A. Srakar | Solution to stochastic partial Loewner eq with several comp. var using Nevanlinna theory
Channel: EIMI, PDMI RAS and Chebyshev Laboratory
Date: 2022-08-01

O.A. Ladyzhenskaya centennial conference on PDE’s, St. Petersburg, July 16 – July 22, 2022
Andrej Srakar (Institute for Economic Research, University of Ljubljana) 
Solution to stochastic partial Loewner equation with several complex variables using Nevanlinna theory

Becker (1973) studied solutions to the Loewner differential equation in one complex variable using Carath´eodory class of holomorphic functions. However, in several complex variables point singularities are removable and other approaches necessary. Pfaltzgraff generalized to higher dimensions the Loewner differential equation and developed existence and uniqueness theorems for its solutions. The existence and regularity theory has been considered by several authors, and applications given to the characterization of subclasses of biholomorphic mappings, univalence criteria, growth theorems and coefficient bounds for restricted classes of biholomorphic mappings. Duren et al. (2010) studied general form of solutions to the Loewner differential equation under common
assumptions of holomorphicity and uniquely determined univalent subordination chains. To our knowledge, to date stochastic Loewner equation has not been studied in a several complex variable setting. We solve the equation in its partial and ordinary differential form by firstly appropriately defining Brownian motion in high dimensions, following Pitman and Yor (2018). We translate the problem in meromorphic form using generalizations of Nevanlinna theory for several complex variables (Noguchi and Winkelman, 2013). Finally, the equation is solved using techniques from rough paths theory (see e.g. Hairer, 2013). Solution allows to study stochastic phenomena such as Schramm-Loewner evolution in high dimensions and we shortly study convergence to a scaling limit for multidimensional lattice models as a power series problem on Hartogs domain. 
References
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Variables and Diophantine Approximation, Springer Japan, Tokyo, 2014.
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