distributions of time averages for weakly chaotic systems the role of infinite invariant density distributions of time averaged observables are investigated using deterministic maps with n indifferent fixed points and n state continuous time random walk processes associated with them in a weakly chaotic phase namely when separation of trajectories is subexponential maps are characterized by an infinite invariant density we find that the infinite density can be used to calculate the distribution of time averages of integrable observables with a formula recently obtained by rebenshtok and barkai as an example we calculate distributions of the average position of the particle and average occupation fractions our work provides the distributional limit theorem for time averages for a wide class of nonintegrable observables with respect to the infinite invariant density in other words it deals with the situation where the darling kac aaronson theorem does not hold

residence time distribution for a class of gaussian markov processes we study the distribution of residence time or equivalently that of mean magnetization for a family of gaussian markov processes indexed by a positive parameter alpha the persistence exponent for these processes is simply given by theta alpha but the residence time distribution is nontrivial the shape of this distribution undergoes a qualitative change as theta increases indicating a sharp change in the ergodic properties of the process we develop two alternate methods to calculate exactly but recursively the moments of the distribution for arbitrary alpha for some special values of alpha we obtain closed form expressions of the distribution function s1063 651x 99 03306 1

self diffusion and visited surface in the droplet condensation problem breath figures abstract we study experimentally and theoretically new aspects of condensation growth and coalescence of droplets on a substrate we address in particular the dynamics of a marked droplet which undergoes coalescences with neighbouring droplets we find that the number of coalescences of the droplet grows as log t t is time and that the traveled distance increases as r t the average radius of the droplets at time t the fraction of the surface which was never covered by any droplet an important quantity for applications such as drug spreading or surface decontamination decays as t showing that completion occurs only slowly heuristic arguments accurate numerical simulations on simplified models which neglect polydispersity and an exact solution reported elsewhere a bray b derrida and c godreche europhys lett 27 1994 175 strongly support these findings and show that this power law decay is a generic feature common to many different situations finally the contour of the ensemble of dry sites appears fractal with dimension d 1 22 an experimental result not reproduced by the simplified models

distributional limit theorems in infinite ergodic theory we present a unified approach to the darling kac theorem and the arcsine laws for occupation times and waiting times for ergodic transformations preserving an infinite measure our method is based on control of the transfer operator up to the first entrance to a suitable reference set rather than on the full asymptotics of the operator we illustrate our abstract results by showing that they easily apply to a significant class of infinite measure preserving interval maps we also show that some of the tools introduced here are useful in the setup of pointwise dual ergodic transformations

the segal bargmann coherent state transform for compact lie groups abstract let k be an arbitrary compact connected lie group we describe on k an analog of the segal bargmann coherent state transform and we prove theorem 1 that this generalized coherent state transform maps l 2 k isometrically onto the space of holomorphic functions in l 2 g where g is the complexification of k and where is an appropriate heat kernel measure on g the generalized coherent state transform is defined in terms of the heat kernel on the compact group k and its analytic continuation to the complex group g we also define a k averaged version of the coherent state transform and we prove a similar result for it theorem 2 in the appendix we describe the coherent states which motivate the definitions of these transforms in section 9 we obtain inversion formulas for both the coherent state transform and the k averaged coherent state transform theorems 3 4 as a consequence we obtain formulas for certain bergman kernels on the complex group g theorems 5 6 in section 10 we derive a general paley wiener theorem for k and we exhibit a family of convolution transforms each of which is an isometric isomorphism of l 2 k onto a certain l 2 space of holomorphic functions on g these transforms include the k averaged coherent state transform as a special case we also discuss the analogous results for r 1 the results of section 10 are theorems 7 10 finally in section 11 we discuss the case of quotient spaces we prove theorem 11 that both the coherent state transform and the k averaged coherent state transform pass in a natural way from k to k h where h is a closed connected subgroup of k

