simulation of stopped diffusions in this work we study standard euler updates for simulating stopped diffusions as an immediate application we discuss the computation of first exit times of diffusions from a domain we focus on one dimensional situations and show how the ideas for the simulation of killed diffusions can be adapted to this problem in particular we give a fully implementable algorithm to compute the first exit time from an interval numerically the brownian motion case is treated in detail and extensions to general diffusions are given special emphasis is given to numerical experiments for every ansatz we include numerical experiments confirming the conjectured accuracy of our methods our algorithm is of order one in a weak sense comparisons with other algorithms are shown results that are superior to those obtained with other methods are presented when approximating a first hitting time distribution the results obtained with our algorithm are much better than those achieved with other methods

on skew brownian motion we consider the stochastic equation x t w t beta l x 0 t where w is a standard wiener process and l x 0 cdot is the local time at zero of the unknown process x there is a unique solution x and it is adapted to the fields of w if beta leq 1 but no solutions exist if beta 1 in the former case setting alpha beta 1 2 the unique solution x is distributed as a skew brownian motion with parameter alpha this is a diffusion obtained from standard wiener process by independently altering the signs of the excursions away from zero each excursion being positive with probability alpha and negative with probability 1 alpha finally we show that skew brownian motion is the weak limit as n rightarrow infty of n 1 2 s lbrack nt rbrack where s n is a random walk with exceptional behavior at the origin

on the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues abstract consider the inverse acoustic scattering problem for spherically symmetric inhomogeneity of compact support define the corresponding homogeneous and inhomogeneous interior transmission problems we study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem these transmission eigenvalues are shown to be zeros of the scattering amplitude and also the set of eigenvalues of a special sturm liouville problem a uniqueness theorem for the potential of the derived sturm liouville problem is proved when the data are the given spectra and partial knowledge of the potential a corollary of this theorem is a uniqueness theorem for the original inverse acoustic scattering problem

the semiclassical resolvent and the propagator for non trapping scattering metrics abstract consider a compact manifold with boundary m with a scattering metric g or equivalently an asymptotically conic manifold m g euclidean r n with a compactly supported metric perturbation is an example of such a space let be the positive laplacian on m g and v a smooth potential on m which decays to second order at infinity in this paper we construct the kernel of the operator h 2 v 0 i 0 2 1 at a non trapping energy 0 0 uniformly for h 0 h 0 h 0 0 small within a class of legendre distributions on manifolds with codimension three corners using this we construct the kernel of the propagator e i t 2 v t 0 t 0 as a quadratic legendre distribution we also determine the global semiclassical structure of the spectral projector poisson operator and scattering matrix

complex scaling and the distribution of scattering poles the purpose of this paper is to establish sharp polynomial bounds on the number of scattering poles for a general class of compactly supported self adjoint perturbations of the laplacian in r n odd we also consider more general types of bounds that give sharper estimates in certain situations the general conclusion can be stated as follows the order of growth of the poles is the same as the order of growth of eigenvalues for corresponding compact problems from the few known cases however the exact asymptotics are expected to be different the scattering poles for compactly supported perturbations were rigorously defined by lax and phillips 13 in a more general setting they correspond to resonances the study of which has a long tradition in mathematical physics in the lax phillips theory they appear as the poles of the meromorphic continuation of the scattering matrix and coincide with the poles of the meromorphic continuation of the resolvent of the perturbed operator because of the latter characterization they can be considered as the analogue of the discrete spectral data for problems on noncompact domains the problem of estimating the counting function

scattering metrics and geodesic flow at infinity any compact manifold with boundary admits a riemann metric on its interior taking the form x 4 dx 2 x 2 h near the boundary where x is a boundary defining function and h is a smooth symmetric 2 cotensor restricting to be positive definite and hence a metric h on the boundary the scattering theory associated to the laplacian for such a scattering metric was discussed by the first author and here it is shown as conjectured that the scattering matrix is a fourier integral operator which quantizes the geodesic flow on the boundary for the metric h at time to prove this the poisson operator of the associated generalized boundary problem is constructed as a fourier integral operator associated to a singular legendre manifold

