hitting properties of parabolic s p d e s with reflection keywords stochastic partial differential equations singular coefficients reflecting nonlinearity stochastic obstacle problem reference prob article 2006 003doi 10 1214 009117905000000792view record in web of science record created on 2008 12 01 modified on 2017 05 12

pinning and wetting transition for 1 1 dimensional fields with laplacian interaction we consider a random field 1 n as a model for a linear chain attracted to the defect line 0 that is the x axis the free law of the field is specified by the density exp iv i with respect to the lebesgue measure on n where is the discrete laplacian and we allow for a very large class of potentials v the interaction with the defect line is introduced by giving the field a reward 0 each time it touches the x axis we call this model the pinning model we consider a second model the wetting model in which in addition to the pinning reward the field is also constrained to stay nonnegative we show that both models undergo a phase transition as the intensity of the pinning reward varies both in the pinning a p and in the wetting a w case there exists a critical value ca such that when ca the field touches the defect line a positive fraction of times localization while this does not happen for ca delocalization the two critical values are nontrivial and distinct 0 cp cw and they are the only nonanalyticity points of the respective free energies for the pinning model the transition is of second order hence the field at cp is delocalized on the other hand the transition in the wetting model is of first order and for cw the field is localized the core of our approach is a markov renewal theory description of the field

practical variance reduction via regression for simulating diffusions the well known variance reduction methods the method of importance sampling and the method of control variates can be exploited if an approximation of the required solution is known here we employ conditional probabilistic representations of solutions together with the regression method to obtain sufficiently inexpensive although rather rough estimates of the solution and its derivatives by using the single auxiliary set of approximate trajectories starting from the initial position these estimates can effectively be used for significant reduction of variance and further accurate evaluation of the required solution the developed approach is supported by numerical experiments

an automatic adaptive importance sampling algorithm for molecular dynamics in reaction coordinates in this article we propose an adaptive importance sampling scheme for dynamical quantities of complex systems which are metastable the main idea of this article is to combine metadynamics an al

variance reduced weak monte carlo simulations of stochastically driven oscillators of engineering interest abstract analytical solutions of nonlinear and higher dimensional stochastically driven oscillators are rarely possible and this leaves the direct monte carlo simulation of the governing stochastic differential equations sdes as the only tool to obtain the required numerical solution engineers in particular are mostly interested in weak numerical solutions which provide a faster and simpler computational framework to obtain the statistical expectations moments of the response functions the numerical integration tools considered in this study are weak versions of stochastic euler and stochastic newmark methods a well known limitation of a monte carlo approach is however the requirement of a large ensemble size in order to arrive at convergent estimates of the statistical quantities of interest presently a simple form of a variance reduction strategy is proposed such that the ensemble size may be significantly reduced without affecting the accuracy of the predicted expectations of any function of the response vector process provided that the function can be adequately represented through a power series approximation the basis of the variance reduction strategy is to appropriately augment the governing system equations and then weakly replace the stochastic forcing function which is typically a filtered white noise process through a set of variance reduced functions in the process the additional computational cost due to system augmentation is far smaller than the accrued advantages due to a drastically reduced ensemble size indeed we show that the proposed method appears to work satisfactorily even in the special case of the ensemble size being just 1 the variance reduction scheme is first illustrated through applications to a nonlinear duffing equation driven by additive and multiplicative white noise processes a problem for which exact stationary solutions are known this is followed up with applications of the strategy to a few higher dimensional systems i e 2 and 3 dof nonlinear oscillators under additive white noises

an algorithmic introduction to numerical simulation of stochastic differential equations a practical and accessible introduction to numerical methods for stochastic differential equations is given the reader is assumed to be familiar with euler s method for deterministic differential equations and to have at least an intuitive feel for the concept of a random variable however no knowledge of advanced probability theory or stochastic processes is assumed the article is built around 10 matlab programs and the topics covered include stochastic integration the euler maruyama method milstein s method strong and weak convergence linear stability and the stochastic chain rule

a reflected stochastic heat equation as symmetric dynamics with respect to the 3 d bessel bridge abstract we prove that a stochastic heat equation with reflection at 0 on the spatial interval 0 1 with dirichlet boundary conditions and additive white noise admits an explicit symmetrizing invariant measure on c 0 1 the 3 d bessel bridge i e the law of the modulus of a 3 dimensional brownian motion conditioned to be 0 at time 1 a classical measure in probability theory also connected with the theory of excursions of brownian motion this is a non trivial example of a gibbs type measure being singular with respect to the reference gaussian measure and concentrated on the convex set of positive continuous functions on 0 1

hydrodynamic limit for an evolutional model of two dimensional young diagrams we construct dynamics of two dimensional young diagrams which are naturally associated with their grandcanonical ensembles by allowing the creation and annihilation of unit squares located at the boundary of the diagrams the grandcanonical ensembles which were introduced by vershik 17 are uniform measures under conditioning on their size or equivalently area we then show that as the averaged size of the diagrams diverges the corresponding height variable converges to a solution of a certain non linear partial differential equation under a proper hydrodynamic scaling furthermore the stationary solution of the limit equation is identified with the so called vershik curve we discuss both uniform and restricted uniform statistics for the young diagrams

