g for g subset mathscr p a the first difficulty encountered in evaluating i mu is that the domain mathscr d is generally not known explicitly in this paper we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain we then apply this general result to diffusion processes with boundaries

quasilimiting behavior for one dimensional diffusions with killing this paper extends and clarifies results of steinsaltz and evans trans amer math soc 359 2007 1285 1234 which found conditions for convergence of a killed one dimensional diffusion conditioned on survival to a quasistationary distribution whose density is given by the principal eigenfunction of the generator under the assumption that the limit of the killing at infinity differs from the principal eigenvalue we prove that convergence to quasistationarity occurs if and only if the principal eigenfunction is integrable when the killing at is larger than the principal eigenvalue then the eigenfunction is always integrable when the killing at is smaller the eigenfunction is integrable only when the unkilled process is recurrent otherwise the process conditioned on survival converges to 0 density on any bounded interval

survival of near critical branching brownian motion consider a system of particles performing branching brownian motion with negative drift mu sqrt 2 varepsilon and killed upon hitting zero initially there is one particle at x 0 kesten stoch process appl 7 9 47 1978 showed that the process survives with positive probability if and only if e 0 here we are interested in the asymptotics as e 0 of the survival probability q x it is proved that if l pi sqrt varepsilon then for all x lim e 0q l x x 0 1 exists and is a traveling wave solution of the fisher kpp equation furthermore we obtain sharp asymptotics of the survival probability when x l and l x the proofs rely on probabilistic methods developed by the authors in berestycki et al in arxiv 1001 2337 2010 this completes earlier work by harris harris and kyprianou ann inst henri poincare probab stat 42 125 145 2006 and confirms predictions made by derrida and simon europhys lett 78 60006 2007 which were obtained using nonrigorous pde methods

branching brownian motion in a strip survival near criticality we consider a branching brownian motion with linear drift in which particles are killed on exiting the interval 0 k and study the evolution of the process on the event of survival as the width of the interval shrinks to the critical value at which survival is no longer possible we combine spine techniques and a backbone decomposition to obtain exact asymptotics for the near critical survival probability this allows us to deduce the existence of a quasi stationary limit result for the process conditioned on survival which reveals that the backbone thins down to a spine as we approach criticality this paper is motivated by recent work on survival of near critical branching brownian motion with absorption at the origin by aidekon and harris near critical survival probability of branching brownian motion with an absorbing barrier 2010 unpublished manuscript as well as the work of berestycki et al ann probab 41 2013 527 618 j stat phys 143 2011 833 854

accuracy of rate coding when shorter time window and higher spontaneous activity help it is widely accepted that neuronal firing rates contain a significant amount of information about the stimulus intensity nevertheless theoretical studies on the coding accuracy inferred from the exact spike counting distributions are rare we present an analysis based on the number of observed spikes assuming the stochastic perfect integrate and fire model with a change point representing the stimulus onset for which we calculate the corresponding fisher information to investigate the accuracy of rate coding we analyze the effect of changing the duration of the time window and the influence of several parameters of the model in particular the level of the presynaptic spontaneous activity and the level of random fluctuation of the membrane potential which can be interpreted as noise of the system the results show that the fisher information is nonmonotonic with respect to the length of the observation period this counterintuitive result is caused by the discrete nature of the count of spikes we observe also that the signal can be enhanced by noise since the fisher information is nonmonotonic with respect to the level of spontaneous activity and in some cases also with respect to the level of fluctuation of the membrane potential

the isometry groups of asymptotically flat asymptotically empty space times with timelike adm four momentum we give a complete classification of all connected isometry groups together with their actions in the asymptotic region in asymptotically flat asymptotically vacuum space times with timelike adm four momentum

doob s conditioned diffusions and their lifetimes on utilise les resultats de donsker et varadhan pour montrer que la queue de la distribution de la duree de vie decroit exponentiellement

the mass of an asymptotically flat manifold we show that the mass of an asymptotically flat n manifold is a geometric invariant the proof is based on harmonic coordinates and to develop a suitable existence theory results about elliptic operators with rough coefficients on weighted sobolev spaces are summarised some relations between the mass scalar curvature and harmonic maps are described and the positive mass theorem for n dimensional spin manifolds is proved

