Related papers: Loewner evolution driven by a stochastic boundary …
This review provides an introduction to two dimensional growth processes. Although it covers a variety processes such as diffusion limited aggregation, it is mostly devoted to a detailed presentation of stochastic Schramm-Loewner evolutions…
We consider radial Loewner evolution driven by unimodular L\'evy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by…
Motivated by the study of trace for Schramm-Loewner evolutions, we consider evolutions of planar domains governed by ordinary differential equations with holomorphic vector fields $F$ defined on the upper half plane $\mathbb{H}$. We show a…
The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to…
This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution…
The symlectic quantum tomography for the general linear quantization is introduced. Using the approach based upon the Wigner function techniques the evolution equation of quantum tomograms is derived for a parametric driven oscillator.
Lamb has identified a certain class of moving space curves with soliton equations. We show that there are two other classes of curve evolution that may be so identified. Hence three distinct classes of curve evolution are associated with a…
Consider the Loewner equation associated to the upper-half plane. Normally this equation is driven by a real-valued function. In this paper, we show that when the driving function is complex-valued with small $1/2$-H\"older norm, the…
We study the convergence of stochastic time-discretization schemes for evolution equations driven by random velocity fields, including examples like stochastic gradient descent and interacting particle systems. Using a unified framework…
The Kepler problem is considered in a space with the Friedmann--Lemaitre--Robertson--Walker metrics of the expanding universe. The covariant differential of the Friedmann coordinates (X=a(t)x) is considered as a possible mechanism of the…
We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular noise. As an illustration, we discuss a…
Equations of the Loewner class subject to non-constant boundary conditions along the real axis, are formulated and solved giving the geodesic paths of slits growing in the upper half complex plane. The problem is motivated by Laplacian…
In this paper we develop an $L_2$-theory for stochastic partial differential equations driven by L\'evy processes. The coefficients of the equations are random functions depending on time and space variables, and no smoothness assumption of…
The Lalanne-Kreweras involution is an involution on the set of Dyck paths which combinatorially exhibits the symmetry of the number of valleys and major index statistics. We define piecewise-linear and birational extensions of the…
We consider the Loewner differential equation generating univalent maps of the unit disk (or of the upper half-plane) onto itself minus a single slit. We prove that the circular slits, tangent to the real axis are generated by H\"older…
The most general local Markovian stochastic model is investigated, for which it is known that the evolution equation is the Fokker-Planck equation. Special cases are investigated where uncorrelated initial states remain uncorrelated.…
The Shcramm-Loewner evolution (SLE) is a correlated exploration process, in which for the chordal set up, the tip of the trace evolves in a self-avoiding manner towards the infinity. The resulting curves are named SLE$_{\kappa}$,…
Loewner chains with Levy drivers have been proposed as models for random dendritic growth in two dimensions, and as candidates for finding extremal multifractal spectra in problems in classical function theory. These processes are not…
Motivated by applications in economics and finance, in particular to the modeling of limit order books, we study a class of stochastic second-order PDEs with non-linear Stefan-type boundary interaction. To solve the equation we transform…
We study the existence and uniqueness of Lp-bounded mild solutions for a class ofsemilinear stochastic evolutions equations driven by a real L\'evy processes withoutGaussian component not square integrable for instance the stable process…