semiclassical analysis of low lying eigenvalues ii tunneling we discuss the leading asymptotics of eigenvalue splittings of 2 x2v in the limit as x o and where v is a non negative potential with several zeros for example if eo x e1 x are the two lowest eigenvalues in a situation where v has precisely two zeros a and b related by a symmetry then limx x lin e1 x eo x is given as the distance from a to b in a certain riemann metric

most probable dynamics of the tumor growth model with immune surveillance under cross correlated noises abstract the noise is inherent and indispensable in the tumor growth system therefore the behavior of the tumor system will display randomness different from the traditional analysis method such as the mean value variance etc the literature presents another deterministic tool i e the most probable trajectories which are defined by computing the spatial maximizers of the probability density function here we will investigate the tumor cell growth system with immune surveillance under correlated white noises from a deterministic point of view then the most probable extinction time is defined by the time when the most probable trajectories first escape to the extinction state from the tumor state afterward the probability ratio of extinction state versus tumor state characterizes treatment effects from the numerical simulation we derive that for the increasing cross correlation intensity of noises the most probable extinction time is enhanced and the therapeutic effect is weaken and conversely for the intensity of multiplicative noise in contrast there exists a critical intensity of additive noise at which the most probable extinction time is the smallest meanwhile treatment effects would be improved with the shrinking intensity of additive noise

a stochastic burgers equation from a class of microscopic interactions we consider a class of nearest neighbor weakly asymmetric mass conservative particle systems evolving on mathbb z which includes zero range and types of exclusion processes starting from a perturbation of a stationary state when the weak asymmetry is of order o n gamma for 1 2 gamma leq1 we show that the scaling limit of the fluctuation field as seen across process characteristics is a generalized ornstein uhlenbeck process however at the critical weak asymmetry when gamma 1 2 we show that all limit points satisfy a martingale formulation which may be interpreted in terms of a stochastic burgers equation derived from taking the gradient of the kpz equation the proofs make use of a sharp boltzmann gibbs estimate which improves on earlier bounds

two scale large deviations for chemical reaction kinetics through second quantization path integral motivated by the study of rare events for a typical genetic switching model in systems biology in this paper we aim to establish the general two scale large deviations for chemical reaction systems we build a formal approach to explicitly obtain the large deviation rate functionals for the considered two scale processes based upon the second quantization path integral technique we get three important types of large deviation results when the underlying two timescales are in three different regimes this is realized by singular perturbation analysis to the rate functionals obtained by the path integral we find that the three regimes possess the same deterministic mean field limit but completely different chemical langevin approximations the obtained results are natural extensions of the classical large volume limit for chemical reactions we also discuss its implication on the single molecule michaelis menten kinetics our framework and results can be applied to understand general multi scale systems including diffusion processes

asymptotically efficient simulation of elliptic problems with small random forcing recent rare event simulations show that the large deviation principle ldp for stochastic problems plays an important role in both theory and simulation for studying rare events induced by small

an adaptive high order minimum action method in this work we present an adaptive high order minimum action method for dynamical systems perturbed by small noise we use the hp finite element method to approximate the minimal action path and nonlinear conjugate gradient method to solve the optimization problem given by the freidlin wentzell least action principle the gradient of the discrete action functional is obtained through the functional derivative and the moving mesh technique is employed to enhance the approximation accuracy numerical examples are given to demonstrate the efficiency and accuracy of the proposed numerical method

large deviations with diminishing rates the theory of large deviations for jump markov processes has been generally proved only when jump rates are bounded below away from zero dupuis and ellis 1995 the large deviations principle for a general class of queueing systems i trans amer math soc 347 2689 2751 ignatiouk robert 2002 sample path large deviations and convergence parameters ann appl probab 11 1292 1329 shwartz and weiss 1995 large deviations for performance analysis chapman hall yet various applications of interest do not satisfy this condition we describe several classes of models where jump rates diminish to zero in a lipschitz continuous way under appropriate conditions we prove that the sample path large deviations principle continues to hold under our conditions the rate function remains an integral over a local rate function which retains its standard representation

