The dimension of the boundary of super-Brownian motion

Mytnik, Leonid; Perkins, Edwin

doi:10.1007/s00440-018-0866-5

Cited by 15 publications

(141 citation statements)

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“…It follows from the work of Bousquet-Mélou and Janson [7,9] that Θ(dx) has a continuous density (L x ) x∈R with respect to Lebesgue measure on R, N 0 a.e. (this fact could also be derived from the earlier work of Sugitani [30] dealing with super-Brownian motion, see in particular the introduction of [27]). We also consider the quantity σ + = Θ([0, ∞)) (resp.…”

mentioning

confidence: 72%

“…An alternative way to derive the previous two corollaries would be to use the known connections between super-Brownian motion or the Brownian snake and partial differential equations. See formula (1.13) in [27], and note that, as a function of a, the right-hand side of (19) solves the differential equation 1 2 u ′′ = 2u 2 − λδ 0 in the sense of distributions. On the other hand, our method provides a better probabilistic understanding of the results and the derivation of (15) in particular relies on Proposition 3 which is of independent interest and will play a significant role in the proofs of the next section.…”

Section: The Local Time Atmentioning

confidence: 99%

“…In particular, we get that the total local time at 0 (defined as the density at 0 of the measure ∞ 0 dt X t ) is distributed as 3 1/2 2 −2/3 T where T is as previously a positive stable variable with index 2/3 (Corollary 6). This is by no means a difficult result (as pointed out to the authors by Edwin Perkins [28], the fact that the total local time is a stable variable with index 2/3 can also be derived by a scaling argument, see formula (2.13) in [27]), but it seems to have remained unnoticed by the specialists of super-Brownian motion. The fact that the same variable T occurs in the Bousquet-Mélou-Janson result suggests the existence of a direct connection between the two results, but we have been unable to find such a connection.…”

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confidence: 92%

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Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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We derive several explicit distributions of functionals of Brownian motion indexed by the Brownian tree. In particular, we give a direct proof of a result of Bousquet-Mélou and Janson identifying the distribution of the density at 0 of the integrated super-Brownian excursion. IntroductionThe main purpose of the present work is to derive certain explicit distributions for the random process which we call Brownian motion indexed by the Brownian tree, which has appeared in a variety of different contexts. As a key tool for the derivation of our main results we use the excursion theory developed in [1] for Brownian motion indexed by the Brownian tree. In many respects, this excursion theory is similar to the classical Itô theory, which applies in particular to linear Brownian motion and has proved a powerful tool for the calculation of exact distributions of Brownian functionals.Let us briefly describe the objects of interest in this work. We define the Brownian tree T ζ as the random compact R-tree coded by a Brownian excursion ζ = (ζ s ) s≥0 distributed according to the (infinite) Itô measure of positive excursions of linear Brownian motion. If σ stands for the duration of the excursion ζ, this coding means that T ζ is the quotient space of [0, σ] for the equivalence relation defined by s ∼ s ′ if and only if ζ s = ζ s ′ = m ζ (s, s ′ ), where m ζ (s, s ′ ) := min{ζ r : s ∧ s ′ ≤ r ≤ s ∨ s ′ }, and this quotient space is equipped with the metric induced by d ζ (s, s ′The volume measure vol(da) on T ζ is defined as the push forward of Lebesgue measure on [0, σ] under the canonical projection, and the root ρ of T ζ is the equivalence class of 0. We note that under the conditioning by σ = 1 (equivalently the total volume is equal to 1) the tree T ζ is Aldous' Brownian Continuum Random Tree (also called the CRT, see [2,3]), up to an unimportant scaling factor 2.Let us turn to Brownian motion indexed by T ζ . Informally, given T ζ , this is the centered Gaussian process (V a ) a∈T ζ such that V ρ = 0 and Var(V a − V b ) = d ζ (a, b) for every a, b ∈ T ζ . This definition is a bit informal since we are dealing with a random process indexed by a random set. These difficulties can be overcome easily by using the Brownian snake approach. We let (W s ) s≥0 be the Brownian snake (whose spatial motion is linear Brownian motion started at 0) driven by the Brownian excursion (ζ s ) s≥0 . Then, for every s ≥ 0, W s is a finite path started at 0 and with lifetime ζ s , and for every a ∈ T ζ we may define V a as the terminal point W s of the path W s , for any s ∈ [0, σ] such that a is the equivalence class of s in T ζ . The Brownian snake approach thus reduces the study of a tree-indexed Brownian motion to that a process indexed by the positive half-line, and we systematically use this approach in the next sections.The total occupation measure Θ(dx) of (V a ) a∈T ζ is the push forward of vol(da) under the mapping a → V a , or equivalently the push forward of Lebesgue measure on [0, σ] under s → W s . Under the special conditioning...

