For two Polish state spaces $E_X$ and $E_Y$, and an operator $G_X$, we obtain existence and uniqueness of a $G_X$-martingale problem provided there is a dual process $Y$ on $E_Y$ solving a $G_Y$-martingale problem. Duality here means the existence of a rich function $H$ and transition kernels $(\mu_t)_{t\geq 0}$ on $E_X$ such that $\mathbf E_y[H(x,Y_t)] = \int \mu_t(x,dx') H(x',y)$ for all $(x,y) \in E_X \times E_Y$ and $t\geq 0$. While duality is well-known to imply uniqueness of the $G_X$-martingale problem, we give here a set of conditions under which duality also implies existence without using approximation techniques. As examples, we treat branching and resampling models, such as Feller's branching diffusion and the Fleming-Viot superprocess.

In this work we extend the quenched local limit theorem obtained by the authors in [BBDS23]. More precisely, we consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+1 with d≥ 1 being the spatial dimension. In [BBDS23] an annealed local central limit theorem was proven for all d≥ 1 and a quenched local limit theorem under the assumption d≥ 3. Here we show that the latter result also holds for all d ≥ 1.

We consider random walks in dynamic random environments which arise naturally as spatial embeddings of ancestral lineages in spatial locally regulated population models. In particular, as the main result, we prove the quenched central limit theorem for a random walk in dynamic random environment generated by time reversal of logistic branching random walks in a regime where the population density is sufficiently high. As an important tool we consider as auxiliary models random walks in dynamic random environments defined in terms of the time-reversal of oriented percolation. We show that the quenched central limit theorem holds if the influence of the random medium on the walks is suitably weak. The proofs of the quenched central limit theorems in these models rely on coarse-graining arguments and a construction of regeneration times for a pair of conditionally independent random walks in the same medium, combined with a coupling that relates them to a pair of independent random walks in two independent copies of the medium.

Consider two ancestral lineages sampled from a system of two-dimensional branching random walks with logistic regulation in the stationary regime. We study the asymptotics of their coalescence time for large initial separation and find that it agrees with well known results for a suitably scaled two-dimensional stepping stone model and also with Malécot's continuous-space approximation for the probability of identity by descent as a function of sampling distance. This can be viewed as a justification for the replacement of locally fluctuating population sizes by fixed effective sizes. Our main tool is a joint regeneration construction for the spatial embeddings of the two ancestral lineages.

We consider a directed random walk on the backbone of the supercritical oriented percolation cluster in dimensions d+1 with d≥3 being the spatial dimension. For this random walk we prove an annealed local central limit theorem and a quenched local limit theorem. The latter shows that the quenched transition probabilities of the random walk converge to the annealed transition probabilities reweighted by a function of the medium centred at the target site. This function is the density of the unique measure which is invariant for the point of view of the particle, is absolutely continuous with respect to the annealed measure and satisfies certain concentration properties.

We survey results on the description of stochastically evolving genealogies of populations and marked genealogies of multitype populations or spatial populations via tree-valued Markov processes on (marked) ultrametric measure spaces. In particular we explain the choice of state spaces and their topologies, describe the dynamics of genealogical Fleming-Viot and branching models by well-posed martingale problems, and formulate the typical results on the longtime behavior. Furthermore we discuss the basic techniques of proofs and sketch as two key tools of analysis the different forms of duality and the Girsanov transformation.

We consider the evolution of the genealogy of the population currently alive in a Feller branching diffusion model. In contrast to the approach via labeled trees in the continuum random tree world, the genealogies are modeled as equivalence classes of ultrametric measure spaces, the elements of the space $\mathbb{U}$. This space is Polish and has a rich semigroup structure for the genealogy. We focus on the evolution of the genealogy in time and the large time asymptotics conditioned both on survival up to present time and on survival forever. We prove existence, uniqueness and Feller property of solutions of the martingale problem for this genealogy valued, i.e., $\mathbb{U}$-valued Feller diffusion. We give the precise relation to the time-inhomogeneous $\mathbb{U}_1$-valued Fleming-Viot process. The uniqueness is shown via Feynman-Kac duality with the distance matrix augmented Kingman coalescent. Using a semigroup operation on $\mathbb{U}$, called concatenation, together with the branching property we obtain a L\'evy-Khintchine formula for $\mathbb{U}$-valued Feller diffusion and we determine explicitly the L\'evy measure on $\mathbb{U}\setminus\{0\}$. From this we obtain for $h>0$ the decomposition into depth-$h$ subfamilies, a representation of the process as concatenation of a Cox point process of genealogies of single ancestor subfamilies. Furthermore, we will identify the $\mathbb{U}$-valued process conditioned to survive until a finite time $T$. We study long time asymptotics, such as generalized quasi-equilibrium and Kolmogorov-Yaglom limit law on the level of ultrametric measure spaces. We also obtain various representations of the long time limits.