an ordered line integral method for computing the quasi potential in the case of variable anisotropic diffusion abstract nongradient stochastic differential equations sdes with position dependent and anisotropic diffusion are often used in biological modeling the quasi potential is a crucial function in the large deviation theory that allows one to estimate transition rates between attractors of the corresponding ordinary differential equation and find the maximum likelihood transition paths unfortunately the quasi potential can rarely be found analytically it is defined as the solution to a certain action minimization problem in this work the recently introduced ordered line integral method olim is extended for computing the quasi potential for 2d sdes with anisotropic and position dependent diffusion scaled by a small parameter on a regular rectangular mesh the presented solver employs the dynamical programming principle at each step a local action minimization problem is solved using straight line path segments and the midpoint quadrature rule the solver is tested on two examples where analytic formulas for the quasi potential are available the dependence of the computational error on the mesh size the update factor k a key parameter of olims as well as the degree and the orientation of anisotropy is established the effect of anisotropy on the quasi potential and the maximum likelihood paths is demonstrated on the maier stein model the proposed solver is applied to find the quasi potential and the maximum likelihood transition paths in a model of the genetic switch in lambda phage between the lysogenic state where the phage reproduces inside the infected cell without killing it and the lytic state where the phage destroys the infected cell

shape fluctuations and random matrices we study a certain random growth model in two dimensions closely related to the one dimensional totally asymmetric exclusion process the results show that the shape fluctuations appropriately scaled converges in distribution to the tracy widom largest eigenvalue distribution for the gaussian unitary ensemble gue

on the distribution of the length of the longest increasing subsequence of random permutations let sn be the group of permutations of 1 2 n if 7r e sn we say that 7 i1 7f ik is an increasing subsequence in 7r if il i2 ik and 7r ii 7r i2 7r ik let 1n r be the length of the longest increasing subsequence for example if n 5 and 7r is the permutation 5 1 3 2 4 in one line notation thus 7r 1 5 7r 2 1 then the longest increasing subsequences are 1 2 4 and 1 3 4 and n 3 equip sn with uniform distribution

shock fluctuations in the asymmetric simple exclusion process we consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively q p ferrari et al 1991 have shown that if the initial measure isv a product measure with densities and to the left and right of the origin respectively then there exists a microscopic shock for the system a shock is a random positionx t such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively uniformly int we compute the diffusion coefficient of the shockd lim t t 1 e x t 2 ex t 2 and findd p q 1 1 1 as conjectured by spohn 1991 we show that in the scale sqrt t the position ofx t is determined by the initial distribution of particles in a region of length proportional tot we prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and this is the so called dynamical phase transition under shock initial conditions we show how the density fluctuation fields depend on the initial configuration

second order fluctuations and current across characteristic for a one dimensional growth model of independent random walks fluctuations from a hydrodynamic limit of a one dimensional asymmetric system come at two levels on the central limit scale n 1 2 one sees initial fluctuations transported along characteristics and no dynamical noise the second order of fluctuations comes from the particle current across the characteristic for a system made up of independent random walks we show that the second order fluctuations appear at scale n 1 4 and converge to a certain self similar gaussian process if the system is in equilibrium this limiting process specializes to fractional brownian motion with hurst parameter 1 4 this contrasts with asymmetric exclusion and hammersley s process whose second order fluctuations appear at scale n 1 3 as has been discovered through related combinatorial growth models

second class particles and cube root asymptotics for hammersley s process we show that for a stationary version of hammersley s process with poisson sources on the positive x axis and poisson sinks on the positive y axis the variance of the length of a longest weakly north east path l t t from 0 0 to t t is equal to 2e t x t where x t is the location of a second class particle at time t this implies that both e t x t and the variance of l t t are of order t2 3 proofs are based on the relation between the flux and the path of a second class particle continuing the approach of cator and groeneboom ann probab 33 2005 879 903

