Andreas Greven | Friedrich-Alexander-Universität Erlangen-Nürnberg (original) (raw)
Papers by Andreas Greven
Electronic Journal of Probability
arXiv (Cornell University), Sep 20, 2022
arXiv (Cornell University), Apr 3, 2019
arXiv (Cornell University), Sep 9, 2012
The idea for this paper arose from discussions with P. Pfaffelhuber and A. Wakolbinger dur-
arXiv: Probability, 2015
We consider a spatial multi-type branching model in which individuals migrate in geographic space... more We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can be sub-, super- or critical depending on the local intensity of individuals of the different types. The model is a Lotka-Volterra type model with a spatial component and is related to two models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlier work in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focus is on the diffusion limit of small mass, locally many individuals and rapid reproduction. This system differs from spatial critical branching systems since it is not density preserving and the densities for large times do not depend on the initial distribution but mainly on the carrying capacities. We prove existence of the infinite particle model and the system of interacting diffusions as solutions of martingale problems or systems of stochastic equat...
We consider a system of interacting diffusions labeled by a geographic space that is given by the... more We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group OmegaN\Omega_NOmegaN of order NinmathbbNN\in\mathbb{N}NinmathbbN. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The migration kernel has a hierarchical structure: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a layered structure: when individuals become dormant they acquire a colour that determines the rate at which they become active again. The latter allows us to model seed-banks whose wake-up times have a fat tail. We analyse a system of coupled stochastic differential equations that describes the population in the large-colony-size limit. For fixed NinmathbbNN\in\mathbb{N}NinmathbbN, the system exhibits a dichotomy between c...
Latin American Journal of Probability and Mathematical Statistics, 2021
The paper has four goals. First, we want to generalize the classical concept of the branching pro... more The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...
arXiv: Probability, 2019
We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more 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 mathbbU\mathbb{U}mathbbU. 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., mathbbU\mathbb{U}mathbbU-valued Feller diffusion. We give the precise relation to the time-inhomogeneous mathbbU_1\mathbb{U}_1mathbbU_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 mathbbU\mathbb{U}mathbbU, called concatenation, together with the...
Electronic Journal of Probability, 2019
Electronic Journal of Probability, 2016
Electronic Journal of Probability, 2016
Journal of Statistical Physics, 1991
Electronic Journal of Probability, 2013
Electronic Communications in Probability, 2011
Canadian Journal of Mathematics, 1995
This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a ... more This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.
When we cut an i.i.d. sequence of letters into words according to an independent renewal process,... more When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential ...
Transactions of the American Mathematical Society, 1995
Probability Theory and Related Fields, 1991
Lecture Notes in Mathematics, 2013
Electronic Journal of Probability
arXiv (Cornell University), Sep 20, 2022
arXiv (Cornell University), Apr 3, 2019
arXiv (Cornell University), Sep 9, 2012
The idea for this paper arose from discussions with P. Pfaffelhuber and A. Wakolbinger dur-
arXiv: Probability, 2015
We consider a spatial multi-type branching model in which individuals migrate in geographic space... more We consider a spatial multi-type branching model in which individuals migrate in geographic space according to random walks and reproduce according to a state-dependent branching mechanism which can be sub-, super- or critical depending on the local intensity of individuals of the different types. The model is a Lotka-Volterra type model with a spatial component and is related to two models studied in \cite{BlathEtheridgeMeredith2007} as well as to earlier work in \cite{Etheridge2004} and in \cite{NeuhauserPacala1999}. Our main focus is on the diffusion limit of small mass, locally many individuals and rapid reproduction. This system differs from spatial critical branching systems since it is not density preserving and the densities for large times do not depend on the initial distribution but mainly on the carrying capacities. We prove existence of the infinite particle model and the system of interacting diffusions as solutions of martingale problems or systems of stochastic equat...
