Exit Distribution Research Articles

A Note on Hyperbolic von Mises Distributions

We define hyperbolic von Mises distributions in any integral dimension as exit distributions of hyperbolic Brownian motion (H(α)t,t≥0) with drift outside hyperbolic balls centred on the starting point H0. Bidimensional unwrapped hyperbolic distributions are also considered.

In vivo dosimetry with a liquid‐filled electronic portal imaging device

The thesis describes the use of a liquid‐filled electronic portal imaging device, EPID, for in vivo dosimetry, i.e., for the verification of transmission, exit, and midplane dose. First, the application of the EPID to measure the transmission dose, i.e., the dose behind the patient, was investigated under various clinical conditions. For this purpose corrections for the nonlinear response of the EPID with dose and the effects of photon scatter in the EPID were performed. Next, a convolution model was developed to convert transmission dose images to exit and midplane dose distributions in the patient. Patient inhomogeneities are taken into account by using the radiological path length through the patient, which is derived from the measured transmission dose, i.e., without using patient data (CT). Finally, the accuracy of EPID exit and midplane dose measurements was determined for several homogeneous and inhomogeneous phantoms in open and wedged beams and for a few patients treated for larynx, breast, lung and prostate cancer. It was found that for most phantoms exit and midplane doses agree within 2% (1 s.d.) with ionization chamber measurements. EPID midplane dose measurements during patient treatment generally agreed within 2.5% with 3D treatment planning calculations. Occasionally larger differences were observed due to differences between the actual patient anatomy during treatment and the patient CT data, used for treatment planning. It is concluded that transmission dosimetry with a liquid‐filled EPID can be used to determine the 2D exit and midplane dose for several treatment sites in clinical practice and to quantify the influence of anatomical changes on the dose delivery.

A characterization of certain diffusions on the Sierpinski gasket by the exit distributions

LetK be the Sierpinski gasket with verticesa 1,a 2,a 3 forming an equilateral triangle. Suppose that (X t ,P x ) and (Y t ,Q x ) are diffusions onK with the same hitting probabilities to verticesa 1,a 2,a 3. We show that ifX is an asymptotically one-dimensional diffusion or ap-stream diffusion, thenY is a time-change ofX.

On the exit law from saddle points

We consider the effects of adding an asymptotically small random (Brownian) perturbation to a planar dynamical system with a saddle point equilibrium. By applying techniques developed for the problem of exit from a stable equilibrium, we obtain a new limit law for the exit time from a neighborhood of the saddle, assuming the initial point is on the stable manifold. The limit law shows that the exit distribution depends on (the logarithm of) the noise parameter in an additive way. This gives a more accurate description of the exit law than the previous (but more general) results of Kifer and Mikami. Generalization to higher dimensions seems likely, although only if the unstable manifold has dimension 1.

Some Markov processes with Brownian exit distributions

LetD be a bounded domain inRd with regular boundary. LetX=(Xt, Px) be a standard Markov process inD with continuous paths up to its lifetime. IfX satisfies some weak conditions, then it is possible to add a non-local part to its generator, and construct the corresponding standard Markov process inD with Brownian exit distributions fromD.

The Hazard Rate Tangent Approximation for Boundary Hitting Times

We consider the problem of estimating first exit distributions for one-dimensional diffusion processes. We provide analytic bounds and an approximation which is shown to be accurate in a range of numerical examples involving Brownian motion.

Asymptotics of the Exit Distribution for Markov Jump Processes; Application to Atm

Abstract We approximate the exit distribution of a Markov jump process into a set of forbidden states and we apply these general results to an ATM multiplexor. In this case the forbidden states represent an overloaded multiplexor. Statistics for this overload or busy period are difficult to obtain since this is such a rare event. Starting from the approximate exit distribution, one may simulate the busy period without wasting simulation time waiting for the overload to occur.

Positive harmonic functions vanishing on the boundary for the Laplacian in unbounded horn-shaped domains

