A Generalized Taylor–Aris Formula and Skew Diffusion

Ramı́rez, Jorge; Thomann, Enrique; Waymire, Edward C.; Haggerty, R.; Wood, Brian D.

doi:10.1137/050642770

Cited by 40 publications

(54 citation statements)

References 19 publications

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“…The resulting process B α behaves like standard Brownian motion when away from the "interface" at zero, but its sample paths are skewed, namely P(B α t > 0 B α 0 = 0) = α for all t 0. While skew Brownian motion has many interesting and sometimes unexpected properties, it is of particular relevance to recent physical experiments involving diffusive transport in heterogeneous media in the presence of a single membrane; e.g., see [6,12,21,15,24,30,5,18,25,2]. The mathematical theory underlying applications to a single interface rests largely on the foundations for skew Brownian motion as developed by [13,11,31,22,18,1].…”

Section: Introductionmentioning

confidence: 99%

Multi-skewed Brownian motion and diffusion in layered media

Ramı́rez

2011

Proc. Amer. Math. Soc.

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Abstract. Multi-skewed Brownian motion B α = {B α t : t 0} with skewness sequence α = {α k : k ∈ Z} and interface set S = {x k : k ∈ Z} is the solution to} and that S has no accumulation points. The process B α generalizes skew Brownian motion to the case of an infinite set of interfaces. Namely, the paths of B α behave like Brownian motion in R S, and on B α 0 = x k , the probability of reaching x k + δ before x k − δ is α k , for any δ small enough, and k ∈ Z. In this paper, a thorough analysis of the structure of B α is undertaken, including the characterization of its infinitesimal generator and conditions for recurrence and positive recurrence. The associated Dirichlet form is used to relate B α to a diffusion process with piecewise constant diffusion coefficient. As an application, we compute the asymptotic behavior of a diffusion process corresponding to a parabolic partial differential equation in a two-dimensional periodic layered geometry.

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Section: Introductionmentioning

confidence: 99%

Multi-skewed Brownian motion and diffusion in layered media

Ramı́rez

2011

Proc. Amer. Math. Soc.

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“…The stationary distribution is P st ðÀw < z < wÞ ¼ 1=2w and k ¼ 1, 2. The transition probability for the interface with a discontinuous diffusion coefficient at z ¼ 0 is known and given by 34,60 …”

Section: Appendix: Short-time Mean Square Displacement (Msd) Expressionmentioning

confidence: 99%

“…59 Also, Brownian motion through step function interfaces with opened space boundaries has been studied intensively and the transition probabilities are known. 34,60 The present system has features of both these two cases which need to be combined to develop an appropriate model for the present system description. The difference is that the walls are not hard and the pH-profile is defined by a differentiable function, which is the physically realizable situation.…”

Section: A Mean Square Displacement (Msd)mentioning

confidence: 99%

Spatially dependent diffusion coefficient as a model for pH sensitive microgel particles in microchannels

2016

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Solute transport and intermixing in microfluidic devices is strongly dependent on diffusional processes. Brownian Dynamics simulations of pressure-driven flow of model microgel particles in microchannels have been carried out to explore these processes and the factors that influence them. The effects of a pH-field that induces a spatial dependence of particle size and consequently the self-diffusion coefficient and system thermodynamic state were focused on. Simulations were carried out in 1D to represent some of the cross flow dependencies, and in 2D and 3D to include the effects of flow and particle concentration, with typical stripe-like diffusion coefficient spatial variations. In 1D, the mean square displacement and particle displacement probability distribution function agreed well with an analytically solvable model consisting of infinitely repulsive walls and a discontinuous pHprofile in the middle of the channel. Skew category Brownian motion and nonGaussian dynamics were observed, which follows from correlations of step lengths in the system, and can be considered to be an example of so-called "diffusing diffusivity." In Poiseuille flow simulations, the particles accumulated in regions of larger diffusivity and the largest particle concentration throughput was found when this region was in the middle of the channel. The trends in the calculated cross-channel diffusional behavior were found to be very similar in 2D and 3D. Published by AIP Publishing. [http://dx

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“…Indeed DFOs appear in the modelization of diffusion phenomena, and the irregularity of the coefficient can reflect the irregularity of the media the particle is evolving in. This is interesting in a wide variety of physical situations, for example in fluid mechanics in porous media (see [RTW05]), in the modelization of the brain (see [Fau99]), and can also be used in finance (see [DDG05]). …”

Section: Introductionmentioning

confidence: 99%

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

Étoré

Lejay

2007

ESAIM: PS

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To cite this version:Pierre Etoré, Antoine Lejay. A Donsker theorem to simulate one-dimensional processes with measurable coefficients. ESAIM: Probability and Statistics, EDP Sciences, 2007, 11, pp.301-326 In this paper, we prove a Donker theorem for one-dimensional processes generated by an operator with measurable coefficients. We construct a random walk on any grid on the state space, using the transition probabilities of the approximated process, and the conditional average times it spends on each cell of the grid. Indeed we can compute these quantities by solving some suitable elliptic PDE problems.

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A Generalized Taylor–Aris Formula and Skew Diffusion

Cited by 40 publications

References 19 publications

Multi-skewed Brownian motion and diffusion in layered media

Multi-skewed Brownian motion and diffusion in layered media

Spatially dependent diffusion coefficient as a model for pH sensitive microgel particles in microchannels

A Donsker theorem to simulate one-dimensional processes with measurable coefficients

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