shift in the velocity of a front due to a cutoff we consider the effect of a small cutoff ensuremath varepsilon on the velocity of a traveling wave in one dimension simulations done over more than ten orders of magnitude as well as a simple theoretical argument indicate that the effect of the cutoff ensuremath varepsilon is to select a single velocity that converges when ensuremath varepsilon ensuremath rightarrow 0 to the one predicted by the marginal stability argument for small ensuremath varepsilon the shift in velocity has the form k mathrm ln ensuremath varepsilon ensuremath 2 and our prediction for the constant k agrees very well with the results of our simulations a very similar logarithmic shift appears in more complicated situations in particular in finite size effects of some microscopic stochastic systems our theoretical approach can also be extended to give a simple way of deriving the shift in position due to initial conditions in the fisher kolmogorov or similar equations

long time existence for signed solutions of the heat equation with a noise term let be the circle 0 j with the ends identified we prove long time existence for the following equation

exponential linear quadratic optimal control with discounting a direct derivation of the solution of the discounted exponential linear quadratic gaussian optimal control problem is presented some extensions to partial observations problems are also given the solution involves time varying riccati equations providing an example of an infinite horizon constant coefficient regulator problem with a time varying optimal control

phenomenological theory giving the full statistics of the position of fluctuating pulled fronts we propose a phenomenological description for the effect of a weak noise on the position of a front described by the fisher kolmogorov petrovsky piscounov equation or any other traveling wave equation in the same class our scenario is based on four hypotheses on the relevant mechanism for the diffusion of the front our parameter free analytical predictions for the velocity of the front its diffusion constant and higher cumulants of its position agree with numerical simulations

hamiltonian systems near relative equilibria abstract we give explicit differential equations for the dynamics of hamiltonian systems near relative equilibria these split the dynamics into motion along the group orbit and motion inside a slice transversal to the group orbit the form of the differential equations that is inherited from the symplectic structure and symmetry properties of the hamiltonian system is analysed and the effects of time reversing symmetries are included the results will be applicable to the stability and bifurcation theories of relative equilibria of hamiltonian systems

random traveling wave and bifurcations of asymptotic behaviors in the stochastic kpp equation driven by dual noises abstract in this paper we study the influences of dually environmental noises on the traveling wave which develops from the deterministic kpp equation we prove that if the strengths of noises satisfy some condition the solution of the stochastic kpp equation with heaviside initial condition develops a random traveling wave whose wave speed is deterministic and depends on the strengths of noises if the strengths of noises satisfy some other conditions the solution tends to zero as time tends to infinity therefore there exist bifurcations of asymptotic behaviors of solution induced by the strengths of dual noises

universal tree structures in directed polymers and models of evolving populations by measuring or calculating coalescence times for several models of coalescence or evolution with and without selection we show that the ratios of these coalescence times become universal in the large size limit and we identify a few universality classes

scattering matrices and dirichlet to neumann maps abstract a general representation formula for the scattering matrix of a scattering system consisting of two self adjoint operators in terms of an abstract operator valued titchmarsh weyl m function is proved this result is applied to scattering problems for different self adjoint realizations of schrodinger operators on unbounded domains schrodinger operators with singular potentials supported on hypersurfaces and orthogonal couplings of schrodinger operators in these applications the scattering matrix is expressed in an explicit form with the help of dirichlet to neumann maps

the mixed boundary value problem krein resolvent formulas and spectral asymptotic estimates abstract for a second order symmetric strongly elliptic operator a on a smooth bounded open set in r n the mixed problem is defined by a neumann type condition on a part of the boundary and a dirichlet condition on the other part we show a kre n resolvent formula where the difference between its resolvent and the dirichlet resolvent is expressed in terms of operators acting on sobolev spaces over this is used to obtain a new weyl type spectral asymptotics formula for the resolvent difference where upper estimates were known before namely s j j 2 n 1 c 0 2 n 1 where c 0 is proportional to the area of in the case where a is principally equal to the laplacian

a collective coordinate framework to study the dynamics of travelling waves in stochastic partial differential equations abstract we propose a formal framework based on collective coordinates to reduce infinite dimensional stochastic partial differential equations spdes with symmetry to a set of finite dimensional stochastic differential equations which describe the shape of the solution and the dynamics along the symmetry group we study spdes arising in population dynamics with multiplicative noise and additive symmetry breaking noise the collective coordinate approach provides a remarkably good quantitative description of the shape of the travelling front as well as its diffusive behaviour which would otherwise only be available through costly computational experiments we corroborate our analytical results with numerical simulations of the full spde

