Diffusing particles with electrostatic repulsion

Summary. We study a diffusion model of an interacting particles system with general drift and diffusion coefficients, and electrostatic inter-particles repulsion. More precisely, the finite particle system is shown to be well defined thanks to recent results on multivalued stochastic differential equations (see [2]), and then we consider the behaviour of this system when the number of particles goes to infinity (through the empirical measure process). In the particular case of affine drift and constant diffusion coefficient, we prove that a limiting measure-valued process exists and is the unique solution of a deterministic PDE. Our treatment of the convergence problem (as ) is partly similar to that of T. Chan [3] and L.C.G. Rogers - Z. Shi [5], except we consider here a more general case allowing collisions between particles, which leads to a second-order limiting PDE. Received: 5 August 1996 / In revised form: 17 October 1996     

Adaptive estimation of an ergodic diffusion process based on sampled data

We consider adaptive maximum likelihood type estimation of both drift and diffusion coefficient parameters for an ergodic diffusion process based on discrete observations. Two kinds of adaptive maximum likelihood type estimators are proposed and asymptotic properties of the adaptive estimators, including convergence of moments, are obtained.     

On homogenization of time-dependent random flows

We study a diffusion with a random, time dependent drift. We prove the invariance principle when the spectral measure of the drift satisfies a certain integrability condition. This result generalizes the results of [13, 7]. Received: 25 February 2000 / Revised version: 11 December 2000 /?Published online: 14 June 2001     

Diffusions with singular drift related to wave functions

Summary Schrödinger equations are equivalent to pairs of mutually time-reversed non-linear diffusion equations. Here the associated diffusion processes with singular drift are constructed under assumptions adopted from the theory of Schrödinger operators, expressed in terms of a local space-time Sobolev space.By means of Nagasawa's multiplicative functionalN _s ^t , a Radon-Nikodym derivative on the space of continuous paths, a transformed process is obtained from Wiener measure. Its singular drift is identified by Maruyama's drift transformation. For this a version of Itô's formula for continuous space-time functions with first and second order derivatives in the sense of distributions satisfying local integrability conditions has to be derived.The equivalence is shown between weak solutions of a diffusion equation with singular creation and killing term and the solutions of a Feynman-Kac integral equation with a locally integrable potential function.     

Two Brownian particles with rank-based characteristics and skew-elastic collisions

We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems.     

Estimates for the density of functionals of SDEs with irregular drift

We obtain upper and lower bounds for the density of a functional of a diffusion whose drift is bounded and measurable. The argument consists of using Girsanov’s theorem together with an Itô–Taylor expansion of the change of measure. One then applies Malliavin calculus techniques in a non-trivial manner so as to avoid the irregularity of the drift. An integration by parts formula for this set-up is obtained.     

Long time asymptotics for constrained diffusions in polyhedral domains

We study long time asymptotic properties of constrained diffusions that arise in the heavy traffic analysis of multiclass queueing networks. We first consider the classical diffusion model with constant coefficients, namely a semimartingale reflecting Brownian motion (SRBM) in a d$d$-dimensional positive orthant. Under a natural stability condition on a related deterministic dynamical system [P. Dupuis, R.J. Williams, Lyapunov functions for semimartingale reflecting brownian motions, Annals of Probability 22 (2) (1994) 680–702] showed that an SRBM is ergodic. We strengthen this result by establishing geometric ergodicity for the process. As consequences of geometric ergodicity we obtain finiteness of the moment generating function of the invariant measure in a neighborhood of zero, uniform time estimates for polynomial moments of all orders, and functional central limit results. Similar long time properties are obtained for a broad family of constrained diffusion models with state dependent coefficients under a natural condition on the drift vector field. Such models arise from heavy traffic analysis of queueing networks with state dependent arrival and service rates.     

Convergence of a queueing system in heavy traffic with general patience-time distributions

We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity.     

Criteria for ergodicity of Lévy type operators in dimension one

Some sufficient conditions for the recurrence, the positive recurrence and the exponential ergodicity of one-dimensional Lévy type operators are presented. The conditions are classified according to different conditions on the ranges and integrability of the Lévy measure, based on the drift inequalities for the extended generator, and on a comparison with diffusion operators. A number of examples are illustrated, including the fractional Laplacian operator and the Ornstein–Uhlenbeck type operator.     

Multiscale diffusion approximations for stochastic networks in heavy traffic

Stochastic networks with time varying arrival and service rates and routing structure are studied. Time variations are governed by, in addition to the state of the system, two independent finite state Markov processes X and Y. The transition times of X are significantly smaller than typical inter-arrival and processing times whereas the reverse is true for the Markov process Y. By introducing a suitable scaling parameter one can model such a system using a hierarchy of time scales. Diffusion approximations for such multiscale systems are established under a suitable heavy traffic condition. In particular, it is shown that, under certain conditions, properly normalized buffer content processes converge weakly to a reflected diffusion. The drift and diffusion coefficients of this limit model are functions of the state process, the invariant distribution of X, and a finite state Markov process which is independent of the driving Brownian motion.     

