Boundary Local Time Research Articles

Skorokhod decomposition for a reflected -strong Feller diffusion with singular drift

We construct Skorokhod decompositions for diffusions with singular drift and reflecting boundary behaviour on open subsets of with -smooth boundary except for a sufficiently small set. This decomposition holds almost surely under the path measures of the process for every starting point from an explicitly known set. This set is characterized by the boundary smoothness and the singularities of the drift term. We apply modern methods of Dirichlet form theory and -strong Feller processes. These tools have been approved as useful for the pointwise analysis of stochastic processes with singular drift and various boundary conditions. Furthermore, we apply Sobolev space theorems and elliptic regularity results to prove regularity properties of potentials related to surface measures. These are important ingredients for the pointwise construction of the boundary local time of the diffusions under consideration. As an application we construct stochastic dynamics for particle systems with hydrodynamic and pair interaction. Our approach allows highly singular potentials like Lennard–Jones potentials and position-dependent diffusion coefficients and thus the treatment of physically reasonable models.

Reflected Brownian motion: selection, approximation and linearization

We construct a family of SDEs whose solutions select a reflected Brownian flow as well as a stochastic damped transport process (W_t). The latter gives a representation for the solutions to the heat equation for differential 1-forms with the absolute boundary conditions; it evolves pathwise by the Ricci curvature in the interior, by the shape operator on the boundary and driven by the boundary local time, and has its normal part erased on the boundary. On the half line this construction selects the Skorohod solution (and its derivative with respect to initial points), not the Tanaka solution. On the half space this agrees with the construction of N. Ikeda and S. Watanabe \cite{Ikeda-Watanabe} by Poisson point processes. This leads also to an approximation for the boundary local time in the topology of uniform convergence; not in the semi-martingale topology, indicating the difficulty for the convergence of solutions of a family of random ODE's, with nice coefficients, to the solution of an equation with jumps and driven by the local time. In addition, We note that (W_t) is the weak derivative of a family of reflected Brownian motions with respect to the starting point.

Computation of the local time of reflecting Brownian motion and the probabilistic representation of the Neumann problem

In this paper, we propose numerical methods for computing the boundary local time of reflecting Brownian motion (RBM) in R and its use in the probabilistic representation of the solution of the Laplace equation with the Neumann boundary condition. Approximations of the RBM based on a walk-on-spheres (WOS) and random walk on lattices are discussed and tested for sampling the RBM paths and their applicability in finding accurate approximation of the local time and discretization of the probabilistic formula. Numerical tests for several types of domains (cube, sphere, and ellipsoid) have shown the convergence of the numerical methods as the length of the RBM path and number of paths sampled increase.

Numerical Solution of the Robin Problem of Laplace Equations with a Feynman–Kac Formula and Reflecting Brownian Motions

In this paper, we present numerical methods to implement the probabilistic representation of third kind (Robin) boundary problem for the Laplace equations. The solution is based on a Feynman---Kac formula for the Robin problem which employs the standard reflecting Brownian motion (SRBM) and its boundary local time arising from the Skorokhod problem. By simulating SRBM paths through Brownian motion using Walk on Spheres method, approximation of the boundary local time is obtained and the Feynman---Kac formula is then calculated by evaluating the average of all path integrals over the boundary under a measure defined through the local time. Numerical results demonstrate the accuracy and efficiency of the proposed method for finding a local solution of the Laplace equations with Robin boundary conditions.

Discrete approximations to local times for reflected diffusions

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

Functional central limit theorem for Brownian particles in domains with Robin boundary condition

We rigorously derive non-equilibrium space-time fluctuation for the particle density of a system of reflected diffusions in bounded Lipschitz domains in $\mathbb R^d$. The particles are independent and are killed by a time-dependent potential which is asymptotically proportional to the boundary local time. We generalize the functional analytic framework introduced by Kotelenez [19, 20] to deal with time-dependent perturbations. Our proof relies on Dirichlet form method rather than the machineries derived from Kotelenez's sub-martingale inequality. Our result holds for any symmetric reflected diffusion, for any bounded Lipschitz domain and for any dimension $d\geq 1$.

Large deviations for the boundary local time of doubly reflected Brownian motion

We compute a closed-form expression for the moment generating function fˆ(x;λ,α)=1λEx(eαLτ), where Lt is the local time at zero for standard Brownian motion with reflecting barriers at 0 and b, and τ∼Exp(λ) is independent of W. By analyzing how and where fˆ(x;⋅,α) blows up in λ, a large-time large deviation principle (LDP) for Lt/t is established using a Tauberian result and the Gärtner–Ellis Theorem.

Concentration for multidimensional diffusions and their boundary local times

We prove that probability laws of certain multidimensional semimartingales which includes time-inhomogenous diffusions, under suitable assumptions, satisfy quadratic transportation cost inequality under the uniform metric. From this we derive concentration properties of Lipschitz functions of process paths that depend on the entire history. In particular, we estimate concentration of boundary local time of reflected Brownian motions on a polyhedral domain. We work out explicit applications of consequences of measure concentration for the case of Brownian motion with rank-based drifts.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

Asymptotic variance parameters for the boundary local times of reflected Brownian motion on a compact interval

A direct derivation is given of a formula for the normalized asymptotic variance parameters of the boundary local times of reflected Brownian motion (with drift) on a compact interval. This formula was previously obtained by Berger and Whitt using an M/M/1/C queue approximation to the reflected Brownian motion. The bivariate Laplace transform of the hitting time of a level and the boundary local time up to that hitting time, for a one-dimensional reflected Brownian motion with drift, is obtained as part of the derivation.

Some Potential Theory for Reflecting Brownian Motion in Holder and Lipschitz Domains

Bounds are found on the transition densities and Green functions for Brownian motion with normal reflection in Holder and Lipschitz domains. For Lipschitz domains, reflecting Brownian motion and boundary local time are constructed, a Harnack inequality valid up to the boundary is proved, a probabilistic solution to the Neumann problem is given and the Kuramochi boundary is identified.

Probabilistic approach to the Neumann problem

In this paper, we consider the Neumann boundary value problem of Schrodinger operator with measure potential μ. First, a martingale formulation of the Neumann problem and an analytic characterization of the martingale formulation are given. Then, by using the Dirichlet forms and Stochastic analysis we obtain an explicit formula for the unique weak solution of this problem in terms of reflecting Brownias motion and it's boundary local time.

Boundary Local Time and Small Parameter Exit Problems with Characteristic Boundaries

The exit problem for an asymptotically small random perturbation of a stable dynamical system $x(t)$ in a region D is considered. The connection between the distribution of the position of first exit and the equilibrium density of the perturbed system subject to reflection from the boundary of D is developed. Earlier work treated the case in which $x(t)$ enters D nontangentially. Here the case in which $x(t)$ is everywhere tangent to the boundary is examined. The “small-noise” asymptotics of the boundary local time turn out to be of primary importance.

Probabilistic approach to the neumann problem

AbstractThe basic problem considered in this paper is to solve the following Neumann boundary value problem probabilistically:magnified image where we assume that q is in a certain functional class to be specified below, and φ is a bounded measurable function on the boundary. We give a martingale formulation of the Neumann problem and show that this formulation is essentially equivalent to the classical formulation. The paper culminates in an explicit formula for the solution of this problem in terms of reflecting Brownian motion and its boundary local time.