The Intersection Probability of Brownian Motion and SLE κ

By using excursion measure Poisson kernel method, we obtain a second-order differential equation of the intersection probability of Brownianmotion and SLE κ . 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 SLE κ and planar Brownian motion started from distinct points in an upper half-planeH.


Introduction
Shortly after Schramm [1] in 2000 introduced the percolation exploration path and SLE curve, Lawler et al. [2][3][4][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 SLE  are in the half-plane.Not only is this question significant itself, but it also can probe more into the property of SLE  .Kozdron [6,7] derived the intersection probability of Brownian motion and SLE 2 .The aim of this paper is to derive the intersection probability of Brownian motion and SLE  in a half-plane.
Chordal   .The chordal Schramm-Loewner evolution SLE  [8,9] are growth process defined via conformal maps [10,11] which are solutions of the ordinary differential equation as follows: where  0 () =  and   is a one-dimensional Brownian motion.For any fixed  > 0,  0 ∈ , and all  in a neighborhood of  0 , either there exists a solution of (1) for  ∈ [0, ] or, for some  0 ∈ (0, ], there exists a solution for  ∈ [0, ) such that lim ↗ 0   () =   0 .Let Ω  denote the (open) set of  ∈  such that the first holds and   the set of  ∈  such that the second is true.We call the process (  ,  ≥ 0) the chordal Schramm-Loewner evolution process with parameter  (abbreviated as SLE  ).It is easy to see that the hulls of chordal SLE  are the hulls of a continuous path: which is called the trace of the SLE  process.

Intersection Probability
In this section, our main goal is now to compute the intersection probability of SLE  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 {[0, ∞) ∩ [0,   ] = ⌀}, which is a hypergeometric differential equation.Then, we obtain the general solution of the differential equation.

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   =  \ (0, ] for any 0 <  < ∞.This implies that Letting  → ∞, this yields By using expression (7) and conformal covariance, we can obtain We define as Hence, it yields Recalling   =   () +   and   =   () +   , we have We also note that (15) Then, we can obtain Advances in Mathematical Physics 3 Because of we finally obtain Hence, we have This completes the proof of the theorem.

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  satisfies where b = √ 9 2 − 8 + 16/.
Proof.In [8], the authors obtained the following deep theorem.
When  ∈ [8, ∞), these random curves are the spacefilling 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  ∈ (0, 8).Recalling by using Markov property, it is easy to see that . Hence, we have By using Itô formula [12] at  = 0, this implies that Because the intersection probability in our question only depends on the ratio /, we define the transformation H * (, ) = (/), where  is some second-order derivative function.Thus, we have Multiplying by 2 2 and letting  = /, this yields Multiplying by  2 and combining terms, we obtain Or equivalently (because 0 <  < 1) 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 [13] by writing (28) as In 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  = 1/2 − 2/ + √ 9 It is easy to see that ( 32) is now a well-known hypergeometric differential equation whose general solution is given by Hence, we finally obtain that the general solution of (28) is Furthermore, physical considerations imply that () → 0 as  → 0 + (or  → ∞) and () → 1 as  → 1 − (or  → ).This yields 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.