the geometric minimum action method a least action principle on the space of curves freidlin wentzell theory of large deviations for the description of the effect of small random perturbations on dynamical systems is exploited as a numerical tool specifically a numerical algorithm is proposed to compute the quasi potential in the theory which is the key object to quantify the dynamics on long time scales when the effect of the noise becomes ubiquitous the equilibrium distribution of the system the pathways of transition between metastable states and their rate etc can all be expressed in terms of the quasi potential we propose an algorithm to compute these quantities called the geometric minimum action method gmam which is a blend of the original minimum action method mam and the string method it is based on a reformulation of the large deviations action functional on the space of curves that allows one to easily perform the double minimization of the original action required to compute the quasi potential the theoretical background of the gmam in the context of large deviations theory is discussed in detail as well as the algorithmic aspects of the method the gmam is then illustrated on several examples a finite dimensional system displaying bistability and modeled by a nongradient stochastic ordinary differential equation an infinite dimensional analogue of this system modeled by a stochastic partial differential equation and an example of a bistable genetic switch modeled by a markov jump process

nested stochastic simulation algorithms for chemical kinetic systems with multiple time scales we present an efficient numerical algorithm for simulating chemical kinetic systems with multiple time scales this algorithm is an improvement of the traditional stochastic simulation algorithm ssa also known as gillespie s algorithm it is in the form of a nested ssa and uses an outer ssa to simulate the slow reactions with rates computed from realizations of inner ssas that simulate the fast reactions the algorithm itself is quite general and seamless and it amounts to a small modification of the original ssa our analysis of such multi scale chemical kinetic systems allows us to identify the slow variables in the system derive effective dynamics on the slow time scale and provide error estimates for the nested ssa efficiency of the nested ssa is discussed using these error estimates and illustrated through several numerical examples

current fluctuations of a system of one dimensional random walks in random environment we study the current of particles that move independently in a common static random environment on the one dimensional integer lattice a two level fluctuation picture appears on the central limit scale the quenched mean of the current process converges to a brownian motion on a smaller scale the current process centered at its quenched mean converges to a mixture of gaussian processes these gaussian processes are similar to those arising from classical random walks but the environment makes itself felt through an additional brownian random shift in the spatial argument of the limiting current process

order of current variance and diffusivity in the asymmetric simple exclusion process we prove that the variance of the current across a characteristic is of order t 2 3 in a stationary asymmetric simple exclusion process and that the diffusivity has order t 1 3 the proof proceeds via couplings to show the corresponding moment bounds for a second class particle

koszul duality in deformation quantization and tamarkin s approach to kontsevich formality let be a quadratic poisson bivector on a vector space v then one can also consider as a quadratic poisson bivector on the vector space v 1 fixed a universal deformation quantization prediction of some complex weights to all kontsevich graphs 12 we have deformation quantization of the both algebras s v and v these are graded quadratic algebras and therefore koszul algebras we prove that for some universal deformation quantization independent on these two algebras are koszul dual we characterize some deformation quantizations for which this theorem is true in the framework of the tamarkin s theory 19

optimal path to epigenetic switching we use large deviation methods to calculate rates of noise induced transitions between states in multistable genetic networks we analyze a synthetic biochemical circuit the toggle switch and compare the results to those obtained from a numerical solution of the master equation

relative formality theorem and quantisation of coisotropic submanifolds we prove a relative version of kontsevich s formality theorem this theorem involves a manifold m and a submanifold c and reduces to kontsevich s theorem if c m it states that the dgla of multivector fields on an infinitesimal neighbourhood of c is l quasiisomorphic to the dgla of multidifferential operators acting on sections of the exterior algebra of the conormal bundle applications to the deformation quantisation of coisotropic submanifolds are given the proof uses a duality transformation to reduce the theorem to a version of kontsevich s theorem for supermanifolds which we also discuss in physical language the result states that there is a duality between the poisson sigma model on a manifold with a d brane and the poisson sigma model on a supermanifold without branes or more properly with a brane which extends over the whole supermanifold