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confidence: 72%

Section: The Local Time Atmentioning

confidence: 99%

mentioning

confidence: 92%

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Excursion theory for Brownian motion indexed by the Brownian tree

Abraham

Gall

2018

J. Eur. Math. Soc.

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“…Proof. By Theorem 2.2 in [MP17], for any 0 < ξ 0 < 1, with P δ 0 -probability one, there is some 0 < ρ(ω) ≤ 1 such that |L y − L x | < |y − x| ξ 0 , for x, y > 0 with |y − x| < ρ.…”

Section: Now We Discuss the Differentiability Ofmentioning

confidence: 96%

“…Note we may set ε 0 = 0 in Theorem 2.2 of [MP17] due to the global continuity of L x in d = 1. Pick ξ 0 = 1/2, then (2.13) in Theorem 2.3 holds for N ≥ N ξ 0 (ω) = 1 ∨ log 2 (ρ(ω) −1 ).…”

Section: Now We Discuss the Differentiability Ofmentioning

confidence: 99%

Improved Hölder continuity near the boundary of one-dimensional super-Brownian motion

Hong¹

2019

Electron. Commun. Probab.

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Fractional Laplacian with Hardy potential

Bogdan

Grzywny

Jakubowski

et al. 2019

Communications in Partial Differential Equations

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We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on R d , including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.The subject of the paper can be tracked down to Baras and Goldstein [3], who proved the existence of nontrivial nonnegative solutions of the classical heat equation ∂ t = ∆ + κ|x| −2 in R d for 0 ≤ κ ≤ (d − 2) 2 /4, and nonexistence of such solutions, that is explosion, for bigger constants κ. Vazquez and Zuazua [36] studied the Cauchy problem and spectral properties of the operator in bounded subsets of R d using the improved Hardy-Poincaré inequality, and they used weighted Hardy-Poincaré inequality in the more delicate case of the whole of R d . Sharp upper and lower bounds for the heat kernel of the Schrödinger operator ∆ + κ|x| −2 were obtained by Liskevich and Sobol [26, p. 365, Example 3.8, 4.4 and 4.10] for 0 < κ < (d − 2) 2 /4. Milman and Semenov proved the upper and lower bounds for κ ≤ (d − 2) 2 /4, see [28, Theorem 1], [29] and [22, Section 10.4]. Note that Moschini and Tesei [30, Theorem 3.10] gave estimates for the subcritical case in bounded domains, and Filippas, Moschini and Tertikas [18] obtained the critical case in bounded domains. Recently Metafune, Sobajima and Spina [27] extended the results of Milman and Semenov to sigular gradient-and-Schrödinger perturbations of ∆. Because of the borderline singularity of the function R dx → κ|x| −2 at the origin, the choice of κ influences the growth rate of the heat kernel at the origin.The operators ∆+κ|x| −2 play distinctive roles in limiting and self-similar phenomena in probability [32] and partial differential equations [33]. This results in part from the scaling of the corresponding heat kernel, which is similar to that of the Gauss-Weierstrass kernel. Analogous applications are expected for the Hardy perturbations of the fractional Laplacian, see [34] for first attempts. We also note that the effect of Schrödinger perturbations of ∆ is much milder if κ|x| −2 is replaced by functions in appropriate Kato classes. We refer to Bogdan and Szczypkowski [15, Section 1, 4] for references and results on Gaussian bounds for Schrödinger heat kernels along with an approach based on the so-called 4G inequality. The case when even the Gaussian constants do not deteriorate is discussed by Bogdan, Dziubański and Szczypkowski [11]. Estimates of Schrödinger perturbations of general operators and their transition semigroups are given by Bogdan, Hansen and Jakubowski in [12], with focus on situations with 3G inequality. Bogdan, Jakubowski and Sydor [14] and Bogdan, Butko and Szczypkowski [8] estimate Schrödinger perturbations of integral kernels which are not necessarily semigroups. The results of these paper give comparability or near comparability of the perturbed kernel with the original one. In fact, the explicit estimate in [14, Theorem 3] suffices for many applications.The operator (1.1) cannot be ...

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The dimension of the boundary of super-Brownian motion

Abstract: We show that the Hausdorff dimension of the boundary of d-dimensional super-Brownian motion is 0, if d = 1, 4 − 2 √ 2, if d = 2, and (9 − √ 17)/2, if d = 3. September 13, 2018 AMS 2000 subject classifications. Primary 60H15, 60G57. Secondary 28A78, 35J65, 60J55, 60H40, 60J80.

Cited by 15 publications

References 36 publications

Excursion theory for Brownian motion indexed by the Brownian tree

Excursion theory for Brownian motion indexed by the Brownian tree

Improved Hölder continuity near the boundary of one-dimensional super-Brownian motion

Fractional Laplacian with Hardy potential

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