Plane Brownian motion has strictlyn-multiple points

Adelman, Omer; Dvoretzky, Aryeh

doi:10.1007/bf02774087

Cited by 8 publications

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“…Among the consequences of this theorem and a weak converse (Theorem 2) is a recent result of Adelman and Dvoretzky [1] that the inclusions The last part of the paper gives versions of the theorems of Part 2 for the case of Brownian motion in IR 3 .…”

Section: Introductionmentioning

confidence: 96%

Which sets contain multiple points of Brownian motion?

Tongring

1988

Math. Proc. Camb. Phil. Soc.

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In 1950, Dvoretzky, Erdös and Kakutani [2] showed that in ℝ3 almost all paths of Brownian motion X have double points, or self-intersections of order 2 (there are no triple points [4]); later the same authors proved that almost all sample paths of Brownian motion in the plane have points of arbitrarily high multiplicity (a point x in ℝ2 is a k-tuple point for the path ω, or a self-intersection of order k, if there are times tl < t2 < … < tk such that x = X(t1, ω) = X(t2, ω) = … X(tk, ω)).

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Section: Introductionmentioning

confidence: 96%

Which sets contain multiple points of Brownian motion?

Tongring

1988

Math. Proc. Camb. Phil. Soc.

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“…Taylor [146] proved that that for all r almost surely the set of r-multiple points of planar Brownian motion has Hausdorff dimension 2, and later Wolpert [154] provided an alternative proof for this result. Adelman and Dvoretzky [2] generalized a result of Dvoretzky et al…”

Section: We Will See In Section 3 That Almost Surely Spanmentioning

confidence: 78%

The spans in Brownian motion

Evans¹,

Pitman²,

Tang³

2017

Ann. Inst. H. Poincaré Probab. Statist.

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For $d \in \{1,2,3\}$, let $(B^d_t;~ t \geq 0)$ be a $d$-dimensional standard Brownian motion. We study the $d$-Brownian span set $Span(d):=\{t-s;~ B^d_s=B^d_t~\mbox{for some}~0 \leq s \leq t\}$. We prove that almost surely the random set $Span(d)$ is $\sigma$-compact and dense in $\mathbb{R}_{+}$. In addition, we show that $Span(1)=\mathbb{R}_{+}$ almost surely; the Lebesgue measure of $Span(2)$ is $0$ almost surely and its Hausdorff dimension is $1$ almost surely; and the Hausdorff dimension of $Span(3)$ is $\frac{1}{2}$ almost surely. We also list a number of conjectures and open problems.Comment: 33 pages, 4 figures. This paper is published by http://projecteuclid.org/euclid.aihp/150062403

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“…Proposition! of [1]). Further, define a ;r(f)-valued, o(X(s): s e ;r(f))-measurable random variable r(t) by the following prescription.…”

Section: Definementioning

confidence: 95%

“…The question now arises as to whether the processes which we have shown to have r-tuple points have strict r-tuple points. Using techniques akin to those used for Brownian motion in [1] we can show that this will often be so. THEOREM (10.3).…”

Section: X R~\ T R~l ) -mentioning

confidence: 96%

Sample Path Properties of Gaussian Stochastic Processes Indexed by a Local Field

Evans¹

1988

Proceedings of the London Mathematical Society

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Plane Brownian motion has strictlyn-multiple points

Cited by 8 publications

References 0 publications

Which sets contain multiple points of Brownian motion?

Which sets contain multiple points of Brownian motion?

The spans in Brownian motion

Sample Path Properties of Gaussian Stochastic Processes Indexed by a Local Field

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