hydrodynamics and large deviation for simple exclusion processes we prove the hydrodynamical limit for weakly asymmetric simple exclusion processes a large deviation property with respect to this limit is established for the symmetric case we treat also the situation where a slow reaction creation and annihilation of particles is present

existence and stability for fokker planck equations with log concave reference measure we study markov processes associated with stochastic differential equations whose non linearities are gradients of convex functionals we prove a general result of existence of such markov processes and a priori estimates on the transition probabilities the main result is the following stability property if the associated invariant measures converge weakly then the markov processes converge in law the proofs are based on the interpretation of a fokker planck equation as the steepest descent flow of the relative entropy functional in the space of probability measures endowed with the wasserstein distance

white noise driven quasilinear spdes with reflection we study reflected solutions of the heat equation on the spatial interval 0 1 with dirichlet boundary conditions driven by an additive space time white noise roughly speaking at any point x t where the solutionu x t is strictly positive it obeys the equation and at a point x t whereu x t is zero we add a force in order to prevent it from becoming negative this can be viewed as an extension both of one dimensional sdes reflected at 0 and of deterministic variational inequalities an existence and uniqueness result is proved which relies heavily on new results for a deterministic variational inequality

bessel spdes and renormalised local times in this article we prove integration by parts formulae ibpfs for the laws of bessel bridges from 0 to 0 over the interval 0 1 of dimension smaller than 3 as an application we construct a weak version of a stochastic pde having the law of a one dimensional bessel bridge i e the law of a reflected brownian bridge as reversible measure the dimension 1 being particularly relevant in view of applications to scaling limits of dynamical critical pinning models we also exploit the ibpfs to conjecture the structure of the stochastic pdes associated with bessel bridges of all dimensions smaller than 3

scaling limits of 1 1 dimensional pinning models with laplacian interaction we consider a random field 1 n with laplacian interaction of the form i v i where is the discrete laplacian and the potential v is symmetric and uniformly strictly convex the pinning model is defined by giving the field a reward e 0 each time it touches the x axis that plays the role of a defect line it is known that this model exhibits a phase transition between a delocalized regime e e c where 0 e c oo in this paper we give a precise pathwise description of the transition extracting the full scaling limits of the model we show in particular that in the delocalized regime the field wanders away from the defect line at a typical distance n 3 2 while in the localized regime the distance is just o log n 2 a subtle scenario shows up in the critical regime e e c where the field suitably rescaled converges in distribution toward the derivative of a symmetric stable levy process of index 2 5 our approach is based on markov renewal theory

efficient numerical solution of stochastic differential equations using exponential timestepping we present an exact timestepping method for brownian motion that does not require gaussian random variables to be generated time is incremented in steps that are exponentially distributed random variables boundaries can be explicitly accounted for at each timestep the method is illustrated by numerical solution of a system of diffusing particles

sequential monte carlo methods for diffusion processes in this paper we show how to use sequential monte carlo methods to compute expectations of functionals of diffusions at a given time and the gradients of these quantities w r t the initial condition of the process in some cases via the exact simulation of the diffusion there is no time discretization error otherwise the methods use euler discretization we illustrate our approach on both high and low dimensional problems from optimal control and establish that our approach substantially outperforms standard monte carlo methods typically adopted in the literature the methods developed here are appropriate for solving a certain class of partial differential equations as well as for option pricing and hedging

equilibrium fluctuations for nabla_ varphi interface model we study the large scale space time fluctuations of an interface which is modeled by a massless scalar field with reversible langevin dynamics for a strictly convex interaction potential we prove that on a large space time scale these fluctuations are governed by an infinite dimensional ornstein uhlenbeck process its effective diffusion type covariance matrix is characterized through a variational formula

robust variance reduction for random walk methods random walk methods are effective for solving linear partial differential equations in many dimensions especially those involving complex geometries they are based on an equivalence given by a feynman kac formula between an expectation of a functional of a stochastic process and the solution at a point of a partial differential equation the drawback is that the error is proportional only to the square root of the reciprocal of the number of trials efficiency depends critically on variance reduction a general strategy for doing this in the case of stochastic differential equations is proposed by milstein the idea is to introduce a bias in the drift term and to exactly compensate for this by unequal weighting of the trials there is an optimal bias defined in terms of the solution of the partial differential equation which reduces the variance to zero in practice an approximation is used this idea has been tested under the name biased brownian dynamics on the problem of calculating rate constants for diffusion limited reactions the approach is successful in some cases but is less successful in more difficult cases due to the occasional occurrence of a well above average weight proposed and tested here is a weight control algorithm which greatly enhances the effectiveness of biased brownian dynamics