We consider random walks in dynamic random environments, with an environment generated by the time-reversal of a Markov process from the oriented percolation universality class.If the influence of the random medium on the walk is small in space-time regions where the medium is typical, we obtain a law of large numbers and an averaged central limit theorem for the walk via a regeneration construction under suitable coarse-graining.Such random walks occur naturally as spatial embeddings of ancestral lineages in spatial population models with local regulation.We verify that our assumptions hold for logistic branching random walks when the population density is sufficiently high.

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral  recombination graph encodes the genealogies at all loci in one graph.  For a continuous genome $\mathbb G$, we study the tree-valued  process $(T^N_u)_{u\in\mathbb{G}}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure  spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting  process for loci which are far apart.

Kingman's coalescent is a random tree that arises from classical population genetic models such as the Moran model. The individuals alive in these models correspond to the leaves in the tree and the following two laws of large numbers concerning the structure of the tree-top are well-known: (i) The (shortest) distance, denoted by $T_n$, from the tree-top to the level when there are $n$ lines in the tree satisfies $nT_n \xrightarrow{n\to\infty} 2$ almost surely; (ii) At time $T_n$, the population is naturally partitioned in exactly $n$ families where individuals belong to the same family if they have a common ancestor at time $T_n$ in the past. If $F_{i,n}$ denotes the size of the $i$th family, then $n(F_{1,n}^2 + \cdots + F_{n,n}^2) \xrightarrow{n\to \infty}2$ almost surely. For both laws of large numbers we prove corresponding large deviations results. For (i), the rate of the large deviations is $n$ and we can give the rate function explicitly. For (ii), the rate is $n$ for downwards deviations and $\sqrt n$ for upwards deviations. For both cases we give the exact rate function.

We consider the tree-valued Fleming-Viot process, $(\mathcal X_t)_{t\geq 0}$, with mutation and selection as studied in Depperschmidt, Greven, Pfaffelhuber (2012). This process models the stochastic evolution of the genealogies and (allelic) types under resampling, mutation and selection in the population currently alive in the limit of infinitely large populations. Genealogies and types are described by (isometry classes of) marked metric measure spaces. The long-time limit of the neutral tree-valued Fleming-Viot dynamics is an equilibrium given via the marked metric measure space associated with the Kingman coalescent. In the present paper we pursue two closely linked goals. First, we show that two well-known properties of the neutral Fleming-Viot genealogies at fixed time $t$ arising from the properties of the dual, namely the Kingman coalescent, hold for the whole path. These properties are related to the geometry of the family tree close to its leaves. In particular we consider the number and the size of subfamilies whose individuals are not further than $\ve$ apart in the limit $\ve\to 0$. Second, we answer two open questions about the sample paths of the tree-valued Fleming-Viot process. We show that for all $t>0$ almost surely the marked metric measure space $\mathcal X_t$ has no atoms and admits a mark function. The latter property means that all individuals in the tree-valued Fleming-Viot process can uniquely be assigned a type. All main results are proven for the neutral case and then carried over to selective cases via Girsanov's formula giving absolute continuity.

We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the "ancestral lineage" of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e. for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.

We consider a directed random walk on the backbone of the infinite cluster generated by supercritical oriented percolation, or equivalently the space-time embedding of the ``ancestral lineage'' of an individual in the stationary discrete-time contact process. We prove a law of large numbers and an annealed central limit theorem (i.e., averaged over the realisations of the cluster) using a regeneration approach. Furthermore, we obtain a quenched central limit theorem (i.e. for almost any realisation of the cluster) via an analysis of joint renewals of two independent walks on the same cluster.

