On the continuity of $$\text{ SLE }_{\kappa }$$ in $$\kappa $$

Johansson Viklund, Fredrik; Rohde, Steffen; Wong, Carto
August 2014
Probability Theory & Related Fields;Aug2014, Vol. 159 Issue 3/4, p413
Academic Journal
We prove that for almost every Brownian motion sample, the corresponding SLE $$_\kappa $$ curves parameterized by capacity exist and change continuously in the supremum norm when $$\kappa $$ varies in the interval $$[0,\kappa _0)$$ , where $$\kappa _0=8(2-\sqrt{3})\approx 2.1$$ . We estimate the $$\kappa $$ -dependent modulus of continuity of the curves and also give an estimate on the modulus of continuity for the supremum norm change with $$\kappa $$ .


Related Articles

  • Loop-Erasure of Planar Brownian Motion. Zhan, Dapeng // Communications in Mathematical Physics;Apr2011, Vol. 303 Issue 3, p709 

    We use a coupling technique to prove that there exists a loop-erasure of the time-reversal of a planar Brownian motion stopped on exiting a simply connected domain, and that the loop-erased curve is a radial SLE curve. This result extends to Brownian motions and Brownian excursions under certain...

  • Empty confidence sets for epidemics, branching processes and Brownian motion. BALL, FRANK G.; BRITTON, TOM; O'NEILL, PHILIP D. // Biometrika;Mar2002, Vol. 89 Issue 1, p211 

    This paper treats some examples where likelihood‐based inference for certain model parameters may produce empty confidence sets. The first example concerns epidemics, and the parameter of interest is the basic reproduction number R0, which is to be estimated from the final size of an...

  • Existence and Uniqueness of Solutions to the Stochastic Porous Media Equations of Saturated Flows. Ciotir, Ioana // Applied Mathematics & Optimization;Feb2010, Vol. 61 Issue 1, p129 

    This paper proves the existence and uniqueness of nonnegative solutions for the stochastic porous media equations with multiplicative noise, infinite jump and discontinuous diffusivity function relevant in description of saturation processes in underground water infiltration in a bounded domain...

  • Nonparametric likelihood inference for general autoregressive models. Bravo, Francesco // Statistical Methods & Applications;2010, Vol. 19 Issue 1, p79 

    This paper shows how nonparametric likelihood inference for autoregressive models can be based on the family of “empirical” Cressie–Read statistics. The results of the paper apply to possibly nonstationary autoregressive models with innovations that form a martingale...

  • Restriction Properties of Annulus SLE. Zhan, Dapeng // Journal of Statistical Physics;Mar2012, Vol. 146 Issue 5, p1026 

    For κ∈(0,4], a family of annulus SLE( κ;Λ) processes were introduced in (Zhan in arXiv:) to prove the reversibility of whole-plane SLE( κ). In this paper we prove that those annulus SLE( κ;Λ) processes satisfy a restriction property, which is similar to that for chordal...

  • Points of increase of the Brownian sheet. Dalang, Robert C.; Mountford, T. // Probability Theory & Related Fields;1997, Vol. 108 Issue 1, p1 

    Summary. It is well-known that Brownian motion has no points of increase. We show that an analogous statement for the Brownian sheet is false. More precisely, for the standard Brownian sheet in the positive quadrant, we prove that there exist monotone curves along which the sheet has a point of...

  • 3G-inequality for planar domains. Hansen, Wolfhard // Probability Theory & Related Fields;Feb2012, Vol. 152 Issue 1/2, p357 

    The 3G-inequality for Green functions g on arbitrary bounded domains in $${{\mathbb{R}}^2}$$ , which Bass and Burdzy (Probab Theory Relat Fields 101(4):479-493, ) obtained by a genuinely probabilistic proof (using loops of Brownian motion around the origin), is proven (in a more precise form)...

  • The First Passage Time and Estimation of the Number of Level-Crossings for a Telegraph Process. Pogorui, A.; Rodríguez-Dagnino, R.; Kolomiets, T. // Ukrainian Mathematical Journal;Dec2015, Vol. 67 Issue 7, p998 

    It is a well-known result that almost all sample paths of a Brownian motion or Wiener process {W( t) } have infinitely many zero-crossings in the interval (0 , δ) for δ > 0. Under the Kac condition, the telegraph process weakly converges to the Wiener process. We estimate the number of...

  • Unbiased estimation of secondary parameters following a sequential test. AIYI LIU; HALL, W. J. // Biometrika;Oct2001, Vol. 88 Issue 3, p895 

    We consider, within a Brownian motion framework, estimation of secondary parameters following a sequential test. Whitehead (1986b) proposed a method for reducing the bias of maximum likelihood estimators. In this paper we present unbiased estimators to eliminate the bias.


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics