scholarly journals Towards an Innovation Theory of Spatial Brownian Motion under Boundary Conditions

2001 ◽  
Vol 8 (2) ◽  
pp. 297-306
Author(s):  
E. Khmaladze
Keyword(s):  
Brownian Motion ◽  
Boundary Conditions ◽  
Gaussian Process ◽  
Conditional Distribution ◽  
Boundary Values ◽  
Innovation Theory ◽  
Unit Square

Abstract Set-parametric Brownian motion b in a star-shaped set G is considered when the values of b on the boundary of G are given. Under the conditional distribution given these boundary values the process b becomes some set-parametrics Gaussian process and not Brownian motion. We define the transformation of this Gaussian process into another Brownian motion which can be considered as “martingale part” of the conditional Brownian motion b and the transformation itself can be considered as Doob–Meyer decomposition of b. Some other boundary conditions and, in particular, the case of conditional Brownian motion on the unit square given its values on the whole of its boundary are considered.

Integrated Fractional white Noise as an Alternative to Multifractional Brownian Motion

2007 ◽  
Vol 44 (02) ◽  
pp. 393-408 ◽  
Author(s):  
Allan Sly
Keyword(s):  
Stochastic Process ◽  
Brownian Motion ◽  
White Noise ◽  
Gaussian Process ◽  
Discrete Data ◽  
Hölder Exponent ◽  
Scaling Properties ◽  
Multifractional Brownian Motion ◽  
Local Properties

Multifractional Brownian motion is a Gaussian process which has changing scaling properties generated by varying the local Hölder exponent. We show that multifractional Brownian motion is very sensitive to changes in the selected Hölder exponent and has extreme changes in magnitude. We suggest an alternative stochastic process, called integrated fractional white noise, which retains the important local properties but avoids the undesirable oscillations in magnitude. We also show how the Hölder exponent can be estimated locally from discrete data in this model.

Thermo-Solutal Convection of a Nanofluid Utilizing Fourier’s-Type Compass Conditions

2020 ◽  
Vol 9 (4) ◽  
pp. 362-374
Author(s):  
J. C. Umavathi ◽  
Ali J. Chamkha
Keyword(s):  
Brownian Motion ◽  
Prandtl Number ◽  
Boundary Conditions ◽  
Volume Fraction ◽  
Temperature Fields ◽  
Physical Parameters ◽  
Surface Boundary ◽  
Perturbation Scheme ◽  
Runge Kutta ◽  
The Impact

Nanotechnology has infiltrated into duct design in parallel with many other fields of mechanical, medical and energy engineering. Motivated by the excellent potential of nanofluids, a subset of materials engineered at the nanoscale, in the present work, a new mathematical model is developed for natural convection in a vertical duct containing nanofluid. Numerical scrutiny for the double-diffusive free and forced convection within a duct encumbered with nanofluid is performed. Buongiorno’s model is deployed to define the nanofluid. Robin boundary conditions are used to define the surface boundary conditions. Thermal and concentration equations envisage the viscous, Brownian motion, thermosphores of the nanofluid, Soret and Dufour effects. Using the Boussi-nesq approximation the solutal buoyancy effect as a result of gradients in concentration are incorporated. The conservation equations which are nonlinear are numerically estimated using fourth order Runge-Kutta methodology and analytically ratifying regular perturbation scheme. The mass, heat, nanoparticle concentration and species concentration fields on eight dimensionless physical parameters such as thermal and mass Grashof numbers, Brownian motion parameter, thermal parameter, Prandtl number, Eckert number, Schmidt parameter, and Soret parameter are calculated. The impact of these parameters are outlined pictorially. The velocity and temperature fields are boosted with the thermal Grashof number. The Soret and the Schemidt parameters reduces the nanoparticle volume fraction but it heightens the momentum, temperature and concentration. At the cold wall thermal and concentration Grashof numbers reduces the Nusselt values but they increase the Nusselt values at the hot wall. The reversal consequence was attained at the hot plate. The perturbation and Runge-Kutta solutions are equal in the nonappearance of Prandtl number. The (E. Zanchini, Int. J. Heat Mass Transfer 41, 3949 (1998)). results are restored for the regular fluid. The heat transfer rate is high for nanofluid when matched with regular fluid.

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (2) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

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].