killing vectors in asymptotically flat space times ii asymptotically translational killing vectors and the rigid positive energy theorem in higher dimensions we show that the borderline cases in the proof of the positive energy theorem for initial data sets on spin manifolds in dimensions n 3 are only possible for initial data arising from embeddings in minkowski space time

spectral analysis of a family of second order elliptic operators with nonlocal boundary condition indexed by a probability measure abstract let d r d be a bounded domain and let l 1 2 a b be a second order elliptic operator on d let be a probability measure on d denote by l the differential operator whose domain is specified by the following nonlocal boundary condition d l f c 2 d d f d f d and which coincides with l on its domain clearly 0 is an eigenvalue for l with the corresponding eigenfunction being constant it is known that l possesses an infinite sequence of eigenvalues and that with the exception of the zero eigenvalue all eigenvalues have negative real part define the spectral gap of l indexed by by 1 sup re 0 is an eigenvalue for l in this paper we investigate the eigenvalues of l in general and the spectral gap 1 in particular the operator l and its spectral gap 1 have probabilistic significance the operator l is the generator of a diffusion process with random jumps from the boundary and 1 measures the exponential rate of convergence of this process to its invariant measure

maximal hypersurfaces in stationary asymptotically flat spacetimes existence of maximal hypersurfaces and of foliations by maximal hypersurfaces is proven in two classes of asymptotically flat spacetimes which possess a one parameter group of isometries whose orbits are timelike near infinity the first class consists of strongly causal asymptotically flat spacetimes which contain no black hole or white hole but may contain ergoregions where the killing orbits fail to be timelike the second class of spacetimes possess a black hole and a white hole with the black and white hole horizons intersecting in a compact 2 surfaces

critical branching brownian motion with absorption survival probability we consider branching brownian motion on the real line with absorption at zero in which particles move according to independent brownian motions with the critical drift of sqrt 2 kesten stoch process 7 9 47 1978 showed that almost surely this process eventually dies out here we obtain upper and lower bounds on the probability that the process survives until some large time t these bounds improve upon results of kesten stoch process 7 9 47 1978 and partially confirm nonrigorous predictions of derrida and simon epl 78 60006 2007

killing vectors in asymptotically flat space times i asymptotically translational killing vectors and the rigid positive energy theorem we study killing vector fields in asymptotically flat space times we prove the following result implicitly assumed in the uniqueness theory of stationary black holes if the conditions of the rigidity part of the positive energy theorem are met then in such space times there are no asymptotically null killing vector fields except if the initial data set can be embedded in minkowski space time we also give a proof of the nonexistence of nonsingular in an appropriate sense asymptotically flat space times that satisfy an energy condition and that have a null adm four momentum under conditions weaker than previously considered

the classification of static electro vacuum space times containing an asymptotically flat spacelike hypersurface with compact interior we show that static electro vacuum black hole space times containing an asymptotically flat spacelike hypersurface with compact interior and with both degenerate and non degenerate components of the event horizon do not exist this is done by a careful study of the near horizon geometry of degenerate horizons which allows us to eliminate the last restriction of the static electro vacuum no hair theory

fuzzy algebras over fuzzy fields redefined abstract first we point out that nanda s 1986 definition of fuzzy field was irrational hence the definition of fuzzy algebras over fuzzy fields 1990 was also irrational secondly we redefined fuzzy fields and fuzzy algebras over fuzzy fields finally we prove without any restriction that homomorphic images of fuzzy algebras form are fuzzy algebra

how sample paths of leaky integrate and fire models are influenced by the presence of a firing threshold neural membrane potential data are necessarily conditional on observation being prior to a firing time in a stochastic leaky integrate and fire model this corresponds to conditioning the process on not crossing a boundary in the literature simulation and estimation have almost always been done using unconditioned processes in this letter we determine the stochastic differential equations of a diffusion process conditioned to stay below a level s up to a fixed time t1 and of a diffusion process conditioned to cross the boundary for the first time at t1 this allows simulation of sample paths and identification of the corresponding mean process differences between the mean of free and conditioned processes are illustrated as well as the role of noise in increasing these differences