aging arcsine law in brownian motion and its generalization classical arcsine law states that the fraction of occupation time on the positive or the negative side in brownian motion does not converge to a constant but converges in distribution to the arcsine distribution here we consider how a preparation of the system affects the arcsine law i e aging of the arcsine law we derive an aging distributional theorem for occupation time statistics in brownian motion where the ratio of time when measurements start to the measurement time plays an important role in determining the shape of the distribution furthermore we show that this result can be generalized as an aging distributional limit theorem in renewal processes

large deviations and nontrivial exponents in coarsening systems we investigate the statistics of the mean magnetization of its large deviations and persistent large deviations in simple coarsening systems in particular we consider more specifically the case of the diffusion equation of the ising chain at zero temperature and of the two dimensional voter model for the diffusion equation at large times the mean magnetization has a limit law which is studied analytically using the independent interval approximation the probability of persistent large deviations defined as the probability that the mean magnetization was for all previous times greater than some level x decays algebraically at large times with an exponent continuously varying with x when x 1 is the usual persistence exponent similar behaviour is found for the glauber ising chain at zero temperature for the two dimensional voter model large deviations of the mean magnetization are algebraic while the probability of persistent large deviations seem to behave as the usual persistence probability

phase ordering and persistence in a class of stochastic processes interpolating between the ising and voter models we study the dynamics of a class of two dimensional stochastic processes depending on two parameters which may be interpreted as two different temperatures respectively associated to interfacial and to bulk noise special lines in the plane of parameters correspond to the ising model voter model and majority vote model the dynamics of this class of models may be described formally in terms of reaction diffusion processes for a set of coalescing annihilating and branching random walkers we use the freedom allowed by the space of parameters to measure by numerical simulations the persistence probability of a generic model in the low temperature phase where the system coarsens this probability is found to decay at large times as a power law with a seemingly constant exponent we also discuss the connection between persistence and the nature of the interfaces between domains

on the second solution to a critical growth robin problem abstract we investigate the existence of the second mountain pass solution to a robin problem where the equation is at critical growth and depends on a positive parameter more precisely we determine existence and nonexistence regions for this type of solutions depending both on and on the parameter in the boundary conditions

the dynkin lamperti arc sine laws for measure preserving transformations arc sine laws in the sense of renewal theory are proved for return time processes generated by transformations with infinite invariant measure on sets satisfying a type of darling kac condition and an application to real transformations with indifferent fixed points is discussed

statistics of the occupation time for a class of gaussian markov processes we revisit the work of dhar and majumdar 1999 phys rev e 59 6413 on the limiting distribution of the temporal mean mt t 1 0tdu sign yu for a gaussian markovian process yt depending on a parameter which can be interpreted as brownian motion in the time scale t t2 this quantity the mean magnetization is simply related to the occupation time of the process that is the length of time spent on one side of the origin up to time t using the fact that the intervals between sign changes of the process form a renewal process on the time scale t we determine recursively the moments of the mean magnetization we also find an integral equation for the distribution of mt this allows a local analysis of this distribution in the persistence region mt 1 as well as its asymptotic analysis in the regime where is large finally we put the results thus found in perspective with those obtained by dhar and majumdar by another method based on a formalism due to kac

arcsine laws in stochastic thermodynamics we show that the fraction of time that a thermodynamic current spends above its average value follows the arcsine law a prominent result obtained by levy for brownian motion stochastic currents with long streaks above or below their average are much more likely than those that spend similar fractions of time above and below their average our result is confirmed with experimental data from a brownian carnot engine we also conjecture that two other random times associated with currents obey the arcsine law the time a current reaches its maximum value and the last time a current crosses its average value these results apply to inter alia molecular motors quantum dots and colloidal systems

intrinsic randomness of transport coefficient in subdiffusion with static disorder fluctuations in the time averaged mean square displacement for random walks on hypercubic lattices with static disorder are investigated it is analytically shown that the diffusion coefficient becomes a random variable as a manifestation of weak ergodicity breaking for two and higher dimensional systems the distribution function of the diffusion coefficient is found to be the mittag leffler distribution which is the same as for the continuous time random walk whereas for one dimensional systems a different distribution a modified mittag leffler distribution arises we also present a comparison of these two distributions in terms of an ergodicity breaking parameter and show that the modified mittag leffler distribution has a larger deviation from ergodicity some remarks on similarities between these results and observations in biological experiments are presented

