Brownian motion in a wedge with oblique reflection

Varadhan, S. R. S.; Williams, Ruth

doi:10.1002/cpa.3160380405

Cited by 192 publications

(252 citation statements)

References 11 publications

(16 reference statements)

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“…Proof. This follows from Lemma 2.8 on letting e = R -> 00 in the same manner as Corollary 2.3 of [22] follows from Theorem 2.3 there. 0 2.4 Existence.…”

Section: (259) Et' Y L{0£)(v(w(s)))h(w(s))dsmentioning

confidence: 67%

Reflected Brownian motion in a cone with radially homogeneous reflection field

Kwon¹,

Williams²

1991

Trans. Amer. Math. Soc.

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Abstract. This work is concerned with the existence and uniqueness of a strong Markov process that has continuous sample paths and the following additional properties.(i) The state space is a cone in ¿-dimensions (d > 3), and the process behaves in the interior of the cone like ordinary Brownian motion.(ii) The process reflects instantaneously at the boundary of the cone, the direction of reflection being fixed on each radial line emanating from the vertex of the cone.(iii) The amount of time that the process spends at the vertex of the cone is zero (i.e., the set of times for which the process is at the vertex has zero Lebesgue measure).The question of existence and uniqueness is cast in precise mathematical terms as a submartingale problem in the style used by Stroock and Varadhan for diffusions on smooth domains with smooth boundary conditions. The question is resolved in terms of a real parameter a which in general depends in a rather complicated way on the geometric data of the problem, i.e., on the cone and the directions of reflection. However, a criterion is given for determining whether a > 0. It is shown that there is a unique continuous strong Markov process satisfying (i)-(iii) above if and only if a < 2 , and that starting away from the vertex, this process does not reach the vertex if a < 0 and does reach the vertex almost surely if0 2, there is a unique continuous strong Markov process satisfying (i) and (ii) above; it reaches the vertex of the cone almost surely and remains there. These results are illustrated in concrete terms for some special cases.The process considered here serves as a model for comparison with a reflected Brownian motion in a cone having a nonradially homogeneous reflection field. This is discussed in a subsequent work by Kwon.

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“…Proof. This follows from Lemma 2.8 on letting e = R -> 00 in the same manner as Corollary 2.3 of [22] follows from Theorem 2.3 there. 0 2.4 Existence.…”

Section: (259) Et' Y L{0£)(v(w(s)))h(w(s))dsmentioning

confidence: 67%

Reflected Brownian motion in a cone with radially homogeneous reflection field

Kwon¹,

Williams²

1991

Trans. Amer. Math. Soc.

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“…(−∞, 0)). It is well known (see, e.g., [48] Proof. As in the proof of Theorem 9.1, it suffices to show that the law of K T ∩ A is the same as that ofB τ ∩ A.…”

Section: Formal Calculationsmentioning

confidence: 99%

Conformal restriction: The chordal case

Lawler

Schramm

Werner³

2003

J. Amer. Math. Soc.

237

554

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We characterize and describe all random subsets K of a given simply connected planar domain (the upper half-plane H, say) which satisfy the "conformal restriction" property, i.e., K connects two fixed boundary points (0 and ∞, say) and the law of K conditioned to remain in a simply connected open subset H of H is identical to that of Φ(K), where Φ is a conformal map from H onto H with Φ(0) = 0 and Φ(∞) = ∞. The construction of this family relies on the stochastic Loewner evolution processes with parameter κ ≤ 8/3 and on their distortion under conformal maps. We show in particular that SLE 8/3 is the only random simple curve satisfying conformal restriction and relate it to the outer boundaries of planar Brownian motion and SLE 6 .

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“…≥ x n } and discusses its relation with the solution of the Burgers' equation. For more details on reflected Brownian motions in an orthant, see Varadhan and Williams [37] and Williams [38]. For more on stochastic particle system methods in analyzing Burgers' equation, see Calderoni and Pulvirenti [11].…”

Section: Introductionmentioning

confidence: 99%

A phase transition behavior for Brownian motions interacting through their ranks

Chatterjee

Pal

2009

Probab. Theory Relat. Fields

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Abstract. Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain 'continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law.

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Brownian motion in a wedge with oblique reflection

Cited by 192 publications

References 11 publications

Reflected Brownian motion in a cone with radially homogeneous reflection field

Reflected Brownian motion in a cone with radially homogeneous reflection field

Conformal restriction: The chordal case

A phase transition behavior for Brownian motions interacting through their ranks

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