homogenization of fully nonlinear uniformly elliptic and parabolic partial differential equations in stationary ergodic media we study the homogenization of fully nonlinear uniformly elliptic and parabolic second order partial differential equations in stationary ergodic media c 2004 wiley periodicals inc

variational formula for the time constant of first passage percolation consider first passage percolation with positive stationary ergodic weights on the square lattice in d dimensions let t x be the first passage time from the origin to x in z d the convergence of t nx n to the time constant as n tends to infinity is a consequence of the subadditive ergodic theorem this convergence can be viewed as a problem of homogenization for a discrete hamilton jacobi bellman hjb equation by borrowing several tools from the continuum theory of stochastic homogenization for hjb equations we derive an exact variational formula duality principle for the timeconstant under a symmetry assumption we will use the variational formula to construct an explicit iteration that produces the limit shape

stochastic homogenization of certain nonconvex hamilton jacobi equations abstract in this paper we prove the stochastic homogenization of certain nonconvex hamilton jacobi equations the nonconvex hamiltonians which are generally uneven and inseparable are generated by a sequence of quasiconvex hamiltonians and a sequence of quasiconcave hamiltonians through the min max formula we provide a monotonicity assumption on the contact values between those stably paired hamiltonians so as to guarantee the stochastic homogenization

rates of convergence for the homogenization of fully nonlinear uniformly elliptic pde in random media we establish a logarithmic type rate of convergence for the homogenization of fully nonlinear uniformly elliptic second order pde in strongly mixing media with similar i e logarithmic decorrelation rate the proof consists of two major steps the first which is actually the only place in the paper where probability plays a role establishes the rate for special quadratic data using the methodology developed by the authors and wang to study the homogenization of nonlinear uniformly elliptic pde in general stationary ergodic random media the second is a general argument based on the new notion of viscosity solutions which is introduced in this paper that shows that rates known for quadratic can be extended to general data as an application of this we also obtain here rates of convergence for the homogenization in periodic and almost periodic environments the former is algebraic while the latter depends on the particular equation

rank of a hadamard product abstract we provide a new proof of a sharp lower bound for the rank of the hadamard product of positive semidefinite matrices our proof uses standard matrix analysis tools and involves the rank of one factor and the kruskal rank of the other it leads to a new understanding of when the hadamard exponential of a positive semidefinite matrix can be singular

homogenization of weakly coupled systems of hamilton jacobi equations with fast switching rates we consider homogenization for weakly coupled systems of hamilton jacobi equations with fast switching rates the fast switching rate terms force the solutions to converge to the same limit which is a solution of the effective equation we discover the appearance of the initial layers which appear naturally when we consider the systems with different initial data and analyze them rigorously in particular we obtain matched asymptotic solutions of the systems and the rate of convergence we also investigate properties of the effective hamiltonian of weakly coupled systems and show some examples which do not appear in the context of single equations

homogenization and enhancement of the g equation in random environments we study the homogenization of a g equation which is advected by a divergence free stationary vector field in a general ergodic random environment we prove that the averaged equation is an anisotropic deterministic g equation and we give necessary and sufficient conditions in order to have enhancement since the problem is not assumed to be coercive it is not possible to have uniform bounds for the solutions in addition as we show the associated minimal first passage time function does not satisfy in general the uniform integrability condition which is necessary to apply the sub additive ergodic theorem we overcome these obstacles by i establishing a new reachability controllability estimate for the minimal function and ii constructing for each direction and almost surely a random sequence which has both a long time averaged limit due to the sub additive ergodic theorem and stays in the same sense asymptotically close to the minimal time

stochastic homogenization of a front propagation problem with unbounded velocity we study the homogenization of hamilton jacobi equations which arise in front propagation problems in stationary ergodic media our results are obtained for fronts moving with possible unbounded velocity we show by an example that the homogenized hamiltonian which always exists may be unbounded in this context we show convergence results if we start with a compact initial front on the other hand if the media satisfies a finite range of dependence condition we prove that the effective hamiltonian is bounded and obtain classical homogenization in this context

upper bounds for the critical probability of oriented percolation in two dimensions we give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions this method enables us to prove that the critical probability is at most 0 6863 greatly improving the best published upper bound 0 84 we also prove that our method can be used to give arbitrarily good upper bounds we also use a slight variant of our method to obtain an upper bound of 0 725 99 for the critical probability in oriented site percolation