generalized arcsine laws for fractional brownian motion the three arcsine laws for brownian motion are a cornerstone of extreme value statistics for a brownian b t starting from the origin and evolving during time t one considers the following three observables i the duration t the process is positive ii the time t mathrm last the process last visits the origin and iii the time t mathrm max when it achieves its maximum or minimum all three observables have the same cumulative probability distribution expressed as an arcsine function thus the name arcsine laws we show how these laws change for fractional brownian motion x t a non markovian gaussian process indexed by the hurst exponent h it generalizes standard brownian motion i e h frac 1 2 we obtain the three probabilities using a perturbative expansion in h ensuremath frac 1 2 while all three probabilities are different this distinction can only be made at second order in our results are confirmed to high precision by extensive numerical simulations

separation of trajectories and its relation to entropy for intermittent systems with a zero lyapunov exponent i t where is determined by the nonlinearity of the map in the vicinity of marginally unstable fixed points the mean of is determined by the infinite invariant density using semianalytical arguments we calculate the infinite invariant density for the pomeaumanneville map and with it we obtain excellent agreement between numerical simulation and theory we show that is equal to krengel s entropy and to the complexity calculated by the lempel ziv compression algorithm this generalized pesin s identity shows that and krengel s entropy are the natural generalizations of usual lyapunov exponent and entropy for these systems

the two parameter free unitary segal bargmann transform and its biane gross malliavin identification abstract motivated by a conditional expectation interpretation of the segal bargmann transform we derive the integral kernel for the large n limit of the two parameter segal bargmann hall transform over the unitary group u n and explore its limiting behavior we also extend the notion of circular systems to more general elliptic systems in order to give an alternate construction of our new two parameter free unitary segal bargmann hall transform via a biane gross malliavin type theorem

interaction between the geometry of the boundary and positive solutions of a semilinear neumann problem with critical nonlinearity abstract we consider the problem u u un 2 n 2 u 0 in u v 0 on where is a bounded smooth domain in r n n 3 we show that for large least energy solutions of the above problem have a unique maximum point p on and the limit points of p as are contained in the set of the points of maximum mean curvature we also prove that if has k peaks then the equation has at least k solutions for large

stationary definition of persistence for finite temperature phase ordering for the two dimensional kinetic ising model at finite temperature the local mean magnetization simply related to the fraction of time spent by a given spin in the positive direction has a limiting distribution singular at the onsager spontaneous magnetization the exponent of this singularity defines the persistence exponent we also study first passage exponents associated to persistent large deviations of and their temperature dependence

nonergodicity of a time series obeying levy statistics time averaged autocorrelation functions of a dichotomous random process switching between 1 and 0 and governed by wide power law sojourn time distribution are studied such a process called a levy walk describes dynamical behaviors of many physical systems fluorescence intermittency of semiconductor nanocrystals under continuous laser illumination being one example when the mean sojourn time diverges the process is non ergodic in that case the time average autocorrelation function is not equal to the ensemble averaged autocorrelation function instead it remains random even in the limit of long measurement time several approximations for the distribution of this random autocorrelation function are obtained for different parameter ranges and favorably compared to monte carlo simulations nonergodicity of the power spectrum of the process is briefly discussed and a nonstationary wiener khintchine theorem relating the correlation functions and the power spectrum is presented the considered situation is in full contrast to the usual assumptions of ergodicity and stationarity

random f expansions we consider the asymptotic distribution properties of f expansion digits in particular if x 1 varphi 0 x 1 varphi 1 x cdots etc then frac 1 n sum n 1 k 0 varphi k rightarrow 3 text in measure