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

exactly soluble noisy traveling wave equation appearing in the problem of directed polymers in a random medium we calculate exactly the velocity and diffusion constant of a microscopic stochastic model of n evolving particles which can be described by a noisy traveling wave equation with a noise of order n 1 2 our model can be viewed as the infinite range limit of a directed polymer in random medium with n sites in the transverse direction despite some peculiarities of the traveling wave equations in the absence of noise our exact solution allows us to test the validity of a simple cutoff approximation and to show that in the weak noise limit the position of the front can be completely described by the effect of the noise on the first particle

last passage percolation and traveling fronts we consider a system of n particles with a stochastic dynamics introduced by brunet and derrida phys rev e 70 016106 2004 the particles can be interpreted as last passage times in directed percolation on 1 n of mean field type the particles remain grouped and move like a traveling front subject to discretization and driven by a random noise as n increases we obtain estimates for the speed of the front and its profile for different laws of the driving noise as shown in brunet and derrida phys rev e 70 016106 2004 the model with gumbel distributed jumps has a simple structure we establish that the scaling limit is a levy process in this case we study other jump distributions we prove a result showing that the limit for large n is stable under small perturbations of the gumbel in the opposite case of bounded jumps a completely different behavior is found where finite size corrections are extremely small

stochastic generalised kpp equations we classify multiplicative white noise perturbations k dw of generalised kpp equations and their effects on deterministic approximate travelling wave solutions by the behaviour of the solutions of the stochastic generalised kpp equations converge to deterministic approximate travelling waves and if being an associated potential energy s a solution of the corresponding classical mechanical equations of newton d being a certain domain in r1 rr then the white noise perturbations essentially destroy the wave structure and force the solutions to die down for the case suppose the existence of the limit we show that there is a residual wave form but propagating at a different speed from that of the unperturbed equations numerical solutions are included and give good agreement with theoretical results

stochastic p d e s arising from the long range contact and long range voter processes a long range contact process and a long range voter process are scaled so that the distance between sites decreases and the number of neighbors of each site increases the approximate densities of occupied sites under suitable tine scaling converge to continuous space time densities which solve stochastic p d e s for the contact process the limiting equation is the kolmogorov petrovskii piscuinov equation driven by branching white noise for the voter process the limiting equation is the heat equation driven by fisher wright white noise

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

travelling waves for reaction diffusion equations forced by translation invariant noise abstract inspired by applications we consider reaction diffusion equations on r that are stochastically forced by a small multiplicative noise term that is white in time coloured in space and invariant under translations we show how these equations can be understood as a stochastic partial differential equation spde forced by a cylindrical q wiener process and subsequently explain how to study stochastic travelling waves in this setting in particular we generalize the phase tracking framework that was developed in hamster and hupkes 2018 2019 for noise processes driven by a single brownian motion the main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed we illustrate our approach by two fully worked out examples which highlight the predictive power of our expansions

spde in hilbert space with locally monotone coefficients the aim of this paper is to extend the usual framework of spde with monotone coecients to include a large class of cases with merely locally monotone coecients this new framework is conceptually not more involved than the classical one but includes many more fundamental examples not included previously thus our main result can be applied to various types of spdes such as stochastic reaction diusion equations stochastic burgers type equation stochastic 2 d navier stokes equation stochastic p laplace equation and stochastic porous media equation with non monotone perturbations

a remark on discrete wave scattering we consider the spectral problem for the adjacency matrix of a discrete star graph composed of a compact cluster and a few semi infinite periodic leads attached based on the spectral properties of the adjacency matrix we develop the lax phillips scattering theory for the corresponding discrete wave equation

