We derive central limit theorems for the Wasserstein distance between the empirical distributions of Gaussian samples. The cases are distinguished whether the underlying laws are the same or different. Results are based on the (quadratic) Fréchet differentiability of the Wasserstein distance in the gaussian case. Extensions to elliptically symmetric distributions are discussed as well as several applications such as bootstrap and statistical testing.

The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space U which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of decompositions of subfamilies of fixed kinship h (described as ultrametric measure spaces), for every depth h as a measurable functional of the genealogy.Technically the elements of the semigroup are those um-spaces which have diameter less or equal to 2h called h-forests (h > 0). They arise from a given ultrametric measure space by applying maps called h−truncation. We can define a concatenation of two h-forests as binary operation. The corresponding semigroup is a Delphic semigroup and any h-forest has a unique prime factorization in h-trees (um-spaces of diameter less than 2h). Therefore we have a nested R + -indexed consistent (they arise successively by truncation) collection of Delphic semigroups with unique prime factorization.Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the h-tops can be represented as concatenation of independent identically distributed h-forests for every h and obtain a Lévy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests.Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations.The results have various applications. In particular the case of the genealogical (U-valued) Feller diffusion and genealogical (U V -valued) super random walk is treated based on the present work in [DG19b] and [GRG].In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forests.

We consider strong uniqueness and thus also existence of strong solutions for the stochastic heat equation with a multiplicative colored noise term. Here, the noise is white in time and colored in q dimensional space (q ≥ 1) with a singular correlation kernel. The noise coefficient is Hölder continuous in the solution. We discuss improvements of the sufficient conditions obtained in Mytnik, Perkins and Sturm (2006) that relate the Hölder coefficient with the singularity of the correlation kernel of the noise. For this we use new ideas of Mytnik and Perkins (2011) who treat the case of strong uniqueness for the stochastic heat equation with multiplicative white noise in one dimension. Our main result on pathwise uniqueness confirms a conjecture that was put forward in their paper. Theorem 1.3. Assume that the assumptions of Theorem 1.2 and therefore also of Theorem 1.1 hold. Let (Ω, F , (F t ) t≥0 , P ) be a filtered probability space with adapted colored noise W and let X 0 ∈ C tem be F 0 -measurable. Then there exists a strong adapted solution X to (1) with respect to the prescribed X 0 and W.Proof. We want to use the terminology of [Kur07]. In order to apply the results we need to specify the space on which the noise W can be realized. One can show, see Lemma 3.3.14 of [Rip12] for details, that the Sobolev space H −q−1 (R q ) is an appropriate space, which is Polish. Now set S 1 = C(R + , H −q−1 (R q )) and S 2 = C(R + , C tem ), which is the sample path space of the solutions, and formulate the SPDE (1) as in Example 3.9 of [Kur07]. By Theorem 1.1 we know that there exist compatible solutions (see Lemma 3.2 of [Kur07] for a compatibility criterion which is applicable for weak solutions) and by Theorem 1.2 we have pointwise uniqueness for compatible solutions. So we can apply Theorem 3.14 a) ⇒ b) of [Kur07]. More details can be found in the proof of Lemma 5.1.1 of [Rip12].We want to conclude this section with a number of remarks regarding the Hölder continuity condition on σ stated in (5):(a) When (4) holds, it suffices to assume (5) for |X − X ′ | ≤ 1. Indeed, (5) (with any γ > 0) is immediate from (4) for |X − X ′ | ≥ 1 with A 0 (T ) = 2c 4 , A 1 = 0, A 2 = 1.

Branching property, generator criterion, genealogical processes, historical processes, ancestral path marked genealogies, processes with values in marked ultrametric measure spaces, genealogical super random walk, ancestral path, processes with values in semigroups.Patric Glöde was in part supported by a Technion Fellowship.

We prove general results about separation and weak # -convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a * -algebra F of bounded complex-valued functions and give conditions for it to be separating or weak # -convergence determining for those boundedly finite measures that integrate all functions in F. For separation, it is sufficient if F separates points, vanishes nowhere, and either consists of only countably many measurable functions, or of arbitrarily many continuous functions. For convergence determining, it is sufficient if F induces the topology of the underlying space, and every bounded set A admits a function in F with values bounded away from zero on A.