the segal bargmann transform on classical matrix lie groups abstract we study the complex time segal bargmann transform b s k n on a compact type lie group k n where k n is one of the following classical matrix lie groups the special orthogonal group so n r the special unitary group su n or the compact symplectic group sp n our work complements and extends the results of driver hall and kemp on the segal bargman transform for the unitary group u n we provide an effective method of computing the action of the segal bargmann transform on trace polynomials which comprise a subspace of smooth functions on k n extending the polynomial functional calculus using these results we show that as n the finite dimensional transform b s k n has a meaningful limit g s which can be identified as an operator on the space of complex laurent polynomials

distributional ergodicity in stored energy driven levy flights we study a class of random walk the stored energy driven l evy flight sedlf whose jump length is determined by a stored energy during a trapped state the sedlf is a continuous time random walk with jump lengths being coupled with the trapping times it is analytically shown that the ensemble averaged mean square displacements exhibit subdiffusion as well as superdiffusion depending on the coupling parameter we find that time averaged mean square displacements increase linearly with time and the diffusion coefficients are intrinsically random a manifestation of distributional ergodicity the diffusion coefficient shows aging in subdiffusive regime whereas it increases with the measurement time in superdiffusive regime

statistics of the occupation time of renewal processes we present a systematic study of the statistics of the occupation time and related random variables for stochastic processes with independent intervals of time according to the nature of the distribution of time intervals the probability density functions of these random variables have very different scalings in time we analyze successively the cases where this distribution is narrow where it is broad with index 1 and finally where it is broad with index 1 2 the methods introduced in this work provide a basis for the investigation of the statistics of the occupation time of more complex stochastic processes see joint paper by g de smedt c godreche and j m luck 26

further study on the effect of boundary conditions abstract this paper concerns the least energy solutions to a semilinear elliptic equation the role of geometry of the boundary is studied when a parameter in the boundary condition tends to a critical index as well as to infinity

the complex time segal bargmann transform abstract we introduce a new form of the segal bargmann transform for a connected lie group k of compact type we show that the heat kernel t x t 0 x k has a space time analytic continuation to a holomorphic function c z re 0 z k c where k c is the complexification of k the new transform is defined by the integral b f z k c z k 1 f k d k z k c if s 0 and d s s the disk of radius s centered at s this integral defines a holomorphic function on k c for each f l 2 k s we construct a heat kernel density s on k c such that for all s as above b s b l 2 k s is an isometric isomorphism from l 2 k s onto the space of holomorphic functions in l 2 k c s when t s the transform b t t coincides with the one introduced by the second author for compact groups and extended by the first author to groups of compact type when t 0 2 s the transform b s t coincides with the one introduced by the first two authors

infinite invariant density in a semi markov process with continuous state variables we report on a fundamental role of a non normalized formal steady state i e an infinite invariant density in a semi markov process where the state is determined by the interevent time of successive renewals the state describes certain observables found in models of anomalous diffusion e g the velocity in the generalized levy walk model and the energy of a particle in the trap model in our model the interevent time distribution follows a fat tailed distribution which makes the state value more likely to be zero because long interevent times imply small state values we find two scaling laws describing the density for the state value which accumulates in the vicinity of zero in the long time limit these laws provide universal behaviors in the accumulation process and give the exact expression of the infinite invariant density moreover we provide two distributional limit theorems for time averaged observables in these nonstationary processes we show that the infinite invariant density plays an important role in determining the distribution of time averages

a random telegraph signal of mittag leffler type a general method is presented to explicitly compute autocovariance functions for non poisson dichotomous noise based on renewal theory the method is specialized to a random telegraph signal of mittag leffler type analytical predictions are compared to monte carlo simulations non poisson dichotomous noise is non stationary and standard spectral methods fail to describe it properly as they assume stationarity