Denote points x ¯ ∈ R d + 1 \bar x \in {R^{d + 1}} , d ≥ 2 d \geq 2 , by x ¯ = ( ρ , θ , z ) \bar x = (\rho ,\theta ,z) , where ρ > 0 \rho > 0 , θ ∈ S d − 1 \theta \in {S^{d - 1}} , and z ∈ R z \in R . Let a : [ 0 , ∞ ) → ( 0 , ∞ ) a:[0,\infty ) \to (0,\infty ) be a nondecreasing C 2 {C^2} -function and define the "horn-shaped" domain Ω = { x ¯ = ( ρ , θ , z ) : | z | > a ( ρ ) } \Omega = \{ \bar x = (\rho ,\theta ,z):|z| > a(\rho )\} and its unit "cylinder" D = { x ¯ = ( ρ , θ , z ) ∈ Ω : ρ > 1 } D = \{ \bar x = (\rho ,\theta ,z) \in \Omega :\rho > 1\} . Under appropriate regularity conditions on a, we prove the following theorem: (i) If ∫ ∞ a ( ρ ) / ρ 2 d ρ = ∞ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho = \infty , then the Martin boundary at infinity for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega is a single point, (ii) If ∫ ∞ a ( ρ ) / ρ 2 d ρ > ∞ {\smallint ^\infty }a(\rho )/{\rho ^2}d\rho > \infty , then the Martin boundary at infinity for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega is homeomorphic to S d − 1 {S^{d - 1}} . More specifically, a sequence { ( ρ n , θ n , z n ) } n = 1 ∞ ⊂ Ω \{ ({\rho _n},{\theta _n},{z_n})\} _{n = 1}^\infty \subset \Omega satisfying lim n → ∞ ρ n = ∞ {\lim _{n \to \infty }}{\rho _n} = \infty is a Martin sequence if and only if lim n → ∞ θ n {\lim _{n \to \infty }}{\theta _n} exists on S d − 1 {S^{d - 1}} . From (i), it follows that the cone of positive harmonic functions in Ω \Omega vanishing continuously on ∂ Ω \partial \Omega is one-dimensional. From (ii), it follows easily that the cone of positive harmonic functions on Ω \Omega vanishing continuously on ∂ Ω \partial \Omega is generated by a collection of minimal elements which is homeomorphic to S d − 1 {S^{d - 1}} . In particular, the above result solves a problem stated by Kesten, who asked what the Martin boundary is for 1 2 Δ \frac {1}{2}\Delta in Ω \Omega in the case a ( ρ ) = 1 + ρ γ a(\rho ) = 1 + {\rho ^\gamma } , 0 > γ > 1 0 > \gamma > 1 . Our method of proof involves an analysis as ρ → ∞ \rho \to \infty of the exit distribution on ∂ D \partial D for Brownian motion starting from ( ρ , θ , z ) ∈ Ω (\rho ,\theta ,z) \in \Omega and conditioned to hit D before exiting Ω \Omega .

Exit distributions for symmetric Markov processes via Gaussian techniques

We derive the distribution of the first exit value for a class of symmetric real-valued Markov processes with finite Green's functions using prediction theory for Gaussian processes and Dynkin's theory which relates Markov and Gaussian processes. For Lévy processes with exponential lifetime this method allows us to easily rederive Rogozin's infinitely divisible factorization and to obtain the Fourier transform of the distribution of the first exit value.

Brownian Exit Distributions from Normal Balls in $S^3 \times H^3$

Brownian Exit Distributions from Normal Balls in $S^3 \times H^3$

A characterization ofh-Brownian motion by its exit distributions

LetX h be anh-Brownian motion in the unit ballD⊃R d withh harmonic, such that the representing measure ofh is not singular with respect to the surface measure on ∂D. IfY is a continuous strong Markov process inD with the same killing distributions asX h , thenY is a time change ofX h . Similar results hold in simply connected domains inC provided with either the Martin or the Euclidean boundary.

On the dirichlet problem for a class of second order pde systems with small parameter

The Dirichlet problems for a class of singularly perturbed systems of PDE are considered. We use the stochastic representation for solutions of such systems that based on the exit distributions for...

Large deviations results for the exit problem with characteristic boundary

We consider the exit problem for an asymptotically small random perturbation of a stable dynamical system in a region D. We show the standard large deviations results for the exit distribution and mean exit time, as obtained by Wentzell and Freidlin under the assumption of nontangential drift 〈 b, n〉 < 0, can be generalized to the case of 〈 b, n〉 = 0 on the boundary. The method is to apply the approach of Wentzell and Freidlin in the context of a conormally reflected version of the small noise diffusion. Some simple examples are described which illustrate the point that large deviations results alone will not be adequate to resolve the asymptotic behavior of the exit distribution in many natural problems.