monotonicity for entrywise functions of matrices abstract we characterize real functions f on an interval for which the entrywise matrix function a ij f a ij is positive monotone and convex respectively in the positive semidefiniteness order fractional power functions are exemplified and related weak majorizations are shown

singular perturbations of nonlinear degenerate parabolic pdes a general convergence result the main result of the paper is a general convergence theorem for the viscosity solutions of singular perturbation problems for fully nonlinear degenerate parabolic pdes partial differential equations with highly oscillating initial data it substantially generalizes some results obtained previously in 2 under the only assumptions that the hamiltonian is ergodic and stabilizing in a suitable sense the solutions are proved to converge in a relaxed sense to the solution of a limit cauchy problem with appropriate effective hamiltonian and initial data in its formulation our convergence result is analogous to the stability property of barles and perthame it should thus reveal a useful tool for studying general singular perturbation problems by viscosity solutions techniques a detailed exposition of ergodicity and stabilization is given with many examples applications to homogenization and averaging are also discussed

complete characterization of hadamard powers preserving loewner positivity monotonicity and convexity abstract entrywise powers of symmetric matrices preserving positivity monotonicity or convexity with respect to the loewner ordering arise in various applications and have received much attention recently in the literature following fitzgerald and horn 1977 8 it is well known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness similar phenomena have also recently been shown by hiai 2009 to occur for monotonicity and convexity in this paper we complete the characterization of all the entrywise powers below and above the critical exponents that are positive monotone or convex on the cone of positive semidefinite matrices we then extend the original problem by fully classifying the positive monotone or convex powers in a more general setting where additional rank constraints are imposed on the matrices we also classify the entrywise powers that are super sub additive with respect to the loewner ordering finally we extend all the previous characterizations to matrices with negative entries our analysis consequently allows us to answer a question raised by bhatia and elsner 2007 regarding the smallest dimension for which even extensions of the power functions do not preserve loewner positivity

stochastic homogenization of hamilton jacobi bellman equations we study the homogenization of some hamilton jacobi bellman equations with a vanishing second order term in a stationary ergodic random medium under the hyperbolic scaling of time and space imposing certain convexity growth and regularity assumptions on the hamiltonian we show the locally uniform convergence of solutions of such equations to the solution of a deterministic effective first order hamilton jacobi equation the effective hamiltonian is obtained from the original stochastic hamiltonian by a minimax formula our homogenization results have a large deviations interpretation for a diffusion in a random environment c 2006 wiley periodicals inc

error estimates and convergence rates for the stochastic homogenization of hamilton jacobi equations we present exponential error estimates and demonstrate an algebraic convergence rate for the homogenization of level set convex hamilton jacobi equations in i i d random environments the first quantitative homogenization results for these equations in the stochastic setting by taking advantage of a connection between the metric approach to homogenization and the theory of first passage percolation we obtain estimates on the fluctuations of the solutions to the approximate cell problem in the ballistic regime away from flat spot of the effective hamiltonian in the sub ballistic regime on the flat spot we show that the fluctuations are governed by an entirely different mechanism and the homogenization may proceed without further assumptions at an arbitrarily slow rate we identify a necessary and sufficient condition on the law of the hamiltonian for an algebraic rate of convergence to hold in the sub ballistic regime and show under this hypothesis that the two rates may be merged to yield comprehensive error estimates and an algebraic rate of convergence for homogenization our methods are novel and quite different from the techniques employed in the periodic setting although we benefit from previous works in both first passage percolation and homogenization the link between the rate of homogenization and the flat spot of the effective hamiltonian which is related to the nonexistence of correctors is a purely random phenomenon observed here for the first time

stochastic homogenization of nonconvex hamilton jacobi equations in one space dimension abstract we prove stochastic homogenization for a general class of coercive nonconvex hamilton jacobi equations in one space dimension some properties of the effective hamiltonian arising in the nonconvex case are also discussed