We consider a system of interacting diffusions labeled by a geographic space that is given by the... more We consider a system of interacting diffusions labeled by a geographic space that is given by the hierarchical group OmegaN\Omega_NOmegaN of order NinmathbbNN\in\mathbb{N}NinmathbbN. Individuals live in colonies and are subject to resampling and migration as long as they are active. Each colony has a seed-bank into which individuals can retreat to become dormant, suspending their resampling and migration until they become active again. The migration kernel has a hierarchical structure: individuals hop between colonies at a rate that depends on the hierarchical distance between the colonies. The seed-bank has a layered structure: when individuals become dormant they acquire a colour that determines the rate at which they become active again. The latter allows us to model seed-banks whose wake-up times have a fat tail. We analyse a system of coupled stochastic differential equations that describes the population in the large-colony-size limit. For fixed NinmathbbNN\in\mathbb{N}NinmathbbN, the system exhibits a dichotomy between c...
Latin American Journal of Probability and Mathematical Statistics, 2021
The paper has four goals. First, we want to generalize the classical concept of the branching pro... more The paper has four goals. First, we want to generalize the classical concept of the branching property so that it becomes applicable for historical and genealogical processes (using the coding of genealogies by ($V$-marked) ultrametric measure spaces leading to state spaces mathbbU\mathbb{U}mathbbU resp. mathbbUV\mathbb{U}^VmathbbUV). The processes are defined by well-posed martingale problems. In particular we want to complement the corresponding concept of infinite divisibility developed in \cite{infdiv} for this context. Second one of the two main points, we want to find a corresponding characterization of the generators of branching processes more precisely their martingale problems which is both easy to apply and general enough to cover a wide range of state spaces. As a third goal we want to obtain the branching property of the mathbbU\mathbb{U}mathbbU-valued Feller diffusion respectively mathbbUV\mathbb{U}^VmathbbUV-valued super random walk and the historical process on countable geographic spaces the latter as two examples of...
arXiv: Probability, 2019
We consider the evolution of the genealogy of the population currently alive in a Feller branchin... more 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 mathbbU\mathbb{U}mathbbU. 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., mathbbU\mathbb{U}mathbbU-valued Feller diffusion. We give the precise relation to the time-inhomogeneous mathbbU_1\mathbb{U}_1mathbbU_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 mathbbU\mathbb{U}mathbbU, called concatenation, together with the...
Electronic Journal of Probability, 2019
Electronic Journal of Probability, 2016
Electronic Journal of Probability, 2016
Journal of Statistical Physics, 1991
Electronic Journal of Probability, 2013
Electronic Communications in Probability, 2011
Canadian Journal of Mathematics, 1995
This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a ... more This paper analyzes the n-fold composition of a certain non-linear integral operator acting on a class of functions on [0,1 ]. The attracting orbit is identified and various properties of convergence to this orbit are derived. The results imply that the space-time scaling limit of a certain infinite system of interacting diffusions has universal behavior independent of model parameters.
When we cut an i.i.d. sequence of letters into words according to an independent renewal process,... more When we cut an i.i.d. sequence of letters into words according to an independent renewal process, we obtain an i.i.d. sequence of words. In the annealed large deviation principle (LDP) for the empirical process of words, the rate function is the specific relative entropy of the observed law of words w.r.t. the reference law of words. In the present paper we consider the quenched LDP, i.e., we condition on a typical letter sequence. We focus on the case where the renewal process has an algebraic tail. The rate function turns out to be a sum of two terms, one being the annealed rate function, the other being proportional to the specific relative entropy of the observed law of letters w.r.t. the reference law of letters, with the former being obtained by concatenating the words and randomising the location of the origin. The proportionality constant equals the tail exponent of the renewal process. Earlier work by Birkner considered the case where the renewal process has an exponential ...
Transactions of the American Mathematical Society, 1995
Probability Theory and Related Fields, 1991
Lecture Notes in Mathematics, 2013