generalized q functions and dirichlet to neumann maps for elliptic differential operators the classical concept of q functions associated to symmetric and selfadjoint operators due to m g krein and h langer is extended in such a way that the dirichletto neumann map in the theory of elliptic differential equations can be interpreted as a generalized q function for couplings of uniformly elliptic second order differential expression on bounded and unbounded domains explicit krein type formulas for the difference of the resolvents and trace formulas in an h 2 framework are obtained

on the location of the weyl circles the paper deals with explicit estimates concerning certain circles in the complex plane which were associated with sturm liouville problems by h weyl by the use of riccati equations instead of linear integral equations improvements are obtained for results of everitt and halvorsen concerning the behaviour of the titchmarsh weyl m coefficient

the scaling limit for a stochastic pde and the separation of phases we investigate the problem of singular perturbation for a reaction diffusion equation with additive noise or a stochastic partial differential equation of ginzburg landau type under the situation that the reaction term is determined by a potential with double wells of equal depth as the parameter e the temperature of the system tends to 0 the solution converges to one of the two stable phases and consequently the phase separation is formed in the limit we derive a stochastic differential equation which describes the random movement of the phase separation point the proof consists of two main steps we show that the solution stays near a manifoldm e of minimal energy configurations based on a lyapunov type argument then the limit equation is identified by introducing a nice coordinate system in a neighborhood ofm e

dirichlet to neumann maps on bounded lipschitz domains the dirichlet to neumann map associated to an elliptic partial differential equation becomes multivalued when the underlying dirichlet problem is not uniquely solvable the main objective of this paper is to present a systematic study of the dirichlet to neumann map and its inverse the neumann to dirichlet map in the framework of linear relations in hilbert spaces our treatment is inspired by abstract methods from extension theory of symmetric operators utilizes the general theory of linear relations and makes use of some deep results on the regularity of the solutions of boundary value problems on bounded lipschitz domains

extension theory and kreĭn type resolvent formulas for nonsmooth boundary value problems for a strongly elliptic second order operator a on a bounded domainr n it has been known for many years how to interpret the general closed l2 realizations of a as representing boundary conditions generally nonlocal when the domain and coe cients are smooth the purpose of the present paper is to extend this representation to nonsmooth domains and coecients including the case of hol der c 3 2 smoothness in such a way that pseudodierential methods are still avail able for resolvent construc tions and ellipticity considerations we show how it can be done for domains with

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

a mild ito formula for spdes this article introduces a new somehow mild ito type formula for the solution process of a stochastic partial differential equation of evolutionary type

spectral theory for perturbed krein laplacians in nonsmooth domains abstract we study spectral properties for h k the krein von neumann extension of the perturbed laplacian v defined on c 0 where v is measurable bounded and nonnegative in a bounded open set r n belonging to a class of nonsmooth domains which contains all convex domains along with all domains of class c 1 r r 1 2 in particular in the aforementioned context we establish the weyl asymptotic formula j n k j 2 n v n n 2 o n 1 2 2 as where v n n 2 n 2 1 denotes the volume of the unit ball in r n and k j j n are the non zero eigenvalues of h k listed in increasing order according to their multiplicities we prove this formula by showing that the perturbed krein laplacian i e the krein von neumann extension of v defined on c 0 is spectrally equivalent to the buckling of a clamped plate problem and using an abstract result of kozlov from the mid 1980s our work builds on that of grubb in the early 1980s who has considered similar issues for elliptic operators in smooth domains and shows that the question posed by alonso and simon in 1980 pertaining to the validity of the above weyl asymptotic formula continues to have an affirmative answer in this nonsmooth setting we also study certain exterior type domains r n k n 3 with k r n compact and vanishing bessel capacity b 2 2 k 0 to prove equality of friedrichs and krein laplacians in l 2 d n x that is c 0 has a unique nonnegative self adjoint extension in l 2 d n x