stochastic homogenization of interfaces moving with changing sign velocity abstract we are interested in the averaging behavior of interfaces moving in stationary ergodic environments with oscillatory normal velocity which changes sign the problem can be reformulated as the homogenization of a hamilton jacobi equation with a positively homogeneous of degree one non coercive hamiltonian the periodic setting was studied earlier by cardaliaguet lions and souganidis 2009 16 here we concentrate in the random media and show that the solutions of the oscillatory hamilton jacobi equation converge in l weak to a linear combination of the initial datum and the solutions of several initial value problems with deterministic effective hamiltonian s determined by the properties of the random media

large deviations for random walks in a random environment we prove two large deviation principles for random walks in a random environment one for quenched walks in general ergodic environments and the other for averaged walks in a product environment 2003 wiley periodicals inc

homogenization for stochastic hamilton jacobi equations homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales we prove homogenization for certain stochastic hamilton jacobi partial differential equations the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale separation limit in some cases we also establish a central limit theorem

stochastic homogenization of l variational problems abstract we present a homogenization result for l variational problems in general stationary ergodic random environments by introducing a generalized notion of distance function a special solution of an associated eikonal equation and demonstrating a connection to absolute minimizers of the variational problem we obtain the homogenization result as a consequence of the fact that the latter homogenizes in random environments

stability and bifurcation analysis of a delayed predator prey model of prey dispersal in two patch environments abstract in this paper a class of delayed predator prey model of prey dispersal in two patch environments is considered by analyzing the associated characteristic transcendental equation its linear stability is investigated and hopf bifurcation is demonstrated some explicit formulae determining the stability and the direction of the hopf bifurcation periodic solutions bifurcating from hopf bifurcations are obtained by using the normal form theory and center manifold theory some numerical simulation for justifying the theoretical analysis are also provided finally biological explanations and main conclusions are given

the variational approach to hamilton jacobi equations driven by a gaussian noise abstract the global existence and uniqueness of viscosity solutions to the cauchy problem for the hamilton jacobi equations in r n driven by additive and multiplicative wiener processes are studied for convex hamiltonians via variational techniques the finite speed of propagation is also established in the multiplicative noise case for equations with lipschitzian hamiltonians

rate of convergence of the mean for sub additive ergodic sequences abstract for sub additive ergodic processes x m n with weak dependence we analyze the rate of convergence of e x 0 n n to its limit g we define an exponent given roughly by e x 0 n n g n and assuming existence of a fluctuation exponent that gives var x 0 n n 2 we provide a lower bound for of the form the main requirement is that 1 2 in the case 1 2 and under the assumption var x 0 n o n log n for some 0 we prove c for a dependent constant c these results show in particular that non diffusive fluctuations are associated to non trivial various models including first passage percolation directed polymers the minimum of a branching random walk and bin packing fall into our general framework and the results apply assuming exists in the case of first passage percolation in z d we provide a version of 1 2 without assuming existence of

hadamard powers of some positive matrices abstract positivity properties of the hadamard powers of the matrix 1 x i x j for distinct positive real numbers x 1 x n and the matrix cos i j n are studied in particular it is shown that the n n matrix 1 x i x j r is positive semidefinite if and only if r is a nonnegative integer or r n 2 and for every odd integer n 3 the n n matrix cos i j n r is positive semidefinite if and only if r is a nonnegative even integer or r n 3

almost periodic solution for a volterra model with mutual interference and beddington deangelis functional response in this paper we consider a volterra model with mutual interference and beddington deangelis functional response by applying the comparison theorem of the differential equation and constructing a suitable lyapunov functional sufficient conditions which guarantee the permanence and existence of a unique globally attractive positive almost periodic solution of the system are obtained

stochastic homogenization of nonconvex hamilton jacobi equations a counterexample this paper provides a counterexample to hamilton jacobi homogenization in the nonconvex case for general stationary ergodic environments

multiscale problems and homogenization for second order hamilton jacobi equations we prove a general convergence result for singular perturbations with an arbitrary number of scales of fully nonlinear degenerate parabolic pdes as a special case we cover the iterated homogenization for such equations with oscillating initial data explicit examples among others are the two scale homogenization of quasilinear equations driven by a general hypoelliptic operator and the n scale homogenization of uniformly parabolic fully nonlinear pdes