The Fleming-Viot measure-valued diffusion is a Markov process describing the evolution of (allelic) types under mutation, selection and random reproduction. We enrich this process by genealogical relations of individuals so that the random type distribution as well as the genealogical distances in the population evolve stochastically. The state space of this tree-valued enrichment of the Fleming-Viot dynamics with mutation and selection (TFVMS) consists of marked ultrametric measure spaces, equipped with the marked Gromov-weak topology and a suitable notion of polynomials as a separating algebra of test functions. The construction and study of the TFVMS is based on a well-posed martingale problem. For existence, we use approximating finite population models, the tree-valued Moran models, while uniqueness follows from duality to a function-valued process. Path properties of the resulting process carry over from the neutral case due to absolute continuity, given by a new Girsanov-type theorem on marked metric measure spaces. To study the long-time behavior of the process, we use a duality based on ideas from Dawson and Greven [On the effects of migration in spatial Fleming-Viot models with selection and mutation (2011c) Unpublished manuscript] and prove ergodicity of the TFVMS if the Fleming-Viot measure-valued diffusion is ergodic. As a further application, we consider the case of two allelic types and additive selection. For small selection strength, we give an expansion of the Laplace transform of genealogical distances in equilibrium, which is a first step in showing that distances are shorter in the selective case.

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate γ. Extending our previous approach (Depperschmidt and Pfaffelhuber in Stoch Processes Appl 120:901–925, 2010) we allow the ratcheting molecules to dissociate (at rate δ) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speeds of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation. This speed gives an approximation for the time it takes to translocate a protein of given length.

We consider two reflecting diffusion processes (X_t)_t ≥ 0 with a moving reflection boundary given by a non-decreasing pure jump Markov process (R_t)_t ≥ 0. Between the jumps of the reflection boundary the diffusion part behaves as a reflecting Brownian motion with negative drift or as a reflecting Ornstein-Uhlenbeck process. In both cases at rate γ(X_t -R_t) for some γ≥ 0 the reflection boundary jumps to a new value chosen uniformly in [R_t-,X_t]. Since after each jump of the reflection boundary the diffusions are reflected at a higher level we call the processes Brownian ratchet and Ornstein-Uhlenbeck ratchet. Such diffusion ratchets are biologically motivated by passive protein transport across membranes. The processes considered here are generalisations of the Brownian ratchet (without drift) studied in (Depperschmidt and Pfaffelhuber, 2010). For both processes we prove a law of large numbers, in particular each of the ratchets moves to infinity at a positive speed which can be computed explicitly, and a central limit theorem.

A marked metric measure space (mmm-space) is a triple (X,r,mu), where (X,r) is a complete and separable metric space and mu is a probability measure on XxI for some Polish space I of possible marks. We study the space of all (equivalence classes of) marked metric measure spaces for some fixed I. It arises as state space in the construction of Markov processes which take values in random graphs, e.g. tree-valued dynamics describing randomly evolving genealogical structures in population models. We derive here the topological properties of the space of mmm-spaces needed to study convergence in distribution of random mmm-spaces. Extending the notion of the Gromov-weak topology introduced in (Greven, Pfaffelhuber and Winter, 2009), we define the marked Gromov-weak topology, which turns the set of mmm-spaces into a Polish space. We give a characterization of tightness for families of distributions of random mmm- spaces and identify a convergence determining algebra of functions, called polynomials.

We study a model for the translocation of proteins across membranes through a nanopore using a ratcheting mechanism. When the protein enters the nanopore it diffuses in and out of the pore according to a Brownian motion. Moreover, it is bound by ratcheting molecules which hinder the diffusion of the protein out of the nanopore, i.e. the Brownian motion is reflected such that no ratcheting molecule exits the pore. New ratcheting molecules bind at rate gamma. Extending our previous approach (Depperschmidt and Pfaffelhuber, 2010) we allow the ratcheting molecules to dissociate (at rate delta) from the protein (Model I). We also provide an approximate model (Model II) which assumes a Poisson equilibrium of ratcheting molecules on one side of the current reflection boundary. Using analytical methods and simulations we show that the speed of both models are approximately the same. Our analytical results on Model II give the speed of translocation by means of a solution of an ordinary differential equation.

Protein translocation in cells has been modelled by Brownian ratchets. In such models, the protein diffuses through a nanopore. On one side of the pore, ratcheting molecules bind to the protein and hinder it to diffuse out of the pore. We study a Brownian ratchet by means of a reflected Brownian motion (Xt)t≥0 with a changing reflection point (Rt)t≥0. The rate of change of Rt is γ(Xt−Rt) and the new reflection boundary is distributed uniformly between Rt− and Xt. The asymptotic speed of the ratchet scales with γ1/3 and the asymptotic variance is independent of γ.