Safety Topics

A final rule has been issued by the FAA which prohibits aircraft manufacturers and air carriers from increasing the distance between emergency exits to more than 60 feet. Existing regulations do not limit the distance and this rule is intended to ensure an opportunity for safe passenger evacuation during an emergency. The location of emergency eixts has been regulated by the FAA since 1967 by requiring the provision of an exit for every specified number of passengers and the exit being located where it would allow the most effective means of evacuation. The regulation also requires the exits to be distributed as uniformly as practicable and this has led to the underlying assumption that this results in a reasonable seat‐to‐exit and exit‐to‐exit distance. Some recent exit configurations do however, have distances that are greater than those originally envisaged. For example, the exit‐to‐exit distance that was originally 50 feet in an earlier version of the L—1011 increased to nearly 70 feet in a later model. Also, the exit‐to‐exit distance of 44 feet in the Boeing 747–100, –200 and –300 increases to nearly 70 feet in the 747–200 and 747–300 models with the No 3 exits deactivated. A recent certification request (which was refused by the FAA) proposed a derivative configuration of an aircraft with a distance substantially greater than 80 feet. These examples indicate the exit distribution requirement alone is ineffective in preventing increases in escape path distances.

Solution of non-coherent spectral line transfer problem for finite atmosphere by theF n -method

By use of the orthogonality and normalization integrals developed by McCormick and Siewert (1970) a set of singular integral equations suitable forFn-method is derived for non-coherent spectral line formation problem in finite media.Fn-equations for exit distributions are used to develop some algebraic equations with suitable recursion relations.

Recent progress on the small parameter exit problem†

In this paper we will bring together some recent results of S.-J. Sheu, T.A. Darden and ourselves and develop their application to the small parameter exit problem of A.D. Wentzell and M.I. Freidlin. This problem concerns the asymptotic behavior of the exit distribution from a domain of attraction for an exponentially stable critical point of a dynamical system with an asymptotically small random perturbation. Recent results of Day and Darden on regularity properties of the so-called quasipotential function allow certain improvements and generalizations to be made in the work fo S.-J. Sheu on the asyptotic behavior of the equilibrium density. Applying these results to the exit problem through its connection with the equilibrium density, [2], we obtain a new theorem on the exit problem: Theorem 4 below. This theorem subsumes previous results and generalizes the conclusion of the Matkowksi-Schuss-Kamin approach from smooth to nonsmooth quasipotental functions. In all cases the exit problem is reduced to the...

Multigroup linear transport in the exponential half-space—II

The multigroup half-space exit distribution problem in an exponential atmosphere is considered by two separate approaches. The first approach consists of an expansion of the multigroup angular flux solution in a half-range complete set of eigenfunctions. Then, through an orthogonality relation between these eigenfunctions and those of the adjoint transport equation, a set of singular integral equations is derived for the multigroup exit distributions. The half-range completeness of the eigenfunctions is then shown to imply that these singular integral equations possess a unique solution. The second approach consists of deriving this same set of singular integral equations by Laplace transforming the set of integral equations for the scalar group fluxes. Uniqueness of the solution is also shown in an appropriate function space. A multigroup version of the F-N method is then formulated and applied to the solution of the set of singular integral equations. Convergence is proven and the numerical results for the two group albedo is seen to be in good agreement with those obtained from the standard S-N method calculations.

Multigroup linear transport in the exponential half-space—I

A continuum set of singular eigenfunctions is constructed for the multigroup linear transport equation in the exponential atmosphere. This eigenfunction set is then shown to be complete for solving the subcritical half-space exit distribution problem.

Performance of Two Transonic Axial Compressor Rotors Incorporating Inlet Counterswirl

A single-stage axial-flow compressor which incorporates rotor inlet counterswirl to improve stage performance is discussed. Results for two rotor configurations are presented, including design and experimental test data. In this compressor design, inlet guide vanes were used to add counterswirl to the inlet of the rotor. The magnitude of the counterswirl was radially distributed to maximize the overall stage efficiency by minimizing the rotor combined losses (diffusion losses and shock losses). The shock losses were minimized by simultaneously optimizing the rotor blade section geometry, through-blade static pressure distribution, and leading edge aerodynamic/geometric shock sweep angles. Results from both the design and experimental performance analyses are presented and comparisons are made between the experimental data and the analyses and between the performance of both rotor designs. The computation of the flow field for both rotor designs and for the analysis of both tests was performed in an identical fashion using an axisymmetric, streamline-curvature-type code. Results presented include tip section blade-to-blade static pressure distributions and rotor through-blade and exit distributions of various aerodynamic parameters. The performance of this compressor stage represents a significant improvement in axial compressor performance compared to previous attempts to use rotor inlet counterswirl and to current, more conventional, state-of-the-art axial compressors operating under similar conditions.

A remark on the attainable distributions of controlled diffusions

It is proved that for controlled diffusions, the sets of attainable laws at a given time and attainable exit distributions, starting from a given initial law, are the same under the family of all nonanticipative controls and its subfamily of Markov controls.