feynman kac kernels in markovian representations of the schrodinger interpolating dynamics probabilistic solutions of the so called schrodinger boundary data problem provide for a unique markovian interpolation between any two strictly positive probability densities designed to form the input output statistics data for the process taking place in a finite time interval the key issue is to select the jointly continuous in all variables positive feynman kac kernel appropriate for the phenomenological physical situation we extend the existing formulations of the problem to cases when the kernel is not a fundamental solution of a parabolic equation and prove the existence of a continuous markovian interpolation in this case next we analyze the compatibility of this stochastic evolution with the original parabolic dynamics which is assumed to be governed by the temporally adjoint pair of parabolic partial differential equations and prove that the pertinent random motion is a diffusion process in particular in conjunction with born s statistical interpretation postulate in quantum theor

proof of the positive mass theorem ii the positive mass theorem states that for a nontrivial isolated physical system the total energy which includes contributions from both matter and gravitation is positive this assertion was demonstrated in our previous paper in the important case when the space time admits a maximal slice here this assumption is removed and the general theorem is demonstrated abstracts of the results of this paper appeared in 11 and 13

quasi stationary distributions for discrete state models this paper contains a survey of results related to quasi stationary distributions which arise in the setting of stochastic dynamical systems that eventually evanesce and which may be useful in describing the long term behaviour of such systems before evanescence we are concerned mainly with continuous time markov chains over a finite or countably infinite state space since these processes most often arise in applications but will make reference to results for other processes where appropriate next to giving an historical account of the subject we review the most important results on the existence and identification of quasi stationary distributions for general markov chains and give special attention to birth death processes and related models the question of under what circumstances a quasi stationary distribution given its existence is indeed a good descriptor of the long term behaviour of a system before evanescence is addressed as well we conclude with a discussion of computational aspects with more details given in a web appendix accompanying this paper

the boost problem in general relativity we show that any asymptotically flat initial data for the einstein field equations have a development which includes complete spacelike surfaces boosted relative to the initial surface furthermore the asymptotic fall off is preserved along these boosted surfaces and there exists a global system of harmonic coordinates on such a development we also extend former results on global solutions of the constraint equations by virtue of this extension the constraint and evolution parts of the problem fit together exactly several theorems are given which concern the behaviour in the large of general classes of linear and quasilinear differential systems this paper contains in addition a systematic exposition of the functional spaces employed

quasi stationary distributions and diffusion models in population dynamics in this paper we study quasi stationarity for a large class of kolmogorov diffusions the main novelty here is that we allow the drift to go to infty at the origin and the diffusion to have an entrance boundary at infty these diffusions arise as images by a deterministic map of generalized feller diffusions which themselves are obtained as limits of rescaled birth death processes generalized feller diffusions take nonnegative values and are absorbed at zero in finite time with probability 1 an important example is the logistic feller diffusion we give sufficient conditions on the drift near 0 and near infty for the existence of quasi stationary distributions as well as rate of convergence in the yaglom limit and existence of the q process we also show that under these conditions there is exactly one quasi stationary distribution and that this distribution attracts all initial distributions under the conditional evolution if and only if infty is an entrance boundary in particular this gives a sufficient condition for the uniqueness of quasi stationary distributions in the proofs spectral theory plays an important role on l 2 of the reference measure for the killed process

an invariance principle for branching diffusions in bounded domains we study branching diffusions in a bounded domain d of mathbb r d in which particles are killed upon hitting the boundary partial d it is known that any such process undergoes a phase transition when the branching rate beta exceeds a critical value a multiple of the first eigenvalue of the generator of the diffusion we investigate the system at criticality and show that the associated genealogical tree when the process is conditioned to survive for a long time converges to aldous continuum random tree under appropriate rescaling the result holds under only a mild assumption on the domain and is valid for all branching mechanisms with finite variance and a general class of diffusions