geometric bounds on the density of resonances for semiclassical problems in this paper we shall give upper bounds on the number of resonances in certain regions in the complex plane close to the real axis for semiclassical operators like h 2 v x when h is small

efficient random walk particle tracking algorithm for advective dispersive transport in media with discontinuous dispersion coefficients and water contents 1 random walk particle tracking rwpt is a well established and efficient alternative to grid based eulerian approaches when simulating the advection dispersion transport problem in highly heterogeneous porous media however rwpt methods suffer from a lack of accuracy when the dispersion tensor or the water content is spatially discontinuous we present improvements to the concept of a partially reflecting barrier used to account for these discontinuities 1 the nonlinear time splitting with that corrects for the systematic overestimation of the second dispersion displacement across an element interface when linear time splitting is used 2 the one sided reflection coefficient that correctly represents the effect of discontinuous dispersion coefficients and water content but eliminates redundant reflections of the two sided reflection coefficient and limits the error for discrete and 3 the transformation of the dispersive displacement across the element interface for complex multidimensional transport problems the proposed improvements are verified numerically by comparison with an analytical solution and a reference rwpt method the results indicate an increased efficiency and accuracy of the new rwpt algorithm because the new algorithm efficiently simulates both advection and dispersion dominated transport conditions it enhances the applicability of rwpt to scenarios in which both conditions occur as for example in the highly transient unsaturated zone the algorithm is easily implemented and it is shown that the computational benefit increases with increasing variability of the hydraulic parameter field

scattering asymptotics for riemann surfaces in this article we prove the optimal polynomial lower bound for the number of resonances of a surface with hyperbolic ends we also give weyl asymptotics for the relative scattering phase of such a surface the proofs are based on trace formulae analogous to those of the euclidean odd dimensional scattering the main technical ingredient is a new proof of the poisson formula theorem 5 7 which is applicable in the euclidean case as well our lower bound seems to be the first example of an optimal polynomial lower bound for the number of resonances holding for a general class of higher dimensional elliptic operators with no symmetries the previous general lower bounds or asymptotics were either nonoptimal 25 58 9 one dimensional or radial 65 67 and 54 41 1 or they required some degeneracy of the

decay of correlations for normally hyperbolic trapping we prove that for evolution problems with normally hyperbolic trapping in phase space correlations decay exponentially in time normally hyperbolic trapping means that the trapped set is smooth and symplectic and that the flow is hyperbolic in directions transversal to it flows with this structure include contact anosov flows classical flows in molecular dynamics and null geodesic flows for black holes metrics the decay of correlations is a consequence of the existence of resonance free strips for green s functions cut off resolvents and polynomial bounds on the growth of those functions in the semiclassical parameter

local smoothing for scattering manifolds with hyperbolic trapped sets we prove a resolvent estimate for the laplace beltrami operator on a scattering manifold with a hyperbolic trapped set and as a corollary deduce local smoothing we use a result of nonnenmacher zworski to provide an estimate near the trapped region a result of burq and cardoso vodev to provide an estimate near infinity and the microlocal calculus on scattering manifolds to combine the two

distribution of resonances for asymptotically euclidean manifolds in this paper we discuss meromorphic continuation of the resolvent and bounds on the number of resonances for scattering manifolds a class of manifolds generalizing euclidian n space subject to the basic assumption of analyticity near infinity we show that resolvent of the laplacian has a meromorphic continuation to a conic neighborhood of the continuous spectrum this involves a geometric interpretation of the complex scaling method in terms of deformations in the grauert tube of the manifold we then show that the number of resonances poles of the meromorphic continuation of the resolvent in a conic neighborhood of mathbb r of absolute value less than r 2 is mathcal o r n under the stronger assumption of global analyticity and hyperbolicity of the geodesic flow we prove a finer weyl type upper bound for the counting function for resonances in small neighborhoods of the real axis this estimate has an exponent which involves the dimension of the trapped set of the geodesic flow

propagation of singularities for the wave equation on conic manifolds for the wave equation associated to the laplacian on a compact manifold with boundary with a conic metric with respect to which the boundary is metrically a point the propagation of singularities through the boundary is analyzed under appropriate regularity assumptions the diffracted non direct wave produced by the boundary is shown to have sobolev regularity greater than the incoming wave