Uniform concentration inequality for ergodic diffusion processes observed at discrete times

In this paper a concentration inequality is proved for the deviation in the ergodic theorem for diffusion processes in the case of discrete time observations. The proof is based on geometric ergodicity of diffusion processes. We consider as an application the nonparametric pointwise estimation problem of the drift coefficient when the process is observed at discrete times.     

On non-degenerate quasi-linear stochastic partial differential equations

We prove a limit theorem for non-degenerate quasi-linear parabolic SPDEs driven by space-time white noise in one space-dimension, when the diffusion coefficient is Lipschitz continuous and the nonlinear drift term is only measurable. Hence we obtain an existence and uniqueness and a comparison theorem, which generalize those in [2], [4], [5] to the case of non-degenerate SPDEs with measurable drift and Lipschitz continuous diffusion coefficients.Research supported by the Hungarian National Foundation of Scientific Research No. 2290.     

On quasi-linear stochastic partial differential equations

Summary We prove existence and uniqueness of the solution of a parabolic SPDE in one space dimension driven by space-time white noise, in the case of a measurable drift and a constant diffusion coefficient, as well as a comparison theorem.and INRIAPartially supported by DRET under contract 901636/A000/DRET/DS/SR     

Large deviations for small noise diffusions with discontinuous statistics

This paper proves the large deviation principle for a class of non-degenerate small noise diffusions with discontinuous drift and with state-dependent diffusion matrix. The proof is based on a variational representation for functionals of strong solutions of stochastic differential equations and on weak convergence methods. Received: 26 May 1998 / Revised version: 24 February 1999     

On diffusivity of a tagged particle in asymmetric zero-range dynamics

Consider a distinguished, or tagged particle in zero-range dynamics on Zd with rate g whose finite-range jump probabilities p possess a drift ∑jp(j)≠0. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all d?1, and at most order t in d=1 and d?3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and the rate g(k) increases, and g(k)/k either decreases or increases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion with a homogenized diffusion coefficient in the sense of finite-dimensional distributions. Some characterizations of the tagged particle variance are also given.     

Reflected BSDEs with nonpositive jumps,and controller-and-stopper games

We study a class of reflected backward stochastic differential equations with nonpositive jumps and upper barrier. Existence and uniqueness of a minimal solution are proved by a double penalization approach under regularity assumptions on the obstacle. In a suitable regime switching diffusion framework, we show the connection between our class of BSDEs and fully nonlinear variational inequalities. Our BSDE representation provides in particular a Feynman–Kac type formula for PDEs associated to general zero-sum stochastic differential controller-and-stopper games, where control affects both drift and diffusion term, and the diffusion coefficient can be degenerate. Moreover, we state a dual game formula of this BSDE minimal solution involving equivalent change of probability measures, and discount processes. This gives in particular a new representation for zero-sum stochastic differential controller-and-stopper games.     

Convergence of the Grünwald–Letnikov scheme for time-fractional diffusion

Using bivariate generating functions, we prove convergence of the Grünwald–Letnikov difference scheme for the fractional diffusion equation (in one space dimension) with and without central linear drift in the Fourier–Laplace domain as the space and time steps tend to zero in a well-scaled way. This implies convergence in distribution (weak convergence) of the discrete solution towards the probability of sojourn of a diffusing particle. The difference schemes allow also interpretation as discrete random walks. For fractional diffusion with central linear drift we show that in the Fourier–Laplace domain the limiting ordinary differential equation coincides with that for the solution of the corresponding diffusion equation.     

On mean-square stability properties of a new adaptive stochastic Runge–Kutta method

We analyze the mean-square (MS) stability properties of a newly introduced adaptive time-stepping stochastic Runge–Kutta method which relies on two local error estimators based on drift and diffusion terms of the equation [A. Foroush Bastani, S.M. Hosseini, A new adaptive Runge–Kutta method for stochastic differential equations, J. Comput. Appl. Math. 206 (2007) 631–644]. In the same spirit as [H. Lamba, T. Seaman, Mean-square stability properties of an adaptive time-stepping SDE solver, J. Comput. Appl. Math. 194 (2006) 245–254] and with applying our adaptive scheme to a standard linear multiplicative noise test problem, we show that the MS stability region of the adaptive method strictly contains that of the underlying stochastic differential equation. Some numerical experiments confirms the validity of the theoretical results.     

Strong convergence rates for backward Euler on a class of nonlinear jump-diffusion problems

We generalise the current theory of optimal strong convergence rates for implicit Euler-based methods by allowing for Poisson-driven jumps in a stochastic differential equation (SDE). More precisely, we show that under one-sided Lipschitz and polynomial growth conditions on the drift coefficient and global Lipschitz conditions on the diffusion and jump coefficients, three variants of backward Euler converge with strong order of one half. The analysis exploits a relation between the backward and explicit Euler methods.     

Measure-valued branching diffusions with spatial interactions

Summary A strong equation driven by a historical Brownian motion is used to construct and characterize measure-valued branching diffusions in which the spatial motions obey an Itô equation with drift and diffusion depending on the position of an individual and the entire population.