Distribution Of Brownian Motion Research Articles

Establishment of Mathematical Model of Random Motion Distribution of Brownian Motion Based on Dynamic Management Decision of Tourists

Establishment of Mathematical Model of Random Motion Distribution of Brownian Motion Based on Dynamic Management Decision of Tourists

Optimal transport and Skorokhod embedding

The Skorokhod embedding problem is to represent a given probability as the distribution of Brownian motion at a chosen stopping time. Over the last 50 years this has become one of the important classical problems in probability theory and a number of authors have constructed solutions with particular optimality properties. These constructions employ a variety of techniques ranging from excursion theory to potential and PDE theory and have been used in many different branches of pure and applied probability. We develop a new approach to Skorokhod embedding based on ideas and concepts from optimal mass transport. In analogy to the celebrated article of Gangbo and McCann on the geometry of optimal transport, we establish a geometric characterization of Skorokhod embeddings with desired optimality properties. This leads to a systematic method to construct optimal embeddings. It allows us, for the first time, to derive all known optimal Skorokhod embeddings as special cases of one unified construction and leads to a variety of new embeddings. While previous constructions typically used particular properties of Brownian motion, our approach applies to all sufficiently regular Markov processes.

What Exactly does a Random Walk Simulation Simulate?

One of the most common statements about random walks or Brownian motion is that they are statistically self-similar. Self-similarity is commonly depicted as an individual trajectory at different magnifications or plots of square displacement versus time at different magnifications. The first question is, what exactly does self-similarity mean here? That the trajectory itself and certain functionals of it are self-similar? That all functionals of it are self-similar? That all functionals of a random walk are self-similar in the limit as the random walk approaches Brownian motion? This problem is a form of the standard question of how a random walk (finite step length) approaches the limit of Brownian motion (infinitesimal step length). This problem is well understood mathematically but for applications it is useful to understand how the approach to the Brownian limit affects simulations. When does the histogram of a functional of a random walk approximate the corresponding distribution for Brownian motion? Second, what exactly does a random walk simulation yield? In the simulation the step size is chosen from a Rayleigh distribution with mean-square radius = 2d D₀ Δt, and the angle is random. This is exactly what the propagator for Brownian motion yields for a time interval Δt. Thus the usual random walk simulation is periodically sampled Brownian motion. Periodic sampling — in engineering terminology, downsampling — is a form of low-pass filtering. Likewise, an experimental single-particle tracking trajectory is to a first approximation periodically sampled Brownian motion, but there are deviations. At short distances, the dynamics is Newtonian, not Brownian, and at longer distances, complex behavior may occur, such as the Alder-Wainwright long-time tails from hydrodynamics, and the collective motions found by Vattulainen and collaborators. Supported in part by NIH grant GM038133.

Free convolution operators and free Hall transform

We define an extension of the polynomial calculus on a W⁎-probability space by introducing an algebra C{Xi:i∈I} which contains polynomials. This extension allows us to define transition operators for additive and multiplicative free convolution. It also permits us to characterize the free Segal–Bargmann transform and the free Hall transform introduced by Biane, in a manner which is closer to classical definitions. Finally, we use this extension of polynomial calculus to prove two asymptotic results on random matrices: the convergence for each fixed time, as N tends to ∞, of the ⁎-distribution of the Brownian motion on the linear group GLN(C) to the ⁎-distribution of a free multiplicative circular Brownian motion, and the convergence of the classical Hall transform on U(N) to the free Hall transform.

Fractional diffusion equations and processes with randomly varying time

In this paper the solutions $u_{\nu}=u_{\nu}(x,t)$ to fractional diffusion equations of order $0<\nu \leq 2$ are analyzed and interpreted as densities of the composition of various types of stochastic processes. For the fractional equations of order $\nu =\frac{1}{2^n}$, $n\geq 1,$ we show that the solutions $u_{{1/2^n}}$ correspond to the distribution of the $n$-times iterated Brownian motion. For these processes the distributions of the maximum and of the sojourn time are explicitly given. The case of fractional equations of order $\nu =\frac{2}{3^n}$, $n\geq 1,$ is also investigated and related to Brownian motion and processes with densities expressed in terms of Airy functions. In the general case we show that $u_{\nu}$ coincides with the distribution of Brownian motion with random time or of different processes with a Brownian time. The interplay between the solutions $u_{\nu}$ and stable distributions is also explored. Interesting cases involving the bilateral exponential distribution are obtained in the limit.