anomalous diffusion in a quenched trap model on fractal lattices models with mixed origins of anomalous subdiffusion have been considered important for understanding transport in biological systems here one such mixed model the quenched trap model qtm on fractal lattices is investigated it is shown that both ensemble and time averaged mean square displacements msds show subdiffusion with different scaling exponents i e this system shows weak ergodicity breaking moreover time averaged msd exhibits aging and converges to a random variable following the modified mittag leffler distribution it is also shown that the qtm on a fractal lattice cannot be reduced to the continuous time random walks if the spectral dimension of the fractal lattice is less than 2

phase diagram in stored energy driven levy flight phase diagram based on the mean square displacement msd and the distribution of diffusion coefficients of the time averaged msd for the stored energy driven levy flight sedlf is presented in the sedlf a random walker cannot move while storing energy and it jumps by the stored energy the sedlf shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion depending on the coupling parameter between storing time trapping time and stored energy this stochastic process can be investigated analytically with the aid of renewal theory here we consider two different renewal processes i e ordinary renewal process and equilibrium renewal process when the mean trapping time does not diverge we analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping time distribution in particular we find that distributional behavior of time averaged msd intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge

stability in a system subject to noise with regulated periodicity the stability of a simple dynamical system subject to multiplicative one side pulse noise with hidden periodicity is investigated both analytically and numerically the stability analysis is based on the exact result for the characteristic functional of the renewal pulse process the influence of the memory effects on the stability condition is analyzed for two cases i the dead time distorted poissonian process and ii the renewal process with pareto distribution we show that for fixed noise intensity the system can be stable when the noise is characterized by high periodicity and unstable at low periodicity

on the kakutani ito segal gross and segal bargmann hall isomorphisms recently gross has shown that the kakutani ito segal isomorphism theorem has an extension from the setting of gaussian measure on a vector space to heat kernel measure pt on a simply connected lie group g of compact type the isomorphism relates l2 pt to a certain completion of the universal enveloping algebra of g lie g gross proves this result using the kakutani ito segal theorem and an infinite dimensional calculus associated to g valued brownian motion hijab has greatly simplified and clarified gross proof hiiab s proof avoids most but not all of the infinite dimensional analysis in the original proof in this paper we will build on hijab s proof to give a completely finite dimensional non probabilistic proof of gross isomorphism theorem the proof given here relies heavily on hall s beautiful extension of the segal bargmann transformation to the setting of compact lie groups this theorem relating l2 pt to a certain l2 space of holomorphic functions l2 gc h on the complexified lie group gc will also be generalized to lie groups of compact type in the process it is shown how to characterize in terms of summability conditions on all the derivatives at the identity in gc those holomorphic functions which are in l2 gc

multiple positive solutions for robin problem involving critical weighted hardy sobolev exponents with boundary singularities abstract this paper deals with the existence of positive solutions for robin elliptic problems involving critical weighted hardy sobolev exponents with boundary singularities using the caffarelli kohn nirenberg inequalities and variational methods we prove the existence and multiplicity of positive solutions

distributional behavior of time averages of non l 1 observables in one dimensional intermittent maps with infinite invariant measures in infinite ergodic theory two distributional limit theorems are well known one is characterized by the mittag leffler distribution for time averages of l 1 m functions i e integrable functions with respect to an infinite invariant measure the other is characterized by the generalized arc sine distribution for time averages of non l 1 m functions here we provide another distributional behavior of time averages of non l 1 m functions in one dimensional intermittent maps where each has an indifferent fixed point and an infinite invariant measure observation functions considered here are non l 1 m functions which vanish at the indifferent fixed point we call this class of observation functions weak non l 1 m function our main result represents a first step toward a third distributional limit theorem i e a distributional limit theorem for this class of observables in infinite ergodic theory to prove our proposition we propose a stochastic process induced by a renewal process to mimic a birkoff sum of a weak non l 1 m function in the one dimensional intermittent maps

