Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant

Franceschi, Sandro

doi:10.1007/s10959-020-01043-8

Cited by 6 publications

(3 citation statements)

References 50 publications

(94 reference statements)

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“…Complex analysis techniques prove to be quite efficient in dimension 2, see [4,8]. In particular, this method leads to explicit expressions for the Laplace transforms of quantities of interest (stationary distribution in the recurrent case [22,8], Green functions in the transient case [21], escape and absorption probabilities [18,20]).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A dual skew symmetry for transient reflected Brownian motion in an orthant

Franceschi¹,

Raschel²

2022

Preprint

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We introduce a transient reflected Brownian motion in a multidimensional orthant, which is either absorbed at the apex of the cone or escapes to infinity. We address the question of computing the absorption probability, as a function of the starting point of the process. We provide a necessary and sufficient condition for the absorption probability to admit an exponential product form, namely, that the determinant of the reflection matrix is zero. We call this condition a dual skew symmetry. It recalls the famous skew symmetry introduced by Harrison [23], which characterizes the exponential stationary distributions in the recurrent case. The duality comes from that the partial differential equation satisfied by the absorption probability is dual to the one associated with the stationary distribution in the recurrent case.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

A dual skew symmetry for transient reflected Brownian motion in an orthant

Franceschi¹,

Raschel²

2022

Preprint

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“…Our first goal is to complete the literature and to show how, in this more complicated non-convex setting, one can solve the problem of finding the stationary distribution. Our techniques could also be applied to the transient case, for example to analyse Green functions or absorption probabilities (see [15,9] for the convex case); however, we do not tackle these problems here.…”

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confidence: 99%

On the stationary distribution of reflected Brownian motion in a non-convex wedge

Fayolle¹,

Franceschi²,

Raschel³

2021

Preprint

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We study the stationary reflected Brownian motion in a non-convex wedge, which, compared to its convex analogue model, has been much rarely analyzed in the probabilistic literature. We prove that its stationary distribution can be found by solving a two dimensional vector boundary value problem (BVP) on a single curve for the associated Laplace transforms. The reduction to this kind of vector BVP seems to be new in the literature. As a matter of comparison, one single boundary condition is sufficient in the convex case. When the parameters of the model (drift, reflection angles and covariance matrix) are symmetric with respect to the bisector line of the cone, the model is reducible to a standard reflected Brownian motion in a convex cone. Finally, we construct a one-parameter family of distributions, which surprisingly provides, for any wedge (convex or not), one particular example of stationary distribution of a reflected Brownian motion.

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“…Recurrence and transience of obliquely reflecting Brownian motion were examined in [21,30], and the process has also been considered in planar domains [14,19] as well as in general dimensions in orthants [18,28,32]. The stationary distribution of obliquely reflecting Brownian motion has been studied in [4,5,13] and its Green's functions have been studied in [11]. Obliquely reflecting Brownian motion has played an important role in applications concerning heavy traffic approximations for open queueing networks ( [15,26]).…”

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Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant

Ernst,

Franceschi,

Huang

2021

Preprint

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We consider an obliquely reflected Brownian motion Z with positive drift in a quadrant stopped at time T , where T := inf{t > 0 : Z(t) = (0, 0)} is the first hitting time of the origin. Such a process can be defined even in the non-standard case where the reflection matrix is not completely-S. We show that in this case the process has two possible behaviors: either it tends to infinity or it hits the corner (origin) in a finite time. Given an arbitrary starting point (u, v) in the quadrant, we consider the escape (resp. absorption) probabilities). We establish the partial differential equations and the oblique Neumann boundary conditions which characterize the escape probability and provide a functional equation satisfied by the Laplace transform of the escape probability. We then give asymptotics for the absorption probability in the simpler case where the starting point in the quadrant is (u, 0). We exhibit a remarkable geometric condition on the parameters which characterizes the case where the absorption probability has a product form and is exponential. We call this new criterion the dual skew symmetry condition due to its natural connection with the skew symmetry condition for the stationary distribution. We then obtain an explicit integral expression for the Laplace transform of the escape probability. We conclude by presenting exact asymptotics for the escape probability at the origin.

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Green’s Functions with Oblique Neumann Boundary Conditions in the Quadrant

Cited by 6 publications

References 50 publications

A dual skew symmetry for transient reflected Brownian motion in an orthant

A dual skew symmetry for transient reflected Brownian motion in an orthant

On the stationary distribution of reflected Brownian motion in a non-convex wedge

Escape and absorption probabilities for obliquely reflected Brownian motion in a quadrant

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