inverse problem with transmission eigenvalues for the discrete schrodinger equation the discrete schrodinger equation with the dirichlet boundary condition is considered on a half line lattice when the potential is real valued and compactly supported the inverse problem of recovery of the potential from the corresponding transmission eigenvalues is analyzed the marchenko method and the gel fand levitan method are used to solve the inverse problem uniquely except in one unusual case where the sum of the transmission eigenvalues is equal to a certain integer related to the support of the potential it is shown that in the unusual case there may be a unique potential corresponding to a given set of transmission eigenvalues there may be a finite number of distinct potentials for a given set of transmission eigenvalues or there may be infinitely many potentials for a given set of transmission eigenvalues the theory presented is illustrated with several explicit examples

reply to comment by d h lim on diffusion processes in composite porous media and their numerical integration by random walks generalized stochastic differential equations with discontinuous coefficients 1 we thank lim 2005 for providing this opportunity to clarify the restrictions and confirm the applicability of the results of labolle et al 2000 labolle et al 2000 develop generalized stochastic differential equations sde that converge to advection dispersion equations with discontinuous dispersion tensor d the problem of interest here relates to equation 15a of labolle et al 2000 and its application in the numerical simulation of random walks converging to advection dispersion equations with discontinuous coefficients in one dimension this equation can be written as

the fbi transform on compact manifolds we present a geometric theory of the fourier bros iagolnitzer transform on a compact c manifold m the fbi transform is a generalization of the classical notion of the wave packet transform we discuss the mapping properties of the fbi transform and its relationship to the calculus of pseudodifferential operators on m we also describe the microlocal properties of its range in terms of the scattering calculus of pseudodifferential operators on the noncompact manifold t m

diffusion processes in composite porous media and their numerical integration by random walks generalized stochastic differential equations with discontinuous coefficients discontinuities in effective subsurface transport properties commonly arise l at abrupt contacts between geologic materials i e in composite porous media and 2 in discrete velocity fields of numerical groundwater flow solutions however standard random walk methods for simulating transport and the theory on which they are based diffusion theory and the theory of stochastic differential equations sdes only apply when effective transport properties are sufficiently smooth limitations of standard theory have precluded development of random walk methods diffusion processes that obey advection dispersion equations in composite porous media in this paper we 1 generalize sdes to the case of discontinuous coefficients i e step functions and 2 develop random walk methods to numerically integrate these equations the new random walk methods obey advection dispersion equations even in composite media the techniques retain many of the computational advantages of standard random walk methods including the ability to efficiently simulate solute mass distributions and arrival times while suppressing errors such as numerical dispersion examples relevant to the simulation of subsurface transport demonstrate the new theory and methods the results apply to problems found in many scientific disciplines and offer a unique contribution to diffusion theory and the theory of sdes

on open scattering channels for manifolds with ends abstract in the framework of time dependent geometric scattering theory we study the existence and completeness of the wave operators for perturbations of the riemannian metric for the laplacian on a complete manifold of dimension n the smallness condition for the perturbation is expressed intrinsically and coordinate free in purely geometric terms using the harmonic radius therefore the size of the perturbation can be controlled in terms of local bounds on the injectivity radius and the ricci curvature as an application of these ideas we obtain a stability result for the scattering matrix with respect to perturbations of the riemannian metric this stability result implies that a scattering channel which interacts with other channels preserves this property under small perturbations

on some properties of reflected skew brownian motions and applications to dispersion in heterogeneous media abstract motivated by the close connection between the skew brownian motion and the random particle motion in heterogeneous media we investigate the reflected skew brownian motion and try to find out its relationship with the corresponding dispersion problem when there exists a reflecting boundary through the use of the knowledge of stochastic analysis we provide some basic properties of reflected skew brownian motions including the transition density the laplace transform of the first passage time and some related results a simple method to generate the sample path is also proposed at the end of this paper we reveal the strong relationship between the reflected skew brownian motion and the solute dispersion in the presence of a sharp interface and a reflecting boundary