On Erlang(2) Risk Process Perturbed by Diffusion

ABSTRACT In this article, we consider an Erlang(2) risk process perturbed by diffusion. From the extreme value distribution of Brownian motion with drift and the renewal theory, we show that the survival probability satisfies an integral equation. We then give the bounds for the ultimate ruin probability and the ruin probability caused by claim. By introducing a random walk associated with the proposed risk process, we define an adjustment-coefficient. The relation between the adjustment-coefficient and the bound is given and the Lundberg-type inequality for the bound is obtained. Also, a formula of Pollaczek–Khinchin type for the bound is derived. Using these results, the bound can be calculated when claim sizes are exponentially distributed.

Repair times for systems that have high early failures

Active repair time is that portion of down time during which the system is worked on to effect a repair. Repair time includes preparation time, diagnostic time, correction time and final checkout time. Systems such as airborne communications transceivers, switching circuits and radar‐missile units usually suffer an initial high rate of wear and failure. Improvement in this area requires actions to reduce the frequency of failure and to increase ease of repair. This paper advances the first passage time distribution of Brownian motion as a repairability model. The paper fits the model to observed active repair time of radar systems, obtaining as a result estimates of the mean first passage time, drift and diffusion parameters of the associated Brownian motion. Hypothesizing the first passage time distribution of Brownian motion for active repair time data of radar systems, the Kolmogorov‐Smirnov test shows that the model is accepted and can be chosen as the parent population.

The distribution of brownian motion on linear stopping boundaries

We consider Brownian motion with drift and stopping boundaries: linear upper and lower boundaries, and possibly a vertical boundary at a truncation point, all under conditions assuring a finite stopping time. T. W. Anderson (Ann. Math. Statist. 31, 1960) derived formulas for the distributions of the stopped process along these boundaries and for the associated expected stopping times. We present simpler formulas, and briefer derivations.

On some examples of quadratic functionals of Brownian motion

During the last few years, several variants of P. Lévy's formula for the stochastic area of complex Brownian motion have been obtained. These are of interest in various domains of applied probability, particularly in relation to polymer studies. The method used by most authors is the diagonalization procedure of Paul Lévy. Here we derive one such variant of Lévy's formula, due to Chan, Dean, Jansons and Rogers, via a change of probability method, which reduces the computation of Laplace transforms of Brownian quadratic functionals to the computations of the means and variances of some adequate Gaussian variables. We then show that with the help of linear algebra and invariance properties of the distribution of Brownian motion, we are able to derive simply three other variants of Lévy's formula.

On some examples of quadratic functionals of Brownian motion

During the last few years, several variants of P. Lévy's formula for the stochastic area of complex Brownian motion have been obtained. These are of interest in various domains of applied probability, particularly in relation to polymer studies. The method used by most authors is the diagonalization procedure of Paul Lévy.Here we derive one such variant of Lévy's formula, due to Chan, Dean, Jansons and Rogers, via a change of probability method, which reduces the computation of Laplace transforms of Brownian quadratic functionals to the computations of the means and variances of some adequate Gaussian variables.We then show that with the help of linear algebra and invariance properties of the distribution of Brownian motion, we are able to derive simply three other variants of Lévy's formula.

Brownian motion in hyperspherical co-ordinates

The skew product representation of the Brownian-Bessel process is used in order to calculate certain semi-group densities and hitting distributions for Brownian motion in p+2 dimensions. The main analytical tool is Hankel's inversion theorem.

The distribution of Brownian motion in Rn at a natural stopping time

For a measure μ on R n let (( B t , P μ ) be Brownian motion in R n with initial distribution μ. Let D be an open subset of R n with exit time ζ ≡ inf { t > 0: B t ∉ D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν( dx) = P μ ( B T ∉ dx, T < ζ), where T is some natural stopping time for ( B t ), and we have applied this characterization to show that a measure ν in D satisfies G ν ⩽ G μ iff ν is of the form ν( dx) = P α ( B T ∉ dx, T < ζ) + β( dx), where T is some natural stopping time for ( B t ) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R 2 and μ is a finite measure on R 2 such that ∫ log + ∥ x∥ du( x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R 2 such that (log + ∥ B T∧ t ∥) 0< t<∞ is P μ-uniformly integrable.