THE EXIT PROBABILITIES OF BROWNIAN MOTION WITH VARIABLE DIMENSION APPLYING TO THE CONTROL OF POPULATION GROWTH

2013 ◽  
Vol 06 (05) ◽  
pp. 1350027
Author(s):  
LIXIN SONG ◽  
WENBIN CHE ◽  
DAWEI LU
Keyword(s):  
Brownian Motion ◽  
Integral Function ◽  
Upper And Lower Bounds ◽  
Log P ◽  
Small Ball ◽  
Variable Dimension ◽  
Biological Population ◽  
Early Results ◽  
Exit Probabilities ◽  
First Time

Using the theory of small ball estimate to study the biological population for keeping ecological balance in an ecosystem, we consider a Brownian motion with variable dimension starting at an interior point of a general parabolic domain Dt in Rd(t)+1 where d(t) ≥ 1 is an increasing integral function as t → ∞, d(t) → ∞. Let τDt denote the first time the Brownian motion exits from Dt. Upper and lower bounds with exact constants of log P(τDt > t) are given as t → ∞, depending on the shape of the domain Dt. The problem is motivated by the early results of Lifshits and Shi, Li, Lu in the exit probabilities. The methods of proof are based on the calculus of variations and early works of Lifshits and Shi, Li, Shao in the exit probabilities of Brownian motion.

scholarly journals Upper and lower bounds of the first exit time of Brownian motion from The Minimum and Maximun Parabolic Domains with variable dimension to research the variation of biological species

2020 ◽  
Author(s):  
Chao Liu ◽  
Wenbin Che ◽  
Jingjun Zhang
Keyword(s):  
Brownian Motion ◽  
Lower Bounds ◽  
Exit Time ◽  
Biological Species ◽  
Integral Function ◽  
Upper And Lower Bounds ◽  
First Exit Time ◽  
Variable Dimension ◽  
First Time ◽  
Exact Constants

Abstract Consider a Brownian motion with variable dimension starting at an interior point of the minimum or maximum parabolic domains Dmax t and Dmin t in Rd(t)+2, d(t) ≥ 1 is an increasing integral function as t →∞,d(t) →∞, and let τDmax t and τDmin t denote the first time the Brownian motion exits from Dmax t and Dmin t , respectively. Upper and lower bounds with exact constants for the asymptotics of logP(τDmax t > t) and logP(τDmin t > t) are given as t → ∞, depending on the shape of the domain Dmax t and Dmin t . The methods of proof are based on Gordon’s inequality and early works of Li, Lifshits and Shi in the single general parabolic domain case.

scholarly journals Some asymptotic formulars for a Brownian motion whose dimension is variable with a regular variation from a parabolic domain to research the variation of biological species

2020 ◽  
Author(s):  
Xiaoming Li ◽  
Wenbin Che ◽  
Jingjun Zhang
Keyword(s):  
Brownian Motion ◽  
Interior Point ◽  
Regular Variation ◽  
Exit Time ◽  
Biological Species ◽  
First Exit Time ◽  
Early Results ◽  
Asymptotically Equivalence ◽  
First Time ◽  
Exact Constants

Abstract Consider a Brownian motion with a regular variation starting at an interior point of a domain D in R d+1 ,d ≥ 1, let τ D denote the first time the Brownian motion exits from D. Estimates with exact constants for the asymptotics of logP(τ D > T) are given for T → ∞, depending on the shape of the domain D and the order of the regular variation. Furthermore, the asymptotically equivalence are obtained. The problem is motivated by the early results of Lifshits and Shi, Li in the first exit time and Karamata in the regular variation. The methods of proof are based on their results and the calculus of variations.