on travelling waves for the stochastic fisher kolmogorov petrovsky piscunov equation this paper is concerned with properties of the wave speed for the stochastically perturbed fisher kolmogorov petrovsky piscunov fkpp equation it was shown in the classical 1937 paper by kolmogorov petrovsky and piscunov that the large time behavior of the solution to the fkpp equation with heaviside initial data is a travelling wave in a seminal 1995 paper mueller and sowers proved that this also holds for a stochastically perturbed fkpp equation the wave speed depends on the strength of the noise in this paper bounds on the asymptotic behavior of the wave speed c as 0 and are obtained

the boundary control approach to the titchmarsh weyl m function i the response operator and the a amplitude we link the boundary control theory and the titchmarsh weyl theory this provides a natural interpretation of the a amplitude due to simon and yields a new efficient method to evaluate the titchmarsh weyl m function associated with the schrodinger operator h x 2 q x on l 2 0 with dirichlet boundary condition at x 0

stabilization by noise for a class of stochastic reaction diffusion equations we prove uniqueness ergodicity and strongly mixing property of the invariant measure for a class of stochastic reaction diffusion equations with multiplicative noise in which the diffusion term in front of the noise may vanish and the deterministic part of the equation is not necessary asymptotically stable to this purpose we show that the l1 norm of the difference of two solutions starting from any two different initial data converges a s to zero as time goes to infinity

interacting particles the stochastic fisher kolmogorov petrovsky piscounov equation and duality abstract the stochastic fisher kolmogorov petrovsky piscunov equation is t u x t d xx u u 1 u e u 1 u x t for 0 u 1 where x t is a gaussian white noise process in space and time here d and e are parameters and the equation is interpreted as the continuum limit of a spatially discretized set of ito equations solutions of this stochastic partial differential equation have an exact connection to the a a a reaction diffusion system at appropriate values of the rate coefficients and particles diffusion constant this relationship is called duality by the probabilists it is not via some hydrodynamic description of the interacting particle system in this paper we present a complete derivation of the duality relationship and use it to deduce some properties of solutions to the stochastic fisher kolmogorov petrovsky piscunov equation

optimal control of partially observable stochastic systems with an exponential of integral performance index the stochastic control problem with linear stochastic differential equations driven by brownian motion processes and as cost functional the exponential of a quadratic form is considered the soluti

on the optimal control of stochastic systems with an exponential of integral performance index abstract in this paper a theory of optimal control is developed for stochastic systems whose performance is measured by the exponential of an integral form such a formulation of the cost function is shown to be not only general and useful but also analytically tractable starting with very general classes of stochastic systems optimality conditions are obtained which exploit the multiplicative decomposability of the exponential of integral form specializing to partially observed systems of stochastic differential equations with brownian motion disturbances optimality conditions are obtained which parallel those for systems with integral costs also treated are the special cases of linear systems with exponential of quadratic costs for which explicit optimal controls are obtainable in addition several general results of independent interest are obtained which concern optimality of stochastic systems

anderson transition on the cayley tree as a traveling wave critical point for various probability distributions for anderson localization on the cayley tree we study the statistics of various observables as a function of the disorder strength w and the number n of generations we first consider the landauer transmission tn in the localized phase its logarithm follows the traveling wave form where i the disorder averaged value moves linearly and the localization length diverges as with loc 1 and ii the variable t is a fixed random variable with a power law tail p t 1 t 1 w for large t with 0 w 1 2 so that all integer moments of tn are governed by rare events in the delocalized phase the transmission tn remains a finite random variable as n and we measure near criticality the essential singularity with t 0 25 we then consider the statistical properties of normalized eigenstates x x 2 1 in particular the entropy s x x 2ln x 2 and the inverse participation ratios ipr iq x x 2q in the localized phase the typical entropy diverges as with s 1 5 whereas it grows linearly as styp n n in the delocalized phase finally for the ipr we explain how closely related variables propagate as traveling waves in the delocalized phase in conclusion both the localized phase and the delocalized phase are characterized by the traveling wave propagation of some probability distributions and the anderson localization delocalization transition then corresponds to a traveling non traveling critical point moreover our results point toward the existence of several length scales that diverge with different exponents at criticality

