Occupation and local times for skew Brownian motion with applications to dispersion across an interface

Appuhamillage, Thilanka; Bokil, Vrushali A.; Thomann, Enrique; Waymire, Edward C.; Wood, Brian D.

doi:10.1214/10-aap691

Cited by 74 publications

(103 citation statements)

References 31 publications

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“…A problem related to the case where U is perpendicular to the surface of discontinuity of D x is treated in [1].…”

Section: 3mentioning

confidence: 99%

“…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%

“…By Knight's theorem (see [26, p. 172]), there exist two independent Brownian motions W (1) and W (2) such that…”

mentioning

confidence: 99%

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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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“…A problem related to the case where U is perpendicular to the surface of discontinuity of D x is treated in [1].…”

Section: 3mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multi-skewed Brownian motion and diffusion in layered media

Ramı́rez

2011

Proc. Amer. Math. Soc.

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“…The work [11] contains results such as Theorem 3 for the instant of hitting by solution of homogeneous stochastic equation [19] and [20].…”

Section: Introductionmentioning

confidence: 99%

The Arc-Sine Laws for the Skew Brownian Motion and Their Interpretation

Krykun¹

2018

JAMP

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“…The kernel q(t, x, y) is the fundamental solution (also called heat kernel) q(t, x, y) of L. Skew Brownian motions are a subclass of diffusion processes X of (1.3) with a 1 = a 2 = a 3 = 1 and ρ 2 = ρ 3 . It is first introduced by Ito and McKean in 1963, and has since been studied extensively by many authors; see, for example, [1,2,13,15] and the references therein. However we do not know the explicit formula for q(t, x, y) in the general case.…”

Section: Introductionmentioning

confidence: 99%

One-dimensional heat equation with discontinuous conductance

Chen

Zili

2014

Sci. China Math.

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We study a second-order parabolic equation with divergence form elliptic operator, having piecewise constant diffusion coefficients with two points of discontinuity. Such partial differential equations appear in the modelization of diffusion phenomena in medium consisting of three kind of materials. Using probabilistic methods, we present an explicit expression of the fundamental solution under certain conditions. We also derive small-time asymptotic expansion of the PDE's solutions in the general case. The obtained results are directly usable in applications.

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Occupation and local times for skew Brownian motion with applications to dispersion across an interface

Cited by 74 publications

References 31 publications

Multi-skewed Brownian motion and diffusion in layered media

Multi-skewed Brownian motion and diffusion in layered media

The Arc-Sine Laws for the Skew Brownian Motion and Their Interpretation

One-dimensional heat equation with discontinuous conductance

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