Research Article | Open Access
The Intersection Probability of Brownian Motion and SLEκ
By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownian motion and . Moreover, we find a transformation such that the second-order differential equation transforms into a hypergeometric differential equation. Then, by solving the hypergeometric differential equation, we obtain the explicit formula of the intersection probability for the trace of the chordal and planar Brownian motion started from distinct points in an upper half-plane .
Shortly after Schramm  in 2000 introduced the percolation exploration path and curve, Lawler et al. [2–5] proved in succession the intersection exponents of Brownian motion in the physics literature: half-plane exponents, plane exponents, and two-side exponents. That is, they proved that the intersection exponents of the two independent planar Brownian motions started from distinct points. Naturally, one can ask what the intersection exponents (or probability) of Brownian motion and are in the half-plane. Not only is this question significant itself, but it also can probe more into the property of . Kozdron [6, 7] derived the intersection probability of Brownian motion and . The aim of this paper is to derive the intersection probability of Brownian motion and in a half-plane.
Chordal . The chordal Schramm-Loewner evolution [8, 9] are growth process defined via conformal maps [10, 11] which are solutions of the ordinary differential equation as follows:where and is a one-dimensional Brownian motion. For any fixed , , and all in a neighborhood of , either there exists a solution of (1) for or, for some , there exists a solution for such that . Let denote the (open) set of such that the first holds and the set of such that the second is true. We call the process the chordal Schramm-Loewner evolution process with parameter (abbreviated as ). It is easy to see that the hulls of chordal are the hulls of a continuous path:which is called the trace of the process.
Brownian Excursions Poisson Kernel. For a simply connected domain and when is locally analytic at points and , the Brownian excursion Poisson kernel is defined as the normal derivative of the usual Poisson kernel. Since Brownian excursion measure is the scaling limit of simple random walk excursion measure, the excursion Poisson kernel is also the mass of the Brownian excursion measure. More precisely speaking, the excursion Poisson kernel is defined as where is the unit normal at pointing into the domain and is the usual Poisson kernel. Brownian excursion Poisson kernel satisfied the conformal covariance property as follows.
Lemma 1 ([7, Proposition ]). Suppose is a conformal transformation where is a simply connected Jordan domain and is locally analytic at points and ; thenParticularly, when (the unit disc), And when (the upper half-plane),
2. Intersection Probability
In this section, our main goal is now to compute the intersection probability of and Brownian motion started from distinct points in a half-plane . In order to determinethe probability, our strategy for establishing this result will be as follows. By using the methods of the excursion measure Poisson kernel, we will first determine an explicit differential equation for , which is a hypergeometric differential equation. Then, we obtain the general solution of the differential equation.
2.1. The Explicit Differential Equation of Intersection Probability
We note that if is a standard one-dimensional Brownian motion, is also a standard one-dimensional Brownian motion. Hence, satisfies the ordinary differential equation as follows:Letting and , we have the following.
Theorem 2. The intersection probability for the trace of the chordal and planar Brownian motion started from distinct points in a half-plane satisfies
Proof. Suppose that is the slit-plane for any . This implies thatLetting , this yields By using expression (7) and conformal covariance, we can obtainWe define asHence, it yieldsRecalling and , we haveWe also note thatThen, we can obtainBecause ofwe finally obtainHence, we haveThis completes the proof of the theorem.
2.2. Intersection Probability
Theorem 3. The intersection probability for the trace of the chordal and planar Brownian motion started from distinct points in a half-plane satisfieswhere .
Proof. In , the authors obtained the following deep theorem.
In the range , there exists a random simple curve with and . In the range , there exists a random simple curve . These curves have double points and they hit , but they never cross themselves. In the range , there exists a random simple curve . These curves are the space-filling curves, but they never cross themselves.
When , these random curves are the space-filling curves. It is easy to obtain that In order to prove Theorem 2, therefore, it is sufficient that we only need to prove the theorem when .
Recallingby using Markov property, it is easy to see that is a martingale. That is, is a martingale if we define . Hence, we haveBy using Itô formula  at , this implies thatBecause the intersection probability in our question only depends on the ratio , we define the transformation , where is some second-order derivative function. Thus, we haveMultiplying by and letting , this yieldsMultiplying by and combining terms, we obtainOr equivalently (because )It is easy to know that the second-order differential equation (28) has regular singular points at 0, 1, and . Hence, it is possible to transform it into a hypergeometric differential equation  by writing (28) asIn order to solve (29), our main goal is to find a transformation such that it is a hypergeometric differential equation. We find that the transformation as follows just is our expectation:where .
By (30), we haveBy substituting (30) and (31) into (29), we finally obtainIt is easy to see that (32) is now a well-known hypergeometric differential equation whose general solution is given byHence, we finally obtain that the general solution of (28) isFurthermore, physical considerations imply that as (or ) and as (or ). This yields Thus, This completes the proof of Theorem 3.
By using the knowledge of the conformal transformation, For any bounded, simply connected planar domain, we have a similar theorem.
Theorem 4. Let be a bounded, simply connected domain and let , , , and be four points ordered counterclockwise around . The intersection probability for the trance of the chordal from to in and planar Brownian motion started from to in satisfieswhere and is the conformal transformation with , , and .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (no. 11161004) and the Natural Science Foundation of Guangxi (no. 2013GXNSFAA019015).
- O. Schramm, “Scaling limits of loop-erased random walks and uniform spanning trees,” Israel Journal of Mathematics, vol. 118, pp. 221–288, 2000.
- G. F. Lawler, O. Schramm, and W. Werner, “One-arm exponent for critical 2D percolation,” Electronic Journal of Probability, vol. 7, no. 2, pp. 1–13, 2002.
- G. F. Lawler, O. Schramm, and W. Werner, “Values of Brownian intersection exponents, I: half-plane exponents,” Acta Mathematica, vol. 187, no. 2, pp. 237–273, 2001.
- G. F. Lawler, O. Schramm, and W. Werner, “Values of Brownian intersection exponents. II. Plane exponents,” Acta Mathematica, vol. 187, no. 2, pp. 275–308, 2001.
- G. F. Lawler, O. Schramm, and W. Werner, “Values of Brownian intersection exponents III: two-sided exponents,” Annales de l'Institut Henri Poincare, vol. 38, no. 1, pp. 109–123, 2002.
- M. J. Kozdron, “On the scaling limit of simple random walk excursion measure in the plane,” ALEA—Latin American Journal of Probability and Mathematical Statistics, vol. 2, pp. 125–155, 2006.
- M. J. Kozdron, “The scaling limit of Fomin's identity for two paths in the plane,” J. C. R. Math. Rep. Acad. SCI. Canada, vol. 29, no. 3, pp. 65–80, 2007.
- G. F. Lawler, Conformally Invariant Processes in the Plane, vol. 114 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, USA, 2005.
- S. Rohde and O. Schramm, “Basic properties of SLE,” Annals of Mathematics, vol. 161, no. 2, pp. 883–924, 2005.
- R. Langlands, P. Pouliot, and Y. Saint-Aubin, “Conformal invariance in two-dimensional percolation,” Bulletin American Mathematical Society: New Series, vol. 30, no. 1, pp. 1–61, 1994.
- C. Pommerenke, Boundary Behaviour of Conformal Maps, vol. 299 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1992.
- D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, Springer, 2004.
- M. Abramowitz and I. A. Stegun, Eds., Handbook of Mathematical Function with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards, Washington, DC, USA, 1972.
Copyright © 2015 Shizhong Zhou and Shiyi Lan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.