a note on the theoretical foundations of particle tracking methods in heterogeneous porous media 1 in this technical note the stochastic process for the path of particles undergoing convection dispersion in a heterogeneous media with sharp interfaces is derived the derived process is shown to correspond to the numerical particle tracking scheme introduced on a more heuristic basis by hoteit et al 2002 who validated the accuracy of their numerical method in controlled laboratory experiments specifically it is shown that in the direction transverse to the flow u y and across an interface where the diffusion coefficient is discontinuous the stochastic particle path t y t is given by a rescaled version of ito and mckean s 1963 skew brownian motion for a uniquely determined value of a that depends on the values of the diffusion coefficient at either side of the interface these results help supply a theoretical foundation that supports the numerical schemes and experimental findings of hoteit et al 2002

new monte carlo schemes for simulating diffusions in discontinuous media we introduce new monte carlo simulation schemes for diffusions in a discontinuous media divided in subdomains with piecewise constant diffusivity these schemes are higher order extensions of the usual schemes and take into account the two dimensional aspects of the diffusion at the interface between subdomains this is achieved using either stochastic process techniques or an approach based on finite differences numerical tests on elliptic parabolic and eigenvalue problems involving an operator in divergence form show the efficiency of these new schemes

quasimodes and resonances sharp lower bounds we prove that asymptotically any cluster of quasimodes close to each other approximates at least the same number of resonances counting multiplicities as a consequence we get that the counting function of the number of resonances close to the real axis is bounded from below essentially by the counting function of the quasimodes

monte carlo methods for calculating some physical properties of large molecules in this paper we describe monte carlo methods for solving some boundary value problems for elliptic partial differential equations arising in the computation of physical properties of large molecules the constructed algorithms are based on walk on spheres green s function first passage walk in subdomains techniques and finite difference approximations of the boundary condition the methods are applied to calculating the diffusion limited reaction rate the electrostatic energy of a molecule and point values of an electrostatic field

diffusion theory for transport in porous media transition probability densities of diffusion processes corresponding to advection dispersion equations local scale spatial averaging of pore scale advection diffusion equations in porous media leads to advection dispersion equations ades while often used to describe subsurface transport ades may pose special problems in the context of diffusion theory standard diffusion theory applies only when characteristic coefficients velocity porosity and dispersion tensor are smooth functions of space subsurface porous material properties however naturally exhibit spatial variability transitions between material types are often abrupt rather than smooth such as sand in contact with clay in such composite porous media characteristic coefficients in spatially averaged transport equations may be discontinuous although commonly called on to model transport in these cases standard diffusion theory does not apply herein we develop diffusion theory for ades of transport in porous media derivation of ades from probabilistic assumptions yields 1 necessary conditions for convergence of diffusion processes to ades even when coefficients are discontinuous and 2 general probabilistic definitions of physical quantities velocity and dispersion tensor as examples of how the new theory can be applied to theoretical and numerical problems of transport in porous media we evaluate several random walk methods that have appeared in the water resources literature

an exponential timestepping algorithm for diffusion with discontinuous coefficients abstract we present a new monte carlo algorithm to simulate diffusion processes in presence of discontinuous convective and diffusive terms the algorithm is based on the knowledge of close form analytic expressions of the resolvents of the diffusion processes which are usually easier to obtain than close form analytic expressions of the density in the case of diffusion processes with piecewise constant coefficients known as skew diffusions such close form expressions for the resolvent are available then we apply our algorithm to this particular case and we show that the approximate densities of the particles given by the algorithm replicate the particularities of the true densities well discontinuities bimodality besides numerical experiments show a quick convergence

exponential timestepping with boundary test for stochastic differential equations we present new numerical methods for scalar stochastic differential equations successive time increments are independent random variables with an exponential distribution we perform numerical experiments using a double well potential exponential timestepping algorithms are efficient for escape time problems because a simple boundary test can be performed at the end of each step

semiclassical estimates in asymptotically euclidean scattering we consider long range semiclassical perturbations of the laplacian on asymptotically euclidean manifolds we obtain precise resolvent estimates under non trapping assumptions the novelty lies in a systematic use of geometric microlocal methods