renewal modulation and superstatistics in times series we consider two different approaches to which we refer to as renewal and modulation to generate time series with a nonexponential distribution of waiting times we show that different time series with the same waiting time distribution are not necessarily statistically equivalent and might generate different physical properties renewal generates aging and anomalous scaling while modulation yields no significant aging and either ordinary or anomalous diffusion according to the dynamic prescription adopted we show in fact that the physical realization of modulation generates two classes of events the events of the first class are determined by the persistent use of the same exponential time scale for an extended lapse of time and consequently are numerous the events of the second class are identified with the abrupt changes from one to another exponential prescription and consequently are rare the events of the second class although rare determine the scaling of the diffusion process and for this reason we term them as crucial events according to the prescription adopted to produce modulation the distribution density of the time distances between two consecutive crucial events might have or not a diverging second moment in the former case the resulting diffusion process although going through a transition regime very extended in time will eventually become anomalous in conclusion modulation rather than ruling out the action of renewal events produces crucial events hidden by clouds of exponential events thereby setting the challenge for their identification

generalized arcsine law and stable law in an infinite measure dynamical system limit theorems for the time average of some observation functions in an infinite measure dynamical system are studied it is known that intermittent phenomena such as the rayleigh benard convection and belousov zhabotinsky reaction are described by infinite measure dynamical systems we show that the time average of the observation function which is not the l 1 m function whose average with respect to the invariant measure m is finite converges to the generalized arcsine distribution this result leads to the novel view that the correlation function is intrinsically random and does not decay moreover it is also numerically shown that the time average of the observation function converges to the stable distribution when the observation function has the infinite mean

arcsine laws and interval partitions derived from a stable subordinator levy discovered that the fraction of time a standard one dimensional brownian motion b spends positive before time t has arcsine distribution both for a fixed time when b t 0 almost surely and for t an inverse local time when b t 0 almost surely this identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of b away from 0 similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator for instance a bessel process of dimension d for 0 d 2

the large n limit of the segal bargmann transform on un we study the two parameter segal bargmann transform b n on the unitary group un for large n acting on matrix valued functions that are equivariant under the adjoint action of the group the transform has a meaningful limit gs t as n which can be identified as an operator on the space of complex laurent polynomials we introduce the space of trace polynomials and use it to give effective computational methods to determine the action of the heat operator and thus the segal bargmann transform we prove several concentration of measure and limit theorems giving a direct connection from the finite dimensional transform b n to its limit gs t we characterize the operator gs t through its inverse action on the standard polynomial basis finally we show that in the case s t the limit transform gt t is the free hall transform g t introduced by biane

segal bargmann transform functional calculus on matrix spaces and the theory of semi circular and circular systems letmd c be the space ofd dcomplex matrices and xdbe the subspace of hermitian matrices we study the segal bargmann transform of functions on xd with values inmd c which are given by functional calculus we show that whend the transform of such a map becomes close to the space of holomorphic functional calculus onmd c and that this yields in the limit an isometry between thel2space of wigner s semi circle distribution and the hardy space of the disk we relate this isometry to voiculescu s theory of circular and semi circular systems and we study its analogue when the segal bargmann transform is replaced by the hall transform on unitary groups of large dimensions

global structure of solutions for equations of brezis nirenberg type on the unit ball the brezis nirenberg equation and the scalar field equation on the three dimensional unit ball are studied under the robin condition we show the existence and uniqueness of radial solutions in a unified way in particular it is shown that the global structure of solutions changes qualitatively when a parameter in the boundary condition exceeds a certain critical value

mixing rates for intermittent maps of high exponent we obtain higher order theory for the long term behavior of the transfer operator associated with the unit interval map f x x 1 2 alpha x alpha if 0 1 which corresponds to the infinite measure preserving case higher order theory for alpha ge 2 is more challenging and requires new techniques along the way we provide higher order theory for scalar and operator renewal sequences with infinite measure and regular variation although the present work considers the unit interval map mentioned above as a toy model our interest focuses on finding sufficient conditions under which the asymptotic behavior of the transfer operator associated to dynamical systems preserving an infinite measure is almost like the asymptotic behavior of scalar renewal sequences associated to null recurrent markov chains characterized by regular variation