escape problem for irreversible systems the problem of noise induced escape from a metastable state arises in physics chemistry biology systems engineering and other areas the problem is well understood when the underlying dynamics of the system obey detailed balance when this assumption fails many of the results of classical transition rate theory no longer apply and no general method exists for computing the weak noise asymptotic behavior of fundamental quantities such as the mean escape time in this paper we present a general technique for analyzing the weak noise limit of a wide range of stochastically perturbed continuous time nonlinear dynamical systems we simplify the original problem which involves solving a partial differential equation into one in which only ordinary differential equations need be solved this allows us to resolve some old issues for the case when detailed balance holds when it does not hold we show how the formula for the asymptotic behavior of the mean escape time depends on the dynamics of the system along the most probable escape path we also present results on short time behavior and discuss the possibility of ital focusing along the escape path

on the geometry of chemical reaction networks lyapunov function and large deviations in an earlier paper we proved the validity of large deviations theory for the particle approximation of quite general chemical reaction networks in this paper we extend its scope and present a more geometric insight into the mechanism of that proof exploiting the notion of spherical image of the reaction polytope this allows to view the asymptotic behavior of the vector field describing the mass action dynamics of chemical reactions as the result of an interaction between the faces of this polytope in different dimensions we also illustrate some local aspects of the problem in a discussion of wentzell freidlin theory together with some examples

poincare birkhoff witt theorem for quadratic algebras of koszul type abstract in this paper we prove a general poincare birkhoff witt theorem for quadratic koszul algebras the result is similar to that obtained by polischuk and positselsky but the proof is entirely different and uses algebraic deformation theory in particular we obtain a new proof of the classical pbw theorem for lie algebras based on deformation theory

diffusive fluctuations for one dimensional totally asymmetric interacting random dynamics we study central limit theorems for a totally asymmetric one dimensional interacting random system the models we work with are the aldous diaconis hammersley process and the related stick model the a d h process represents a particle configuration on the line or a 1 dimensional interface on the plane which moves in one fixed direction through random local jumps the stick model is the process of local slopes of the a d h process and has a conserved quantity the results describe the fluctuations of these systems around the deterministic evolution to which the random system converges under hydrodynamic scaling we look at diffusive fluctuations by which we mean fluctuations on the scale of the classical central limit theorem in the scaling limit these fluctuations obey deterministic equations with random initial conditions given by the initial fluctuations of particular interest is the effect of macroscopic shocks which play a dominant role because dynamical noise is suppressed on the scale we are working

a numerical scheme for optimal transition paths of stochastic chemical kinetic systems we present a new framework for finding the optimal transition paths of metastable stochastic chemical kinetic systems with large system size the optimal transition paths are identified to be the most probable paths according to the large deviation theory of stochastic processes dynamical equations for the optimal transition paths are derived using the variational principle a modified minimum action method mam is proposed as a numerical scheme to solve the optimal transition paths applications to gene regulatory networks such as the toggle switch model and the lactose operon model in escherichia coli are presented as numerical examples

random perturbations of reaction diffusion equations the quasi deterministic approximation on considere des champs aleatoires u e t x u 1 e t x u n e t x definis comme les solutions d un systeme d equations aux derivees partielles u k e t l k u k e f k x u 1 e u n e e k t x les l k sont des operateurs elliptiques lineaires d ordre 2 k sont des champs gaussiens bruit blanc

existence of hydrodynamics for the totally asymmetric simple k exclusion process in a totally asymmetric simple k exclusion process particles take nearest neighbor steps to the right on the lattice z under the constraint that each site contain at most k particles we prove that such processes satisfy hydrodynamic limits under euler scaling and that the limit of the empirical particle profile is the entropy solution of a scalar conservation law with a concave flux function our technique requires no knowledge of the invariant measures of the process which is essential because the equilibria of asymmetric k exclusion are unknown but we cannot calculate the flux function precisely the proof proceeds via a coupling with a growth model on the two dimensional lattice in addition to the basic k exclusion with constant exponential jump rates we treat the site disordered case where each site has its own jump rate randomly chosen but frozen for all time the hydrodynamic limit under site disorder is new even for the simple exclusion process the case k 1 our proof makes no use of the markov property so at the end of the paper we indicate how to treat the case with arbitrary waiting times