ito versus stratonovich a survey is given of the facts and fancies concerning the nonlinear langevin or ito equation actually it is merely a pre equation which becomes an equation when an interpretation rule is added the rules of ito and stratonovich differ but both are mathematically consistent and therefore equally admissible conventions the reason why they seem to lead to physical differences is that the langevin approach used to arrive at the equation involves a tacit assumption for systems with external noise this assumption can be justified and it is then clear that the stratonovich rule applies systems with internal noise however can only be properly described by a master equation and the ito stratonovich controversy never enters afterward one is free to model the resulting fluctuations either with an ito or a stratonovich scheme but that does not lead to any new information

a hamiltonian formulation of risk sensitive linear quadratic gaussian control we consider the optimal control problem with time homogeneous linear and in general non markov dynamics and risk sensitive criterion appeal to the extremal principle prescribed by the risk sensitive certainty equivalence principle rscep yields a symmetric equation system indicating that the extended hamiltonian formulation generalizes naturally to the risk sensitive case the conjugate variable of a hamiltonian formulation now has an interpretation in terms of forecasted process noise and the rscep in fact provides a stochastic maximum principle for which all variables have a clear interpretation and the desired measurability properties in the infinite horizon case meaningful under generalized controllability conditions optimal control is determined explicitly in terms of a canonical factorization for the case of imperfect process observation the rscep leads to coupled equation systems that can again be solved in terms of canonical factorizations in the time invariant stationary case

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

boundary value problems for elliptic partial differential operators on bounded domains for a symmetric operator or relation a with infinite deficiency indices in a hilbert space we develop an abstract framework for the description of symmetric and self adjoint extensions a of a as restrictions of an operator or relation t which is a core of the adjoint a this concept is applied to second order elliptic partial differential operators on smooth bounded domains and a class of elliptic problems with eigenvalue dependent boundary conditions is investigated

krein like extensions and the lower boundedness problem for elliptic operators abstract for selfadjoint extensions a of a symmetric densely defined positive operator a min the lower boundedness problem is the question of whether a is lower bounded if and only if an associated operator t in abstract boundary spaces is lower bounded it holds when the friedrichs extension a has compact inverse grubb 1974 also gorbachuk and mikhailets 1976 this applies to elliptic operators a on bounded domains for exterior domains a 1 is not compact and whereas the lower bounds satisfy m t m a the implication of lower boundedness from t to a has only been known when m t m a we now show it for general t the operator a a corresponding to t a i generalizing the krein von neumann extension a 0 appears here its possible lower boundedness for all real a is decisive we study this krein like extension showing for bounded domains that the discrete eigenvalues satisfy n t a a c a t n 2 m o t n 1 e 2 m for t

on bifurcations in a chaotically stirred excitable medium abstract we study a one dimensional filamental model of a chaotically stirred excitable medium in a numerical simulation we systematically explore its rich bifurcation scenarios involving saddle nodes hopf bifurcations and hysteresis loops the bifurcations are described in terms of two parameters signifying the excitability of the reacting medium and the strength of the chaotic stirring respectively the solution behaviour in particular at the bifurcation points is analytically described by means of a nonperturbative variational method using this method we reduce the partial differential equations to either algebraic equations for stationary solutions and bifurcations or to ordinary differential equations in the case of non stationary solutions and bifurcations we present numerical simulations corroborating our analytical results

computing stochastic traveling waves the aim of this paper is to investigate new numerical methods for computing traveling wave solutions and new ways for estimating characteristic properties such as wave speed for stochastically forced partial differential equations as a particular example we consider the nagumo equation with multiplicative noise which we mainly consider in the stratonovich sense a standard approach for determining the position and hence speed of a wave is to compute the evolution of a level set we compare this approach against an alternative where the wave position is found by minimizing the l 2 norm against a fixed reference profile this approach can be used to freeze or stop the wave and obtain a stochastic partial differential algebraic equation that we then discretize and solve although attractive because it leads to a smaller domain size it can be numerically unstable due to large convection terms we compare numerically the different approaches for estimating the wave speed we use these techniques to investigate the effect of both ito and stratonovich noise on the nagumo equation as correlation length and noise intensity increase

