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Schramm–Loewner Evolution pp 49–67Cite as

Loewner Equation

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Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 24))

Abstract

This chapter develops in detail the connection of growing hulls and the Loewner differential equation satisfied by families of conformal maps.

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Notes

  1. 1.

    Remember the notation \(\mathsf {E}[X ;A] = \int _A X \mathrm {d}\mathsf {P}\).

  2. 2.

    The multiplicative constant in (4.21) is fixed by evaluating both sides of (4.20) and its conjugate at 0.

References

  1. Beffara, V.: Schramm-Loewner evolution and other conformally invariant objects. In: Probability and Statistical Physics in Two and More Dimensions, pp. 1–48. American Mathematical Society, Providence, RI (2012)

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  2. Lawler, G.F.: Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs, vol. 114. American Mathematical Society, Providence, RI (2005)

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  3. Lawler, G.F.: Schramm-Loewner Evolution (SLE). In: Statistical mechanics, IAS/Park City Math. Ser., vol. 16, pp. 231–295. Amer. Math. Soc., Providence, RI (2009)

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  4. Lawler, G.F., Schramm, O., Werner, W.: Values of Brownian intersection exponents. I. Half-plane exponents. Acta Math. 187(2), 237–273 (2001). https://doi.org/10.1007/BF02392618

  5. Lind, J., Marshall, D.E., Rohde, S.: Collisions and spirals of Loewner traces. Duke Math. J. 154(3), 527–573 (2010). https://doi.org/10.1215/00127094-2010-045

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland

    Antti Kemppainen

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  1. Antti Kemppainen
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Correspondence to Antti Kemppainen .

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Kemppainen, A. (2017). Loewner Equation. In: Schramm–Loewner Evolution. SpringerBriefs in Mathematical Physics, vol 24. Springer, Cham. https://doi.org/10.1007/978-3-319-65329-7_4

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