Up-to-constants bounds on the two-point Green's function for SLE curves

Lawler, Gregory F.; Rezaei, Mohammad A.

doi:10.1214/ecp.v20-4246

Cited by 6 publications

(3 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In the case n = 1, the right-hand side is C y α−(2−d) |z| α , which agrees with the right-hand side of (1.4). In the case n = 2, the right-hand side is comparable to a sharp estimate of the 2-point Green's function given in [7] up to a constant. Thus, we expect that it holds for all n ∈ N. 4.…”

Section: Introductionmentioning

confidence: 53%

Higher moments of the natural parameterization for SLE curves

Rezaei¹,

Zhan²

2017

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

show abstract

Section: Introductionmentioning

confidence: 53%

Higher moments of the natural parameterization for SLE curves

Rezaei¹,

Zhan²

2017

Ann. Inst. H. Poincaré Probab. Statist.

View full text Add to dashboard Cite

show abstract

“…is obtained in [10]. Combining it with (3.6) and the exact formula for G H;0,∞ (z 0 ), we then get an upper bound for E[G H β ;z 0 ,∞ (z 1 )].…”

Section: − → Gmentioning

confidence: 82%

Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization

Zhan

2018

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

We make use of the fact that a two-sided whole-plane Schramm-Loewner evolution (SLE κ ) curve γ for κ ∈ (0, 8) from ∞ to ∞ through 0 may be parametrized by its ddimensional Minkowski content, where d = 1 + κ 8 , and become a self-similar process of index 1 d with stationary increments. We prove that such γ is locally α-Hölder continuous for any α < 1 d . In the case κ ∈ (0, 4], we show that γ is not locally 1 d -Hölder continuous. We also prove that, for any deterministic closed set A ⊂ R, the Hausdorff dimension of γ(A) almost surely equals d times the Hausdorff dimension of A.

show abstract

“…In this section we want to show the desired lower bound for the multi-point Green's function. The method of the proof is based on the generalization of the method used in [16] and [13] to show the lower bound. We find the best point (almost means the nearest point but we make it precise) to go near first and we consider the event to go near that point before going near other points (as much as possible).…”

Section: Proof Of Theorem 43mentioning

confidence: 99%

Green’s functions for chordal SLE curves

Rezaei

Zhan

2017

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

For a chordal SLE κ (κ ∈ (0, 8)) curve in a domain D, the n-point Green's function valued at distinct points z 1 , . . . , z n ∈ D is defined to bewhere d = 1 + κ 8 is the Hausdorff dimension of SLE κ , provided that the limit converges. In this paper, we will show that such Green's functions exist for any finite number of points. Along the way we provide the rate of convergence and modulus of continuity for Green's functions as well. Finally, we give up-to-constant bounds for them.A Proof of Theorem 3.1 46 Lemmas on two-sided radial SLEFor z ∈ H, and r > 0, we use P r z to denote the conditional law P[·|τ z r < ∞], and use P * z to denote the law of a two-sided radial SLE κ curve through z. For z ∈ R \ {0}, we use P * z to denote the law of a two-sided chordal SLE κ curve through z. Let E r z and E * z denote the corresponding expectation. In any case, we have P * z -a.s., T z < ∞. See [15,16] for definitions and more details on these measures. For a random chordal Loewner curve γ, we use (F t ) to denote the filtration generated by γ.

show abstract

Up-to-constants bounds on the two-point Green's function for SLE curves

Abstract: The Green's function for the chordal Schramm-Loewner evolution SLE κ for 0 < κ < 8, gives the normalized probability of getting near points. We give up-to-constant bounds for the two-point Green's function.

Cited by 6 publications

References 9 publications

Higher moments of the natural parameterization for SLE curves

Higher moments of the natural parameterization for SLE curves

Optimal Hölder continuity and dimension properties for SLE with Minkowski content parametrization

Green’s functions for chordal SLE curves

Contact Info

Product

Resources

About