permanence and extinction in logistic and lotka volterra systems with diffusion abstract in this paper we consider the effect of diffusion on the permanence and extinction of single and multiple endangered species that live in changing patch environments differing from former studies our discussion includes the important situation in conservation biology in which species live in a weak patchy environment in the sense that species will become extinct in some of the isolated patches for the single population model we show that identical species can persist for some diffusion rates and can also vanish for another set of restrictions on the diffusion rates although the single endangered species will vanish in some isolated patches without contributions from other patches furthermore we consider the existence uniqueness and global stability of the positive periodic solution for prey predator systems we can make both the prey and the predator species permanent by choosing the diffusion rates appropriately even if the prey species has a negative intrinsic growth rate in some patches moreover we introduce an exotic competitive species y into the habitat occupied by the native species x competitive permanence and competitive exclusion both are considered the implications of these results are significant for the conservation of endangered species

global asymptotic stability of lotka volterra diffusion models with continuous time delay we consider the global asymptotic stability of diffusion models with multiple species heterogeneous patches and incorporate the effect of continuous time delays for species in each patch for two different patches connected by diffusion with multiple species and with patch dynamics governed by lotka volterra models with continuous time delays and constant influxes we give sufficient conditions for boundedness of solutions and the existence of a nonnegative equilibrium point providing also a candidate for a lyapunov function by applying the homotopy function technique for different symbiotic patches with time delays we provide a sufficient condition for the existence of a positive globally asymptotically stable equilibrium point further we discuss the global asymptotic stability for the prey predator model where the prey has its shelter from predation and only the predator can diffuse between patches we consider also the existence condition for the globally asymptotically stable equilibrium for t

lyapunov exponent for the parabolic anderson model with levy noise we consider the asymptotic almost sure behavior of the solution of the equation where y x x z d is a field of independent levy processes and is the discrete laplacian

min max formulas and other properties of certain classes of nonconvex effective hamiltonians this paper is the first attempt to systematically study properties of the effective hamiltonian overline h arising in the periodic homogenization of some coercive but nonconvex hamilton jacobi equations firstly we introduce a new and robust decomposition method to obtain min max formulas for a class of nonconvex overline h secondly we analytically and numerically investigate other related interesting phenomena such as quasi convexification and breakdown of symmetry of overline h from other typical nonconvex hamiltonians finally in the appendix we show that our new method and those a priori formulas from the periodic setting can be used to obtain stochastic homogenization for the same class of nonconvex hamilton jacobi equations some conjectures and problems are also proposed

analysis of nonautonomous predator prey model with nonlinear diffusion and time delay in this paper a nonautonomous predator prey model with nonlinear diffusion and time delay due to gestation is investigated where all parameters are time dependent it is proved that the system is uniformly persistent under appropriate conditions sufficient conditions are derived for the global stability of the system by constructing a suitable lyapunov functional

first passage percolation on mathbb z 2 a simulation study first passage percolation on mathbb z 2 is a model for describing the spread of an infection on the sites of the square lattice the infection is spread via nearest neighbor sites and the time dynamic is specified by random passage times attached to the edges in this paper the speed of the growth and the shape of the infected set is studied by aid of large scale computer simulations with focus on continuous passage time distributions it is found that the most important quantity for determining the value of the time constant which indicates the inverse asymptotic speed of the growth is mathbf e min tau 1 ldots tau 4 where tau 1 ldots tau 4 are i i d passage time variables the relation is linear for a large class of passage time distributions furthermore the directional time constants are seen to be increasing when moving from the axis towards the diagonal so that the limiting shape is contained in a circle with radius defined by the speed along the axes the shape comes closer to the circle for distributions with larger variability

exact and approximate correctors for stochastic hamiltonians the 1 dimensional case we perform a qualitative investigation of critical hamilton jacobi equations with stationary ergodic hamiltonian in dimension 1 we show the existence of approximate correctors give characterizing conditions for the existence of correctors provide lax type representation formulae and establish comparison principles the results are applied to look into the corresponding effective hamiltonian and to study a homogenization problem in the analysis a crucial role is played by tools from stochastic geometry such as for instance closed random stationary sets

a sublinear variance bound for solutions of a random hamilton jacobi equation we estimate the variance of the value function for a random optimal control problem the value function is the solution w of a hamilton jacobi equation with random hamiltonian h p x k p v x in dimension d 2 it is known that homogenization occurs as 0 but little is known about the statistical fluctuations of w our main result shows that the variance of the solution w is bounded by o log the proof relies on a modified poincare inequality of talagrand