field theory of propagating reaction diffusion fronts the problem of velocity selection of reaction diffusion fronts has been widely investigated while the mean field limit results are well known theoretically there is a lack of analytic progress in those cases in which fluctuations are to be taken into account here we construct an analytic theory connecting the first principles of the reaction diffusion process to an effective equation of motion via field theoretic arguments and we arrive at results already confirmed by numerical simulations

spectral analysis of selfadjoint elliptic differential operators dirichlet to neumann maps and abstract weyl functions abstract the spectrum of a selfadjoint second order elliptic differential operator in l 2 r n is described in terms of the limiting behavior of dirichlet to neumann maps which arise in a multi dimensional glazman decomposition and correspond to an interior and an exterior boundary value problem this leads to pde analogs of renowned facts in spectral theory of odes the main results in this paper are first derived in the more abstract context of extension theory of symmetric operators and corresponding weyl functions and are applied to the pde setting afterwards

asymptotic properties of the solutions to stochastic kpp equations a reduction method is used to prove existence and uniqueness of strong solutions to stochastic kpp equations where the initial condition may be anticipating the asymptotic behavior of the solution for large time and space and the random travelling waves are then studied under two different basic assumptions

large deviations for the branching brownian motion in presence of selection or coalescence the large deviation function has been known for a long time in the literature for the displacement of the rightmost particle in a branching random walk brw or in a branching brownian motion bbm more recently a number of generalizations of the bbm and of the brw have been considered where selection or coalescence mechanisms tend to limit the exponential growth of the number of particles here we try to estimate the large deviation function of the position of the rightmost particle for several such generalizations the l bbm the n bbm and the coalescing branching random walk cbrw which is closely related to the noisy fkpp equation our approach allows us to obtain only upper bounds on these large deviation functions one noticeable feature of our results is their non analytic dependence on the parameters such as the coalescence rate in the cbrw

brunet derrida particle systems free boundary problems and wiener hopf equations we consider a branching selection system in r with n particles which give birth independently at rate 1 and where after each birth the leftmost particle is erased keeping the number of particles constant we show that as n the empirical measure process associated to the system converges in distribution to a deterministic measure valued process whose densities solve a free boundary integrodifferential equation we also show that this equation has a unique traveling wave solution traveling at speed c or no such solution depending on whether c a or c a where a is the asymptotic speed of the branching random walk obtained by ignoring the removal of the leftmost particles in our process the traveling wave solutions correspond to solutions of wiener hopf equations

brunet derrida behavior of branching selection particle systems on the line we consider a class of branching selection particle systems on mathbb r similar to the one considered by e brunet and b derrida in their 1997 paper shift in the velocity of a front due to a cutoff based on numerical simulations and heuristic arguments brunet and derrida showed that as the population size n of the particle system goes to infinity the asymptotic velocity of the system converges to a limiting value at the unexpectedly slow rate log n 2 in this paper we give a rigorous mathematical proof of this fact for the class of particle systems we consider the proof makes use of ideas and results by r pemantle and by n gantert y hu and z shi and relies on a comparison of the particle system with a family of n independent branching random walks killed below a linear space time barrier

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

on the nonlinear diffusion equation of kolmogorov petrovsky and piscounov abstract recently bramson proved a theorem that classifies the initial data under which solutions of the k p p equation converge to the appropriate travelling waves in this paper a simplified proof is given by using maximal principles instead of his brownian motion approach the regularity condition on the forcing term is also weakened

last passage time for linear diffusions and application to the emptying time of a box we study the statistics of last passage time for linear diffusions first we present an elementary derivation of the laplace transform of the probability density of the last passage time thus recovering known results from the mathematical literature we then illustrate them on several explicit examples in a second step we study the spectral properties of the schrodinger operator associated to such diffusions in an even potential u x u x unveiling the role played by the so called weyl coefficient indeed in this case our approach allows us to relate the last passage times for dual diffusions i e diffusions driven by opposite force fields and to obtain new explicit formulae for the mean last passage time we further show that for such even potentials the small time t expansion of the mean last passage time on the interval 0 t involves the korteveg de vries invariants which are well known in the theory of schrodinger operators finally we apply these results to study the emptying time of a one dimensional box of size l containing n independent brownian particles subjected to a constant drift in the scaling limit where both n rightarrow infty and l rightarrow infty keeping the density rho n l fixed we show that the limiting density of the emptying time is given by a gumbel distribution our analysis provides a new example of the applications of extreme value statistics to out of equilibrium systems