aging generates regular motions in weakly chaotic systems using intermittent maps with infinite invariant measures we investigate the universality of time averaged observables under aging conditions according to aaronson s darling kac theorem in non aged dynamical systems with infinite invariant measures the distribution of the normalized time averages of integrable functions converges to the mittag leffler distribution this well known theorem holds when the start of observations coincides with the start of the dynamical processes introducing a concept of the aging limit where the aging time t a and the total measurement time t goes to infinity while the aging ratio t a t is kept constant we obtain a novel distributional limit theorem of time averaged observables integrable with respect to the infinite invariant density applying the theorem to the lyapunov exponent in intermittent maps we find that regular motions and a weakly chaotic behavior coexist in the aging limit this mixed type of dynamics is controlled by the aging ratio and hence is very different from the usual scenario of regular and chaotic motions in hamiltonian systems the probability of finding regular motions in non aged processes is zero while in the aging regime it is finite and it increases when system ages

free convolution operators and free hall transform abstract we define an extension of the polynomial calculus on a w probability space by introducing an algebra c x i i i which contains polynomials this extension allows us to define transition operators for additive and multiplicative free convolution it also permits us to characterize the free segal bargmann transform and the free hall transform introduced by biane in a manner which is closer to classical definitions finally we use this extension of polynomial calculus to prove two asymptotic results on random matrices the convergence for each fixed time as n tends to of the distribution of the brownian motion on the linear group gl n c to the distribution of a free multiplicative circular brownian motion and the convergence of the classical hall transform on u n to the free hall transform

operator renewal theory and mixing rates for dynamical systems with infinite measure we develop a theory of operator renewal sequences in the context of innite ergodic theory for large classes of dynamical systems preserving an innite measure we determine the asymptotic behaviour of iterates l n of the transfer operator this was previously an intractable problem examples of systems covered by our results include i parabolic rational maps of the complex plane and ii not necessarily markovian nonuniformly expanding interval maps with indierent xed points in addition we give a particularly simple proof of pointwise dual ergodicity asymptotic behaviour of p n 1 l j for the class of systems under consideration in certain situations including pomeau manneville intermittency maps we obtain higher order expansions for l n and rates of mixing also we obtain error estimates in the associated dynkin lamperti arcsine laws

coarsening and persistence in the voter model the voter model is a simple model for coarsening with a nonconserved scalar order parameter we investigate coarsening and persistence in the voter model by introducing the quantity pn t defined as the fraction of voters who changed their opinion n times up to time t we show that pn t exhibits scaling behavior that strongly depends on the dimension as well as on the initial opinion concentrations exact results are obtained for the average number of opinion changes n and the autocorrelation function a t 21 n pn t 2d 2 in arbitrary dimension d these exact results are complemented by a mean field theory heuristic arguments and numerical simulations for dimensions d 2 the system does not coarsen and the opinion changes follow a nearly poissonian distribution in agreement with mean field theory for dimensions d 2 the distribution is given by a different scaling form which is characterized by nontrivial scaling exponents for unequal opinion concentrations an unusual situation occurs where different scaling functions correspond to the majority and the minority as well as for even and odd n

distributional response to biases in deterministic superdiffusion we report on a novel response to biases in deterministic superdiffusion for its reduced map we show using infinite ergodic theory that the time averaged velocity tav is intrinsically random and its distribution obeys the generalized arcsine distribution a distributional limit theorem indicates that the tav response to a bias appears in the distribution which is an example of what we term a distributional response induced by a bias although this response in single trajectories is intrinsically random the ensemble averaged tav response is linear

existence and nonexistence of least energy solutions of the neumann problem for a semilinear elliptic equation with critical sobolev exponent and a critical lower order perturbation let be a smooth bounded domain in rn with n 5 a 0 0 and 2 2nn 2 we show that the exponent q 2 n 1 n 2 plays a critical role regarding the existence of least energy or ground state solutions of the neumann problem u au u2 1 uq 1in u 0in u 0on namely we prove that when q 2 n 1 n 2 there exists an 0 0 such that the problem has a least energy solution if 0