Local field brownian motion

A local field is any locally compact, non-discrete field other than the field of real numbers or the field of complex numbers. There is a natural notion of Gaussian measures on a local field vector space. We construct and study a specific local field Gaussian stochastic process taking values in a finite dimensional local field vector space and indexed by another finite dimensional local field vector space. This process has a structure that strongly reflects the algebraic and geometric structure of the underlying index space and, as such, plays the same role in the local field setting that standard Brownian motion and the related multiparameter processes such as Lévy's multiparameter Brownian motion play in a Euclidean context. We investigate the theory of additive functionals and the related potential theory for this process and show that it strongly resembles the Euclidean prototype. As a particular consequence of this investigation, we find that a local time process exists when the process hits points. We give two intrinsic constructions of the local time at a given level. These constructions are analogous to the dilation construction of Kingman and the Hausdorff measure construction of Taylor and Wendel in the Euclidean case. Finally, the local time is shown to be continuous as a measure valued stochastic process indexed by the levèl at which it is evaluated.Research supported in part by an NSF Grant and Presidential Young Investigator Award.     

Plane brownian motion in a region with holes

We consider a fixed family of balls with decreasing radii in the plane. We establish a relationship between a Dirichlet problem in a region without the balls and the solution of a Schroedinger equation in the complete region. Then we find upper bounds for the probability that a brownian motion exits the region without touching these balls. This is used to study harmonic measure and entire functions.     

Branching brownian motion with absorption

We consider a branching diffusion {Z_t}_t?0 in which particles move during their life time according to a Brownian motion with drift -μ and variance coefficient σ², and in which each particle which enters the negative half line is instantaneously removed from the population. If particles die with probability c dt+o(dt) in [t,t+dt] and if the mean number of offspring per particle is m>1, then Z_t dies out w.p.l. if $\text{μ?μ}{}_0\text{≡{2σ}{}^2\text{c(m?1)}}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$. If μ<μ₀, then itZ_t grows exponentially with positive probability. Our main concern here is with the critical case where μ=μ₀. Even though $\text{E}\text{{}\text{Z}{}_{\mathrm{T}}\text{}∽}\text{const.}\text{T}{}^{\mathrm{<mtext>?</mtext><mtext>3</mtext><mtext>2</mtext>}}$ in this case, we find that $\text{P}\text{{}\text{Z}{}_{\mathrm{T}}\text{>0}}$ is only exp$\text{{–}\text{const.}\text{T}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{+0(}\text{log}\text{T)}{}^2\text{}}$, and conditionally on {Z_T>0} there are with high probability much fewer particles alive at time T than $\text{E}\text{{Z}{}_{\mathrm{T}}\text{|Z}{}_{\mathrm{T}}\text{0}}$.     

Linear filtering with fractional brownian motion

A Kalman type system of integral equations is obtained for the linear filtering problem in which the noise generating the signal is a fractional Brownian motion with long-range dependence. The error in applying the usual Kalman filter to this problem is determined explicitly for a simple example     

Reflected brownian motion in a wedge: Semimartingale property

In this paper, the object of study is reflected Brownian motion in a two-dimensional wedge with constant direction of reflection on each side of the wedge. The basic question considered here is “When is this process a semimartingale?”. It is first shown that a related process, defined by specifying the corner of the wedge to be an absorbing state, rather than an instantaneous one, is a semimartingale. Conditions for the existence and uniqueness of the process for which the corner is an instantaneous state were given by Vardhan and Williams (“Brownian motion in a wedge with oblique reflection”, Comm. Pure Appl. Math., to appear). Under these conditions, it is shown that starting away from the corner, the process is a semimartingale if and only if there is a convex combination of the directions of reflection that points into the wedge. This equivalence is also shown to hold starting from the corner, except in one unresolved case for which the wedge angle exceeds π and the directions of reflection are exactly opposed.     

A brownian motion model of return migration

This paper considers a continuous time, continuous state stochastic process to determine a theoretical model and empirical parameters for the probability distribution of remigration. A Brownian motion model is used for simplicity, with empirical findings drawn from a study of Israeli return migrants. A negative relationship between remigration (sojourn) time and the probability of return time is used to provide forecasts of remigration which can help governments who seek actively the return of their migrants to reach better decisions regarding the timing of their efforts.     

Conjugate duality in the optimization of the search for a target with generalized conditionally deterministic motion

A target moves in Euclideann-spaceR ⁿ according to the generalized conditionally deterministic law. The search density that accumulates on the target during its route determines the probability of detection. A necessary and sufficient condition for the search density (x, t) to be optimal is first represented, when there are two types of constraints for the search density: pointwise constraints and total-amount constraints. The second part consists of formulation of the dual problem with the aid of sensitivity parameters for the constraints. By using the dual functional, we obtain the maximal error from the minimum value of the primal objective functional for an arbitrary feasible . Finally, we study the discretized case, which is necessary for numerical calculations.     

On generalized local time for the process of brownian motion

We prove that the functionals of a d-dimensional Brownian process are Hida distributions, i.e., generalized Wiener functionals. Here, δ_Γ(·) is a generalization of the δ-function constructed on a bounded closed smooth surface Γ⊂R ^d , k≥1 and acting on finite continuous functions φ(·) in R ^d according to the rule where ι(·) is a surface measure on Γ.     

Optimal ambushing search for a moving target

This paper investigates a search problem for a moving target in which a searcher can anticipate the probabilities of routes selected by the target but does not have any time information about when the target transits the route. If the searcher had some time information, he could develop an efficient search plan by varying allocations of search effort based on time. Due to the lack of time information, the searcher must ambush the target by distributing search effort to places where the target is likely to pass. There are few papers that deal mathematically with this type of search problem with no time information. Employing the criterion of detection probability, we formulate the problem and obtain necessary and sufficient conditions for the optimal solution. By applying the conditions, we propose two methods for solving the problem. The convex programming problem can be easily solved numerically by some well-known methods, e.g. the gradient projection method or the multiplier method. By numerical comparison, it is verified that the proposed methods have the excellent performance in computational time. We also elucidate some properties of the optimal distribution of search effort by some numerical examples.     

Optimal searching for a Helix target motion

This paper discusses a search problem for a Helix target motion in which any information of the target position is not available to the searchers. There exist three searchers start searching for the target from the origin. The purpose of this paper is to formulate a search model and finds the conditions under which the expected value of the first meeting time between one of the searchers and the target is finite. Also, the existence of the optimal search plan that minimizes the expected value of the first meeting time is shown. Furthermore,this optimal search plan is found. The effectiveness of this method is illustrated by using an example with numerical results.