Chordal Schramm–Loewner evolution in the complex upper half-plane from to is the random Loewner chain driven by , where is standard Brownian motion and .
A compact H-hull is a bounded relatively closed subset of the complex upper half-plane such that is a simply connected domain.
A conformal map from an upper-half-plane domain has hydrodynamic normalization when at infinity. For a compact H-hull this normalization makes its mapping-out function of a compact H-hull unique.
If compact H-hulls satisfy , thenThis follows from the Brownian representation of half-plane capacity or from composition of their mapping-out functions.
If is the first exit time of planar Brownian motion from , thenIt follows by applying the optional sampling theorem for a supermartingale to the harmonic function .
There is a universal constant such that every compact H-hull satisfiesTranslate and scale so the hull lies in a unit half-disc, then use the Brownian representation of half-plane capacity and the harmonic measure of that half-disc as viewed from .
For with ,Indeed, the imaginary part at the Brownian exit point is at most one and the probability of reaching the radius- neighbourhood containing the rectangle from is . Consequently has capacity although its diameter tends to two.
A growing hull is parameterized by half-plane capacity when . The factor two makes its Chordal Loewner equation take the conventional form .
A chordal Loewner chain is an increasing family of compact H-hulls, usually parameterized by half-plane capacity, whose mapping-out functions evolve according to the Chordal Loewner equation.
The local growth property says that, after mapping out the hull at time , the new hull grown during a short interval has diameter tending uniformly to zero with the interval length. It ensures that the growth is described by one continuous boundary point.
The Loewner differential equation describes a growing family of simply connected planar domains through ordinary differential equations for their normalized conformal maps.
For a capacity-parameterized locally growing hull family,where the continuous real function is the Loewner driving function.
The Loewner driving function is the real boundary point at which the mapped-out hull grows. Scaling the hulls by and time by transforms it to .
For every , if is , then has the same law. Its driving function is , which has the same law as by Brownian scaling.
Conditionally on an initial segment through time , mapping out that segment by turns the future into an independent in . This follows because its driving function is , and Brownian motion has stationary increments and independent increments.
If , then for every and fixed , there is an almost surely finite random such thatfor and . The proof combines the derivative martingale, Markov inequality, a dyadic lattice, the Borel-Cantelli lemmas, and the Koebe distortion theorem.
For chordal , the centered image of a real boundary point, divided by , evolves as a Bessel process of dimensionIts first hit of zero is the time at which the point is swallowed by the Loewner hull.
For , couple Bessel flows with the same Brownian motion and let . ThenIt is the probability that the SLE hull swallows the chosen point on the negative side before the point on the positive side.
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Schramm–Loewner evolution (SLE) is a mathematical framework used to describe certain conformally invariant processes in statistical physics and complex analysis. It was introduced by Oded Schramm in 2000 as a method for understanding the scaling limits of random planar processes, such as percolation, random walks, and the interfaces of various models in statistical mechanics.