STATISTICAL ORIGIN OF SPECIAL AND DOUBLY SPECIAL RELATIVITY

2013 ◽  
Vol 23 ◽  
pp. 373-378
Author(s):  
PETR JIZBA ◽  
FABIO SCARDIGLI
Keyword(s):  
Brownian Motion ◽  
Special Relativity ◽  
Coarse Graining ◽  
Relativistic Motion ◽  
Relativistic Dynamics ◽  
Compton Wavelength ◽  
Primitive Concept ◽  
Short Scale ◽  
Doubly Special Relativity ◽  
First Time

We show how a Brownian motion on a short scale can originate a relativistic motion on scales larger than particle's Compton wavelength. Special relativity appears to be not a primitive concept, but rather it statistically emerges when a coarse graining average over distances of order, or longer than the Compton wavelength is taken. Our scheme accommodates easily also the doubly special relativistic dynamics. A previously unsuspected, common statistical origin of the two frameworks is brought to light for the first time.

scholarly journals Brownian Motion and Harmonic Analysis on Sierpinski Carpets

1999 ◽  
Vol 51 (4) ◽  
pp. 673-744 ◽  
Author(s):  
Martin T. Barlow ◽  
Richard F. Bass
Keyword(s):  
Brownian Motion ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Heat Kernel ◽  
Lower Bounds ◽  
Harnack Inequality ◽  
Harmonic Functions ◽  
Upper And Lower Bounds ◽  
Isotropic Diffusion ◽  
Sierpinski Carpets

AbstractWe consider a class of fractal subsets of d formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion X and determine its basic properties; and extend some classical Sobolev and Poincaré inequalities to this setting.

scholarly journals Early results from a diagnostic 1.3 cm survey of massive young protostars

2012 ◽  
Vol 8 (S287) ◽  
pp. 497-501
Author(s):  
Crystal L. Brogan ◽  
Todd R. Hunter ◽  
Claudia J. Cyganowski ◽  
Remy Indebetouw ◽  
Rachel Friesen ◽  
...  
Keyword(s):  
Massive Star ◽  
Thermal Emission ◽  
Young Stellar Objects ◽  
Large Array ◽  
Stellar Objects ◽  
Very Large Array ◽  
Early Results ◽  
Wide Range ◽  
First Time ◽  
24 Ghz

AbstractWe have used the recently-upgraded Karl G. Jansky Very Large Array (JVLA) to conduct a K-band (~24 GHz) study of 22 massive young stellar objects in 1.3 cm continuum and a comprehensive set of diagnostic lines. This survey is unique in that it samples a wide range of massive star formation signposts simultaneously for the first time. In this proceeding we present preliminary results for the 11 sources in the 2-4 kpc distance bin. We detect compact NH3 cores in all of the fields, with many showing emission up through the (6,6) transition. Maser emission in the 25 GHz CH3OH ladder is present in 7 of 11 sources. We also detect non-thermal emission in the NH3 (3,3) transition in 7 of 11 sources.

Biharmonic functions and Brownian motion

1967 ◽  
Vol 4 (1) ◽  
pp. 130-136 ◽  
Author(s):  
L. L. Helms
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Boundary Function ◽  
Normal Derivative ◽  
Dimensional Euclidean Space ◽  
Biharmonic Functions ◽  
Neumann Type ◽  
Bounded Open Subset ◽  
Image Position ◽  
First Time

Let R be a bounded open subset of N-dimensional Euclidean space EN,N ≧ 1, let {xt: t ≧ 0} be a separable Brownian motion starting at a point x ɛ R, and let τ = τR be the first time the motion hits the complement of R. It is known [1] that if g is a bounded measurable function on the boundary ∂R of R, then h(x) = Ex[g(xτ)] is a harmonic function of x ɛ R which “solves” the Dirichlet problem for the boundary function g; i.e., Δh = 0 on R, where Δ is the Laplacian. In elastic plate problems, one must solve the biharmonic equation subject to certain boundary conditions. For the more important applications, these boundary conditions involve the values of u and the normal derivative of u at points of ∂R. Even though a treatment of this Neumann type problem is not available at this time, some things can be said about biharmonic functions and their relationship to Brownian motion. We will show, in fact, that u(x)= Ex[τ(xτ)] is a biharmonic function on R which “satisfies” the boundary conditions (i) u=0 on ∂R and (ii) Δu= −2g on ∂R, provided g satisfies certain hypotheses. More generally, we will show that u(x)=Ex[Δkg(XΔ)] is polyharmonic of order k + 1 on R (i.e., Δk + 1u = Δ(Δku) = 0 on R) and that it satisfies certain boundary conditions. A treatment of the special case g ≡ 1 on ∂R can be found in [3].