a scaling theory of bifurcations in the symmetric weak noise escape problem we consider two dimensional overdamped double well systems perturbed by white noise in the weak noise limit the most probable fluctuational path leading from either point attractor to the separatrix the most probable escape path or mpep must terminate on the saddle between the two wells however as the parameters of a symmetric double well system are varied a unique mpep may bifurcate into two equally likely mpeps at the bifurcation point in parameter space the activation kinetics of the system become non arrhenius we quantify the non arrhenius behavior of a system at the bifurcation point by using the maslov wkb method to construct an approximation to the quasistationary probability distribution of the system that is valid in a boundary layer near the separatrix the approximation is a formal asymptotic solution of the smoluchowski equation our construction relies on a new scaling theory which yields critical exponents describing weak noise behavior at the bifurcation point near the saddle

a dynamic solver consistent minimum action method this paper discusses the necessity and strategy to unify the development of a dynamic solver and a minimum action method mam for a spatially extended system when employing the large deviation principle ldp to study the effects of small random perturbations a dynamic solver is used to approximate the unperturbed system and a minimum action method is used to approximate the ldp which corresponds to solving an euler lagrange equation related to but more complicated than the unperturbed system we will clarify possible inconsistencies induced by independent numerical approximations of the unperturbed system and the ldp based on which we propose to define both the dynamic solver and the mam on the same approximation space for spatial discretization the semi discrete ldp can then be regarded as the exact ldp of the semi discrete unperturbed system which is a finite dimensional ode system we achieve this methodology for the two dimensional navier stokes equations using a divergence free approximation space the method developed can be used to study the nonlinear instability of wall bounded parallel shear flows and be generalized straightforwardly to three dimensional cases numerical experiments are presented

deformation quantization with generators and relations abstract in this paper we prove a conjecture of b shoikhet which claims that two quantization procedures arising from fourier dual constructions actually coincide

hammersley s interacting particle process and longest increasing subsequences in a famous paper 8 hammersley investigated the lengthl n of the longest increasing subsequence of a randomn permutation implicit in that paper is a certain one dimensional continuous space interacting particle process by studying a hydrodynamical limit for hammersley s process we show by fairly soft arguments that limn 1 2 el n 2 this is a known result but previous proofs 14 11 relied on hard analysis of combinatorial asymptotics

generator estimation of markov jump processes based on incomplete observations nonequidistant in time markov jump processes can be used to model the effective dynamics of observables in applications ranging from molecular dynamics to finance in this paper we present a different method which allows the inverse modeling of markov jump processes based on incomplete observations in time we consider the case of a given time series of the discretely observed jump process we show how to compute efficiently the maximum likelihood estimator of its infinitesimal generator and demonstrate in detail that the method allows us to handle observations nonequidistant in time the method is based on the work of and bladt and sorensen j r stat soc ser b stat methodol 67 395 2005 but scales much more favorably than it with the length of the time series and the dimension and size of the state space of the jump process we illustrate its performance on a toy problem as well as on data arising from simulations of biochemical kinetics of a genetic toggle switch

the random average process and random walk in a space time random environment in one dimension we study space time fluctuations around a characteristic line for a one dimensional interacting system known as the random average process the state of this system is a real valued function on the integers new values of the function are created by averaging previous values with random weights the fluctuations analyzed occur on the scale n1 4 where n is the ratio of macroscopic and microscopic scales in the system the limits of the fluctuations are described by a family of gaussian processes in cases of known product form invariant distributions this limit is a two parameter process whose time marginals are fractional brownian motions with hurst parameter 1 4 along the way we study the limits of quenched mean processes for a random walk in a space time random environment these limits also happen at scale n1 4 and are described by certain gaussian processes that we identify in particular when we look at a backward quenched mean process the limit process is the solution of a stochastic heat equation