a study of cell models 2 the effect of time delay on the limiting forms of cascade and manifold cell model response functions a simplified cell model was used to investigate the behavior of the impulse response function irf of cell models incorporating time delay elements as the number of subdivisions increases the simplified model used in this study was a single branch model consisting of a number of equal cells arranged in sequence two cell models were studied a cascade model and a manifold model in this study the model representing each cell is modified by the addition of a delay element which is taken to be linearly proportional to the distance of its corresponding cell from the outlet the limiting forms of the irfs are derived for the two models using laplace transforms the cascade cell model irf is found to converge to a rectangle regardless of the values of its parameters while the manifold cell model irf converges to a family of curves whose shapes depend quite sensitively on the values of the parameters chosen the expression derived contains the exponential integral function which has not been used before in surface hydrology for some combinations of the parameters the irf has the classical or intuitively expected shape consisting of a steep rise and mild recession for other values the irf is reminiscent of the kinematic wave solution to the overland flow equations and some comparisons are made the manifold cell model is found to hold promise as a suitably behaved alternative to the cascade cell model in the context of distributed surface runoff systems

a study of cell models 1 a manifold cell model for distributed surface runoff systems as a means of examining the behavior of distributed linear models of catchments and in particular the cell model a series of investigations were carried out on the modeling of a long narrow rectangular hypothetical catchment impulse response functions of three models were studied the conventional cascade a modified cascade which can be modeled in series or in parallel and a new model called the manifold cell model which is characterized by its exclusively parallel nature the response function of the cascade model converges to a rectangular pulse and that of the modified cascade model converges to a modified pulse the impulse response function for the manifold cell model converges to a family of shapes that look like hydrographs i e that are intuitively acceptable the manifold cell model approaches its asymptotic response with a modest number of cell divisions of the same order as the typical discretization along the longest watercourse of a natural watershed

the scaling limit of polymer pinning dynamics and a one dimensional stefan freezing problem we consider the stochastic evolution of a 1 1 dimensional interface or polymer in the presence of a substrate this stochastic process is a dynamical version of the homogeneous pinning model we start from a configuration far from equilibrium a polymer with a non trivial macroscopic height profile and look at the evolution of a space time rescaled interface in two cases we prove that this rescaled interface has a scaling limit on the diffusive scale space rescaled by l in both dimensions and time rescaled by l 2 where l denotes the length of the interface which we describe when the interaction with the substrate is such that the system is unpinned at equilibrium then the scaling limit of the height profile is given by the solution of the heat equation with dirichlet boundary condition when the attraction to the substrate is infinite the scaling limit is given by a free boundary problem which belongs to the class of stefan problems with contracting boundary also referred to as stefan freezing problems in addition we prove the existence and regularity of the solution to this problem until a maximal time where the boundaries collide our result provides a new rigorous link between stefan problems and statistical mechanics

stochastic burgers and kpz equations from particle systems we consider two strictly related models a solid on solid interface growth model and the weakly asymmetric exclusion process both on the one dimensional lattice it has been proven that in the diffusive scaling limit the density field of the weakly asymmetric exclusion process evolves according to the burgers equation and the fluctuation field converges to a generalized ornstein uhlenbeck process we analyze instead the density fluctuations beyond the hydrodynamical scale and prove that their limiting distribution solves the non linear burgers equation with a random noise on the density current for the solid on solid model we prove that the fluctuation field of the interface profile if suitably rescaled converges to the kardar parisi zhang equation this provides a microscopic justification of the so called kinetic roughening i e the non gaussian fluctuations in some non equilibrium processes our main tool is the cole hopf transformation and its microscopic version we also develop a mathematical theory for the macroscopic equations

conservative stochastic cahn hilliard equation with reflection we consider a stochastic partial differential equation with reflection at 0 and with the constraint of conservation of the space average the equation is driven by the derivative in space of a space time white noise and contains a double laplacian in the drift due to the lack of the maximum principle for the double laplacian the standard techniques based on the penalization method do not yield existence of a solution we propose a method based on infinite dimensional integration by parts formulae obtaining existence and uniqueness of a strong solution for all continuous nonnegative initial conditions and detailed information on the associated invariant measure and dirichlet form

unbiased monte carlo evaluation of certain functional integrals abstract a monte carlo algorithm for the evaluation of certain functional integrals is introduced this algorithm is not based on the approximation of the functional integral by a finite dimensional integral unbiased estimators for the functional integrals are constructed and corresponding variance reduction techniques are considered numerical results of simulation studies are presented

scaling limits of weakly asymmetric interfaces we consider three models of evolving interfaces intimately related to the weakly asymmetric simple exclusion process with n particles on a finite lattice of 2n sites our model 1 defines an evolving bridge on 0 1 our model 1 w an evolving excursion on 0 1 while our model 2 consists of an evolving pair of non crossing bridges on 0 1 based on the observation that the invariant measures of the dynamics depend on the area under or between the interface s we characterise the scaling limits of the invariant measures when the asymmetry of the exclusion process scales like n frac 3 2 then we show that the scaling limits of the dynamics themselves are expressed in terms of variants of the stochastic heat equation in particular in model 1 w we obtain the well studied reflected stochastic heat equation introduced by nualart and pardoux probab theory relat fields 93 1 77 89 1992