Generalised Loewner Evolutions and their driving measures
In this thesis we study the interplay between Loewner chains and their driving measures. We prove three versions of a Loewner-Kufarev bijection between Loewner chains, i.e. normalised conformal maps f_t for t >= 0, defined on a fixed domain and whose images are continuously shrinking, and their driving measures, i.e. a time-dependent family of locally finite measures defined on the domain boundaries. There are two variants of Loewner chains where the functions f_t are either normalised at an interior or at a boundary point. Radial Loewner Evolutions are usually defined on the unit disc D = {z in C: |z| < 1} and normalised in the origin. Alternatively, chordal Loewner Evolutions are normally defined on the upper halfplane H := {z in C: Im(z) > 0} and are normalised at infinity. The theory of Loewner chains was originally introduced by Charles Loewner in 1923. Dating back to this paper Loewner chains solve a differential equation containing the driving measure as a time-dependent parameter. For radial Loewner Evolutions it is known that there exists a bijection between Loewner chains and driving measures. Our main results regards the reverse direction: We obtain the driving measure as a (weak) limit of absolutely continuous measures with known densities. Despite Loewner chains being a popular tool in Complex Analysis and Probability Theory this question had not previously been addressed. We prove this result three times. Firstly, we obtain a formula to derive for radial driving measures as locally finite measures defined on the product space [0, infinity) x {z in C : |z| = 1}. Secondly, we show an analogous result for chordal Loewner Evolutions. In this case the measures mu are defined on the product space [0, infinity) x (R cup {infinity}) such that mu([0, T] x (R cup {infinity})) < infinity for all T > 0. Thirdly, there exist locally finite measures mu defined on [0, infinity) x (R cup {infinity}) this finiteness condition. Hence, we study the most general case of (generalised) chordal Loewner chains. In that case there exists a bijection between these generalised Loewner chains and pairs (mu, b) where b: [0, infinity) -> R is continuous and mu is a measure defined on the product space [0, infinity) x R such that the measure nu on [0, infinity) x R defined by nu(-) = int_(-) (mu(du, dt)) / (u^2 + 1) is locally finite. Then mu can be computed as a vague limit of absolutely continuous measures with known densities. In all three cases Loewner chains are compact under Carathéodory convergence, i.e. uniform convergence on compact subsets of [0, T] x D, resp. [0, T] x H. This corresponds to a tightness property of the underlying driving measures. In all cases these two facts imply that the Loewner-Kufarev Map between Loewner chains and driving measures is continuous under appropriate choices of topologies on the space of Loewner chains and driving measures. Moreover, we apply these results to a family of particle aggregation models that can be described as a sequence of radial Loewner chains. We obtain two universality results for a broad class of models. Additionally, we prove that a specific Hastings-Levitov(0) model with stochastic particles converges strongly to a ball. Lastly, as joint work with Prof. Dr. Eveliina Peltola from Aalto University in Helsinki and the University of Bonn we study chordal Loewner Evolutions with driving functions, i.e. driving measures of the form mu_t = delta_{W(t)} where delta denotes the Dirac measure and W: [0, infinity) -> R is a sufficiently regular function. Specifically, we study when these Loewner Evolutions are generated by a function and the regularity of these generating functions. We say that the Loewner chain $f_t)_t is generated by a function eta: [0, infinity) -> H cup R, if f_t(H) is the unbounded component of H setminus eta[0, t]. We prove that hulls are generated by a function if, and only if, the sets K_t cup R are path-connected for all t >= 0, where K_t is the closure of H setminus f_t(H). Moreover, if W(t_-) = lim_{s -> t} W(s) and the limit eta(t) := lim_{y -> 0} f_t (W(t_-) + i y) exists for all t >= 0, then eta is the generating function. And in this case eta has unique right limits. Finally, we prove that if a left-continuous generating function exists, then the corresponding hulls (K_t)_{t >= 0} are locally connected.
Loewner chain, Complex Analysis, Probability Theory, Particle Aggregation Models, Hasting-Levitov Models
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