Approximation to the exit probability of a continuous Gaussian process over a U-shaped boundary of increasing curvature

1995 ◽  
Vol 32 (02) ◽  
pp. 429-442
Author(s):  
A. N. Balabushkin
Keyword(s):  
Brownian Motion ◽  
Gaussian Process ◽  
Gaussian Processes ◽  
Simple Approximation ◽  
Minimum Point ◽  
Second Derivative ◽  
Numerical Examples ◽  
Exit Probability ◽  
Continuous Gaussian Process

A simple approximation to the probability of crossing a U-shaped boundary by a Brownian motion is given. The larger the second derivative of the curve at a minimum point, the higher the accuracy of the approximation. The result is also extended to a class of continuous Gaussian processes with definite properties. Numerical examples are given.

scholarly journals Centre-of-mass like superposition of Ornstein–Uhlenbeck processes: A pathway to non-autonomous stochastic differential equations and to fractional diffusion

2018 ◽  
Vol 21 (5) ◽  
pp. 1420-1435 ◽  
Author(s):  
Mirko D’Ovidio ◽  
Silvia Vitali ◽  
Vittoria Sposini ◽  
Oleksii Sliusarenko ◽  
Paolo Paradisi ◽  
...  
Keyword(s):  
Differential Equation ◽  
Brownian Motion ◽  
Stochastic Differential Equation ◽  
Gaussian Process ◽  
Relaxation Times ◽  
Random Variable ◽  
Fractional Diffusion ◽  
Time Dependent ◽  
Centre Of Mass ◽  
Time Scaling

Abstract We consider an ensemble of Ornstein–Uhlenbeck processes featuring a population of relaxation times and a population of noise amplitudes that characterize the heterogeneity of the ensemble. We show that the centre-of-mass like variable corresponding to this ensemble is statistically equivalent to a process driven by a non-autonomous stochastic differential equation with time-dependent drift and a white noise. In particular, the time scaling and the density function of such variable are driven by the population of timescales and of noise amplitudes, respectively. Moreover, we show that this variable is equivalent in distribution to a randomly-scaled Gaussian process, i.e., a process built by the product of a Gaussian process times a non-negative independent random variable. This last result establishes a connection with the so-called generalized grey Brownian motion and suggests application to model fractional anomalous diffusion in biological systems.

scholarly journals Relaxation methods applied to engineering problems. VIII. Plane-potential problems involving specified normal gradients

1943 ◽  
Vol 182 (989) ◽  
pp. 129-151 ◽  
Keyword(s):  
Harmonic Function ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Mixed Boundary Conditions ◽  
Boundary Values ◽  
Approximate Computation ◽  
Relaxation Methods ◽  
Potential Problems ◽  
Engineering Problems ◽  
Direct Attack

Since every plane-harmonic function is associated with a conjugate, problems in which normal gradients are specified on the boundary can be transformed into problems in which boundary values are specified. There then remains, however, the problem of deducing a function ψ from its conjugate ϕ, and this, when the conjugate has been determined only approximately, entails uncertainties which were exemplified in Part V. To minimize the errors of approximate computation ψ and ϕ should be determined severally and independently, consequently a method of direct attack is still needed on problems in which normal gradients are specified. Recent applications have, moreover, presented cases in which the boundary conditions are ‘mixed’, i.e. values are specified at some parts of the boundary, gradients at others. Here, two methods are propounded for the satisfaction of mixed boundary conditions, the first applicable also to cases in which normal gradients alone are specified. Test examples indicate that the wanted extension of method is now available.

scholarly journals Path properties of the primitives of a Brownian motion

2001 ◽  
Vol 70 (1) ◽  
pp. 119-133 ◽  
Author(s):  
Zhengyan Lin
Keyword(s):  
Brownian Motion ◽  
Positive Integer ◽  
Gaussian Process ◽  
Large Deviation ◽  
Asymptotic Properties ◽  
Standard Brownian Motion ◽  
Moduli Of Continuity ◽  
Large Increments ◽  
Path Properties

AbstractLet {W(t), t ≥ 0} be a standard Brownian motion. For a positive integer m, define a Gaussian process Watanabe and Lachal gave some asymptotic properties of the process Xm(·), m ≥ 1. In this paper, we study the bounds of its moduli of continuity and large increments by establishing large deviation results.

Potential functions with periodicity in one coordinate

1934 ◽  
Vol 30 (3) ◽  
pp. 315-326 ◽  
Author(s):  
R. C. J. Howland
Keyword(s):  
Boundary Conditions ◽  
Potential Functions ◽  
The Other ◽  
Boundary Values ◽  
Angular Coordinate ◽  
General Conception

The general conception underlying the following analysis is that of a field of potential which is invariant under a group of geometrical transformations. The boundary is to consist of a number of parts which transform into each other and the boundary values also transform into each other. To satisfy the conditions we build up functions which are invariant under the same group of transformations and combine them to give the prescribed boundary values over one section of the boundary. The boundary conditions on the other sections are then automatically satisfied. When the groups are simply translations or rotations we get functions periodic in either a Cartesian or an angular coordinate.

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