differential entropy and dynamics of uncertainty we analyze the functioning of gibbs type entropy functionals in the time domain with emphasis on shannon and kullback leibler entropies of time dependent continuous probability distributions the shannon entropy validity is extended to probability distributions inferred from l 2 r n quantum wave packets in contrast to the von neumann entropy which simply vanishes on pure states the differential entropy quantifies the degree of probability de localization and its time development the associated dynamics of the fisher information functional quantifies nontrivial power transfer processes in the mean both in dissipative and quantum mechanical cases

fractional laplacians in bounded domains killed reflected censored and taboo levy flights the fractional laplacian 2 0 2 has many equivalent albeit formally different realizations as a nonlocal generator of a family of stable stochastic processes in r n on the other hand if the process is to be restricted to a bounded domain there are many inequivalent proposals for what a boundary data respecting fractional laplacian should actually be this ambiguity not only holds true for each specific choice of the process behavior at the boundary e g absorbtion reflection conditioning or boundary taboos but extends as well to its particular technical implementation dirichlet neumann etc problems the inferred jump type processes are inequivalent as well differing in their spectral and statistical characteristics which may strongly influence the ability of the formalism if uncritically adopted to provide an unambiguous description of real geometrically confined physical systems with disorder specifically that refers to their relaxation properties and the near equilibrium asymptotic behavior in the present paper we focus on levy flight induced jump type processes which are constrained to stay forever inside a finite domain this refers to a concept of taboo processes imported from brownian to levy stable contexts to so called censored processes and to reflected levy flights whose status still remains to be unequivocally settled as a by product of our fractional spectral analysis with reference to neumann boundary conditions we discuss disordered semiconducting heterojunctions as the bounded domain problem

n 3 differential calculus and gauge theory on a reduced quantum plane we discuss the algebra of n n matrices as a reduced quantum plane a n 3 nilpotent deformed differential calculus involving a complex parameter q is constructed the two cases q 3rd and nth root of unity are completely treated as an application we establish a gauge field theory for the particular cases n 2 and n 3

equality in the spacetime positive mass theorem we affirm the rigidity conjecture of the spacetime positive mass theorem in dimensions less than eight namely if an asymptotically flat initial data set satisfies the dominant energy condition and has e p then e p 0 where e p is the adm energy momentum vector the dimensional restriction can be removed if we assume the positive mass inequality holds previously the result was only known for spin manifolds beig and chru ciel in j math phys37 4 1939 1961 1996 chru ciel and maerten in j math phys 47 2 2006

non negative feynman kac kernels in schrodinger s interpolation problem the local formulations of the markovian interpolating dynamics which is constrained by the prescribed input output statistics data usually utilize strictly positive feynman kac kernels this implies that the related markov diffusion processes admit vanishing probability densities only at the boundaries of the spatial volume confining the process we discuss an extension of the framework to encompass singular potentials and associated non negative feynman kac type kernels it allows us to deal with a class of continuous interpolations admitted by general non negative solutions of the schrodinger boundary data problem the resulting nonstationary stochastic processes are capable of both developing and destroying nodes zeros of probability densities in the course of their evolution also away from the spatial boundaries this observation conforms with the general mathematical theory due to m nagasawa and r aebi that is based on the notion of multiplicative functionals extending in turn the well know

kpp equation and supercritical branching brownian motion in the subcritical speed area application to spatial trees if r t is the position of the rightmost particle at time t in a one dimensional branching brownian motion u t x p r t x is a solution of kpp equation frac partial u partial t frac 1 2 frac partial 2 u partial x 2 f u where f u 1 u g 1 u g is the generating function of the reproduction law and the inverse of the mean lifetime if m g 1 1 and g 0 0 it is known that frac r t t xrightarrow p c 0 sqrt 2a m 1 text when t to infty for the general kpp equation we show limit theorems for u t ct c c 0 t large deviations for r t and probabilities of presence of particles for the branching process are deduced where z t denotes the random point measure of particles living at time t and a yaglom type theorem is proved the conditional distribution of the spatial tree given z t ct 0 is studied in the limit as t