a multiscale analysis of stochastic bistable reaction diffusion equations abstract a multiscale analysis of 1d stochastic bistable reaction diffusion equations with additive noise is carried out w r t travelling waves within the variational approach to stochastic partial differential equations it is shown with explicit error estimates on appropriate function spaces that up to lower order w r t the noise amplitude the solution can be decomposed into the orthogonal sum of a travelling wave moving with random speed and into gaussian fluctuations a stochastic differential equation describing the speed of the travelling wave and a linear stochastic partial differential equation describing the fluctuations are derived in terms of the coefficients our results extend corresponding results obtained for stochastic neural field equations to the present class of stochastic dynamics

stability of the travelling wave solution of the fitzhugh nagumo system on demontre la stabilite relative au systeme complet d equations aux derivees partielles de ces ondes de propagation

interfacial velocity corrections due to multiplicative noise the problem of velocity selection for reaction fronts has been intensively investigated leading to the successful marginal stability ms approach for propagation into an unstable state because the front velocity is controlled by the leading edge which perforce has low density it is interesting to study the role that finite particle number fluctuations have on this picture here we use the well known mapping of discrete markov processes to stochastic differential equations and focus on the front velocity in the simple a a ensuremath leftrightarrows a system our results are consistent with a recent heuristic proposal that v mathrm ms ensuremath v ensuremath sim 1 mathrm ln 2 n

elliptic differential operators on lipschitz domains and abstract boundary value problems this paper consists of two parts in the first part which is of more abstract nature the notion of quasi boundary triples and associated weyl functions is developed further in such a way that it can be applied to elliptic boundary value problems on non smooth domains a key feature is the extension of the boundary maps by continuity to the duals of certain range spaces which directly leads to a description of all self adjoint extensions of the underlying symmetric operator with the help of abstract boundary values in the second part of the paper a complete description is obtained of all self adjoint realizations of the laplacian on bounded lipschitz domains as well as kre n type resolvent formulas and a spectral characterization in terms of energy dependent dirichlet to neumann maps these results can be viewed as the natural generalization of recent results by gesztesy and mitrea for quasi convex domains in this connection we also characterize the maximal range spaces of the dirichlet and neumann trace operators on a bounded lipschitz domain in terms of the dirichlet to neumann map the general results from the first part of the paper are also applied to higher order elliptic operators on smooth domains and particular attention is paid to the second order case which is illustrated with various examples

random travelling waves for the kpp equation with noise consider the stochastic partial differential equation formula where t x is two parameter while noise assume that u0 is a continuous function taking values in 0 1 such that for some constant a 0 we have c1 u0 x 1 for x a let the wavefront b t sup x r u t x 0 we show that for small enough and with probability 1 limt b t t exists and lies in 0 this limit depends only on the law of v t x u t b t x tends toward a stationary limit as t we also analyze the length of the region a t b t which is the smallest closed interval containing the points x at which 0 u t x 1 we show that the length of this region tends toward a stationary distribution thus the wavefront does not degenerate

dirichlet to neumann maps abstract weyl titchmarsh m functions and a generalized index of unbounded meromorphic operator valued functions abstract we introduce a generalized index for certain meromorphic unbounded operator valued functions the class of functions is chosen such that energy parameter dependent dirichlet to neumann maps associated to uniformly elliptic partial differential operators particularly non self adjoint schrodinger operators on bounded lipschitz domains and abstract operator valued weyl titchmarsh m functions and donoghue type m functions corresponding to closed extensions of symmetric operators belong to it the principal purpose of this paper is to prove index formulas that relate the difference of the algebraic multiplicities of the discrete eigenvalues of robin realizations of non self adjoint schrodinger operators and more abstract pairs of closed operators in hilbert spaces with the generalized index of the corresponding energy dependent dirichlet to neumann maps and abstract weyl titchmarsh m functions respectively