reconstruction of a spherically symmetric speed of sound consider the inverse acoustic scattering problem for a spherically symmetric inhomogeneity of compact support that arises among other places in nondestructive testing define the corresponding homogeneous and inhomogeneous interior transmission problems see e g d colton and p monk quart j mech math 41 1988 pp 97 125 here the authors study the subset of transmission eigenvalues corresponding to spherically symmetric eigenfunctions of the homogeneous interior transmission problem it is shown in mclaughlin and polyakov j differential equations to appear that these eigenvalues are the zeros of an average of the scattering amplitude and a uniqueness theorem for the inverse acoustic scattering problem is presented where these eigenvalues are the given data in the present paper an algorithm for finding the solution of the inverse acoustic scattering problem from this subset of transmission eigenvalues is developed and implemented the method given here completely determines the sound

from analytical solutions of solute transport equations to multidimensional time domain random walk tdrw algorithms in this study new multi dimensional time domain random walk tdrw algorithms are derived from approximate one dimensional 1 d two dimensional 2 d and three dimensional 3 d analytical solutions of the advection dispersion equation and from exact 1 d 2 d and 3 d analytical solutions of the pure diffusion equation these algorithms enable the calculation of both the time required for a particle to travel a specified distance in a homogeneous medium and the mass recovery at the observation point which may be incomplete due to 2 d or 3 d transverse dispersion or diffusion the method is extended to heterogeneous media represented as a piecewise collection of homogeneous media the particle motion is then decomposed along a series of intermediate checkpoints located on the medium interface boundaries the accuracy of the multi dimensional tdrw method is verified against i exact analytical solutions of solute transport in homogeneous media and ii finite difference simulations in a synthetic 2 d heterogeneous medium of simple geometry the results demonstrate that the method is ideally suited to purely diffusive transport and to advection dispersion transport problems dominated by advection conversely the method is not recommended for highly dispersive transport problems because the accuracy of the advection dispersion tdrw algorithms degrades rapidly for a low peclet number consistent with the accuracy limit of the approximate analytical solutions the proposed approach provides a unified methodology for deriving multi dimensional time domain particle equations and may be applicable to other mathematical transport models provided that appropriate analytical solutions are available

an euler maruyama method for diffusion equations with discontinuous coefficients and a family of interface conditions abstract numerical methods are developed for linear parabolic equations in one spatial dimension having piecewise constant diffusion coefficients along with a one parameter family of interface conditions at the discontinuity we construct an euler maruyama numerical method for the stochastic differential equation sde corresponding to the alternative divergence formulation of these equations our main result is the construction of an euler scheme that can accommodate specification of any one of a family of interface conditions considered we then prove convergence estimates for the euler scheme to illustrate our method and its theoretical analysis we implement it for the stochastic formulation of the parabolic system

comment on diffusion processes in composite porous media and their numerical integration by random walks generalized stochastic differential equations with discontinuous coefficients by e m labolle j quastel g e fogg and j gravner

simulating diffusion processes in discontinuous media in this article we propose new monte carlo techniques for moving a diffusive particle in a discontinuous media in this framework we characterize the stochastic process that governs the positions of the particle the key tool is the reduction of the process to a skew brownian motion sbm in a zone where the coefficients are locally constant on each side of the discontinuity the new position of the particle after a constant time step is sampled from the exact distribution of the sbm process at the considered time to do so we propose two different but equivalent algorithms a two steps simulation with a stop at the discontinuity and a one step direct simulation of the sbm dynamic some benchmark tests illustrate their effectiveness

semiclassical resonances generated by a closed trajectory of hyperbolic type we determine all the resonances in certain rectangular regions of the complex plane of the schrodinger operator h2 v whenh 0 under the assumption that the set of trapped points of energy 0 for the classical flow form a closed trajectory and that the corresponding poincare map is hyperbolic

numerical approximation of irregular sdes via skorokhod embeddings abstract we provide a new algorithm for approximating the law of a one dimensional diffusion m solving a stochastic differential equation with possibly irregular coefficients the algorithm is based on the construction of markov chains whose laws can be embedded into the diffusion m with a sequence of stopping times the algorithm does not require any regularity or growth assumption in particular it applies to sdes with coefficients that are nowhere continuous and that grow superlinearly we show that if the diffusion coefficient is bounded and bounded away from 0 then our algorithm has a weak convergence rate of order 1 4 finally we illustrate the algorithm s performance with several examples