scholarly journals Early Results from the VIMOS VLT Deep Survey

2005 ◽  
Vol 216 ◽  
pp. 381-389
Author(s):  
O. Le Fèvre ◽  
C. Adami ◽  
O. Ilbert ◽  
V. Le Brun ◽  
C. Marinoni ◽  
...  
Keyword(s):  
Large Scale ◽  
Limited Sample ◽  
Redshift Distribution ◽  
Star Forming ◽  
Large Scale Structures ◽  
Early Results ◽  
Main Challenge ◽  
Deep Survey ◽  
Scale Structures ◽  
First Time

The VIMOS VLT Deep Survey (VVDS) is underway to study the evolution of galaxies, large scale structures and AGNs, from the measurement of more than 100 000 spectra of faint objects. We present here the results from the first epoch observations of more than 20000 spectra. The main challenge of the program, the redshift measurements, is described, in particular entering the “redshift desert” in the range 1.5 < z < 3 for which only very weak features are detected in the observed wavelength range. The redshift distribution of a magnitude limited sample brighter than IAB = 24 is presented for the first time, showing a peak at a low redshift z ∼ 0.7, and a tail extending all the way above z = 4. The evolution of the luminosity function out to z = 1.5 is presented, with the LF of blue star forming galaxies carrying most of the evolution, with L* changing by more than two magnitudes for this sub-sample.

Geometric bounds on certain sublinear functionals of geometric Brownian motion

2003 ◽  
Vol 40 (4) ◽  
pp. 893-905 ◽  
Author(s):  
Per Hörfelt
Keyword(s):  
Distribution Function ◽  
Brownian Motion ◽  
Borel Measure ◽  
Random Variable ◽  
Geometric Brownian Motion ◽  
Upper And Lower Bounds ◽  
Tail Probabilities ◽  
Absolutely Continuous ◽  
Basket Options ◽  
Sublinear Functionals

Suppose that {Xs, 0 ≤ s ≤ T} is an m-dimensional geometric Brownian motion with drift, μ is a bounded positive Borel measure on [0,T], and ϕ : ℝm → [0,∞) is a (weighted) lq(ℝm)-norm, 1 ≤ q ≤ ∞. The purpose of this paper is to study the distribution and the moments of the random variable Y given by the Lp(μ)-norm, 1 ≤ p ≤ ∞, of the function s ↦ ϕ(Xs), 0 ≤ s ≤ T. By using various geometric inequalities in Wiener space, this paper gives upper and lower bounds for the distribution function of Y and proves that the distribution function is log-concave and absolutely continuous on every open subset of the distribution's support. Moreover, the paper derives tail probabilities, presents sharp moment inequalities, and shows that Y is indetermined by its moments. The paper will also discuss the so-called moment-matching method for the pricing of Asian-styled basket options.

A class of singular stochastic control problems

1983 ◽  
Vol 15 (02) ◽  
pp. 225-254 ◽  
Author(s):  
Ioannis Karatzas
Keyword(s):  
Brownian Motion ◽  
Optimal Stopping ◽  
Moving Boundary ◽  
Upper And Lower Bounds ◽  
Control Problems ◽  
Explicit Solutions ◽  
Value Functions ◽  
Singular Stochastic Control ◽  
Optimal Stopping Problem ◽  
The Third

We consider the problem of tracking a Brownian motion by a process of bounded variation, in such a way as to minimize total expected cost of both ‘action' and ‘deviation from a target state 0'. The former is proportional to the amount of control exerted to date, while the latter is being measured by a function which can be viewed, for simplicity, as quadratic. We discuss the discounted, stationary and finite-horizon variants of the problem. The answer to all three questions takes the form of exerting control in asingularmanner, in order not to exit from a certain region. Explicit solutions are found for the first and second questions, while the third is reduced to an appropriate optimal stopping problem. This reduction yields properties, as well as global upper and lower bounds, for the associated moving boundary. The pertinent Abelian and ergodic relationships for the corresponding value functions are also derived.

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