noise enhanced excitability in bistable activator inhibitor media we show that external fluctuations induce excitable behavior in a bistable spatially extended system with activator inhibitor dynamics of the fitzhugh nagumo type this can be understood as a mechanism for sustained signal propagation in bistable media the phase diagram of the stochastic system is analytically obtained and numerically verified for small noise intensities front propagation becomes unstable and excitable pulses arise as the only possible spatiotemporal behavior of the system for large noise intensities on the other hand the system enters an effective regime of oscillatory behavior where it exhibits spontaneous nucleation of pulses and synchronized firing

m operators a generalisation of weyl titchmarsh theory the theory of the weyl titchmarsh m function for second order ordinary differential operators is generalized and applied to partial differential operators of the form q x acting in three space dimensions weyl operators m z are defined as maps from l2 s 1 to l2 s 1 s 1 unit sphere in r3 for exterior and interior boundary value problems and for the partial differential operator acting in l2 r 3 with the standard weyl titchmarsh m function recovered in the special case that q is spherically symmetric the analysis is carried out rather explicitly allowing for the determination of precise norm bounds for m operators and for the proof of higher dimensional analogues of a number of the fundamental results of standard weyl titchmarsh theory

numerical simulation of stochastic pdes for excitable media we discuss the numerical solution of a number of stochastic perturbations of the barkley model of excitable media widely used in the study of spiral waves two numerical methods are considered for solving this equation one based on barkley s original formulation and one based oil spectral methods it is found to be beneficial to modify the nonlinearity describing the reaction kinetics an efficient method of approximating the wiener process is presented the effectiveness of the methods depends on the stochastic pde under consideration

mosaic length and finite interaction range effects in a one dimensional random energy model in this paper we study finite interaction range corrections to the mosaic picture of the glass transition as it emerges from the study of the kac limit of large interaction range for disordered models to this aim we consider point to set correlation functions or overlaps in a one dimensional random energy model as a function of the range of interaction in the kac limit the mosaic length defines a sharp first order transition separating a high overlap phase from a low overlap one correspondingly we find that overlap curves as a function of the window size and different finite interaction ranges cross roughly at the mosaic length nonetheless we find a very slow convergence to the kac limit and discuss why this could be a problem for measuring the mosaic length in realistic models

on the absolutely continuous spectrum of one dimensional schrodinger operators with square summable potentials for continuous and discrete one dimensional schrodinger operators with square summable potentials the absolutely continuous part of the spectrum is essentially supported by 0 and 2 2 respectively this fact is proved by considering a priori estimates for the transmission coefficient

a variational principle for kpp front speeds in temporally random shear flows we establish the variational principle of kolmogorov petrovsky piskunov kpp front speeds in temporally random shear flows with sufficiently decaying correlations a key quantity in the variational principle is the almost sure lyapunov exponent of a heat operator with random potential to prove the variational principle we use the comparison principle of solutions the path integral representation of solutions and large deviation estimates of the associated stochastic flows the variational principle then allows us to analytically bound the front speeds the speed bounds imply the linear growth law in the regime of large root mean square shear amplitude at any fixed temporal correlation length and the sublinear growth law if the temporal decorrelation is also large enough the so called bending phenomenon

variations on a theme of jost and pais abstract we explore the extent to which a variant of a celebrated formula due to jost and pais which reduces the fredholm perturbation determinant associated with the schrodinger operator on a half line to a simple wronski determinant of appropriate distributional solutions of the underlying schrodinger equation generalizes to higher dimensions in this multi dimensional extension the half line is replaced by an open set r n n n n 2 where has a compact nonempty boundary satisfying certain regularity conditions our variant involves ratios of perturbation determinants corresponding to dirichlet and neumann boundary conditions on and invokes the corresponding dirichlet to neumann map as a result we succeed in reducing a certain ratio of modified fredholm perturbation determinants associated with operators in l 2 d n x n n to modified fredholm determinants associated with operators in l 2 d n 1 n 2 applications involving the birman schwinger principle and eigenvalue counting functions are discussed