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

exact asymptotics for the probability of exit from a domain and applications to simulation we study the asymptotics of the exit probability p x s e t where is the exit time from an open set and p x s e is the law of a diffusion process with a small parameter e multiplying the diffusion coefficient we consider the case of the brownian bridge in many dimensions this choice being motivated by applications to numerical simulation the method uses recent results reducing the problem to the solution of a system of linear first order pde s

local smoothing for the schrodinger equation with a prescribed loss we consider a family of surfaces of revolution each with a single periodic geodesic which is degenerately unstable we prove a local smoothing estimate for solutions to the linear schrodinger equation with a loss that depends on the degeneracy and we construct explicit examples to show our estimate is saturated on a weak semiclassical time scale as a byproduct of our proof we obtain a cutoff resolvent estimate with a sharp polynomial loss

random walk simulation of transport in heterogeneous porous media local mass conservation problem and implementation methods the random walk method for simulating solute transport in porous media is typically based on the assumption that the velocity and velocity dependent dispersion tensor vary smoothly in space however in cases where sharp interfaces separate materials with contrasting hydraulic properties these quantities may be discontinuous normally velocities are interpolated to arbitrary particle locations when finite difference or finite element methods are used to solve the flow equation the use of interpolation schemes that preserve discontinuities in velocity at material contacts can result in a random walk model that does not locally conserve mass unless a correction is applied at these contacts test simulations of random walk particle tracking with and without special treatment of material contacts demonstrate the problem techniques for resolving the problem including interpolation schemes and a reflection principle are reviewed and tested results from simulations of transport in porous media with discontinuities in the dispersion tensor show which methods satisfy continuity simulations of transport in two dimensional heterogeneous porous media demonstrate the potentially significant effect of using a nonconservative model to compute spatial moments and breakthrough of a solute plume

the restaurant at the end of the random walk recent developments in the description of anomalous transport by fractional dynamics fractional dynamics has experienced a firm upswing during the past few years having been forged into a mature framework in the theory of stochastic processes a large number of research papers developing fractional dynamics further or applying it to various systems have appeared since our first review article on the fractional fokker planck equation metzler r and klafter j 2000a phys rep 339 1 77 it therefore appears timely to put these new works in a cohesive perspective in this review we cover both the theoretical modelling of sub and superdiffusive processes placing emphasis on superdiffusion and the discussion of applications such as the correct formulation of boundary value problems to obtain the first passage time density function we also discuss extensively the occurrence of anomalous dynamics in various fields ranging from nanoscale over biological to geophysical and environmental systems

a partially reflecting random walk on spheres algorithm for electrical impedance tomography in this work we develop a probabilistic estimator for the voltage to current map arising in electrical impedance tomography this novel so called partially reflecting random walk on spheres estimator enables monte carlo methods to compute the voltage to current map in an embarrassingly parallel manner which is an important issue with regard to the corresponding inverse problem our method uses the well known random walk on spheres algorithm inside subdomains where the diffusion coefficient is constant and employs replacement techniques motivated by finite difference discretization to deal with both mixed boundary conditions and interface transmission conditions we analyze the global bias and the variance of the new estimator both theoretically and experimentally subsequently the variance of the new estimator is considerably reduced via a novel control variate conditional sampling technique which yields a highly efficient hybrid forward solver coupling probabilistic and deterministic algorithms

simulating diffusion processes in discontinuous media benchmark tests abstract we present several benchmark tests for monte carlo methods simulating diffusion in one dimensional discontinuous media these benchmark tests aim at studying the potential bias of the schemes and their impact on the estimation of micro or macroscopic quantities repartition of masses fluxes mean residence time these benchmark tests are backed by a statistical analysis to filter out the bias from the unavoidable monte carlo error we apply them on four different algorithms the results of the numerical tests give a valuable insight into the fine behavior of these schemes as well as rules to choose between them