Classical and Stochastic Löwner–Kufarev Equations Filippo Bracci, Manuel D. Contreras, Santiago Díaz-Madrigal, and Alexander Vasil’ev Abstract In this paper we present a historical and scientific account of the development of the theory of the Löwner–Kufarev classical and stochastic equations spanning the 90-year period from the seminal paper by K. Löwner in 1923 to recent generalizations and stochastic versions and their relations to conformal field theory. Keywords Brownian motion • Conformal mapping • Evolution family • Inte- grable system • Kufarev • Löwner • Pommerenke • Schramm • Subordination chain Mathematics Subject Classification (2000). Primary 01A70 30C35; Secondary 17B68 70H06 81R10 Filippo Bracci () Dipartimento Di Matematica, Università di Roma “Tor Vergata”, Via Della Ricerca Scientifica 1, 00133 Roma, Italy e-mail:
[email protected] M.D. Contreras S. Díaz-Madrigal Departamento de Matemática Aplicada II, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla, Camino de los Descubrimientos, s/n 41092 Sevilla, Spain e-mail:
[email protected];
[email protected] A. Vasil’ev Department of Mathematics, University of Bergen, P.O. Box 7803, Bergen N-5020, Norway e-mail:
[email protected] A. Vasil’ev (ed.), Harmonic and Complex Analysis and its Applications, 39 Trends in Mathematics, DOI 10.1007/978-3-319-01806-5__2, © Springer International Publishing Switzerland 2014 40 F. Bracci et al. 1 Introduction The Löwner theory went through several periods of its development. It was born in 1923 in the seminal paper by Löwner [165], its formation was completed in 1965 in the paper by Pommerenke [189] and was finally formulated thoroughly in his monograph [190] unifying Löwner’s ideas of semigroups and evolution equations and Kufarev’s contribution [144, 146]ofthet-parameter differentiability (in the Carathéodory kernel) of a family of conformal maps of simply connected domains ˝.t/ onto the canonical domain D, the unit disk in particular, and of PDE for subordination Löwner chains of general type.