the rate function of hypoelliptic diffusions let be a hypoelliptic diffusion operator on a compact manifold m given an a priori smooth reference measure on m we can then rewrite l as the sum of a symmetric part l0 and a first order drift part y the paper investigates the effect of the drift y on the donsker varadhan rate function corresponding to the large deviations of the empirical measure of the diffusion when y is in the linear span of the first and second order lie brackets of the xi s we derive an affine bound relating the rate functions associated with l and l0 as soon as one point exists where y is not in the linear span of the first and second order lie brackets of the xi s we show that such an affine bound is impossible 1994 john wiley sons inc

natural boundaries for the smoluchowski equation and affiliated diffusion processes the schrodinger problem of deducing the microscopic dynamics from the input output statistics data is known to admit a solution in terms of markov diffusion processes the uniqueness of the solution is found to be linked to the natural boundaries respected by the underlying random motion by choosing a reference smoluchowski diffusion process we automatically fix the feynman kac potential and the field of local accelerations it induces we generate the family of affiliated diffusion processes with the same local dynamics but different inaccessible boundaries on finite semi infinite and infinite domains for each diffusion process a unique feynman kac kernel is obtained by the constrained dirichlet boundary data wiener path integration as a by product of the discussion we give an overview of the problem of inaccessible boundaries for the diffusion and bring together sometimes viewed from unexpected angles results which are little known and dispersed in publications from scarcely communicating areas of mathematics and physics

avoided crossings of the quartic oscillator the phenomenon of avoided crossings of energy levels in the spectrum of quantum systems is well known however being of an exponentially small order it is hard to calculate in particular this is the case when the potential is generating a schrodinger equation of a type which is beyond the hypergeometric one recently there have been attempts to understand this phenomenon in connection with heun type differential equations the most famous example of this class is the quantum quartic oscillator which is governed by the triconfluent case of heun s differential equation in the following we consider situations where the fourth order potential has two minima and we calculate the avoided crossings of its eigenvalue curves in dependence on the asymmetry and the barrier height between the two wells the results are compared with those obtained from an asymptotic approach of the problem for large values of the control parameter that governs the barrier height

the genealogy of branching brownian motion with absorption we consider a system of particles which perform branching brownian motion with negative drift and are killed upon reaching zero in the near critical regime where the total population stays roughly constant with approximately n particles we show that the characteristic time scale for the evolution of this population is of order logn 3 in the sense that when time is measured in these units the scaled number of particles converges to a variant of neveu s continuous state branching process furthermore the genealogy of the particles is then governed by a coalescent process known as the bolthausen sznitman coalescent this validates the non rigorous predictions by brunet derrida muller and munier for a closely related model

transformations of diffusion and schrodinger processes a transformation by means of a new type of multiplicative functionals is given which is a generalization of doob s space time harmonic transformation in the case of arbitrary non harmonic function t x which may vanish on a subset of a b x d the transformation induces an additional singular drift term like in the case of doob s space time harmonic transformation to handle the transformation an integral equation of singular perturbations and a diffusion equation with singular potentials are discussed and the feynman kac theorem is established for a class of singular potentials the transformation is applied to schrodinger processes which are defined following an idea of e schrodinger 1931

the continuum random tree iii let mathscr r k k geq 1 be random trees with k leaves satisfying a consistency condition removing a random leaf from mathscr r k gives mathscr r k 1 then under an extra condition this family determines a random continuum tree mathscr l which it is convenient to represent as a random subset of l 1 this leads to an abstract notion of convergence in distribution as n rightarrow infty of rescaled random trees mathscr j n on n vertices to a limit continuum random tree mathscr l the notion is based upon the assumption that for fixed k the subtrees of mathscr j n determined by k randomly chosen vertices converge to mathscr r k as our main example under mild conditions on the offspring distribution the family tree of a galton watson branching process conditioned on total population size equal to n can be rescaled to converge to a limit continuum random tree which can be constructed from brownian excursion

universality of the local spacing distribution in certain ensembles of hermitian wigner matrices consider an n n hermitian random matrix with independent entries not necessarily gaussian a so called wigner matrix it has been conjectured that the local spacing distribution i e the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is in the limit as n the same as that of hermitian random matrices from gue we prove this conjecture for a certain subclass of hermitian wigner matrices