A method to describe unresolved processes in meteorological models by physically based stochastic... more A method to describe unresolved processes in meteorological models by physically based stochastic processes (SP) is proposed by the example of an energy budget model (EBM). Contrary to the common approach using additive white noise, a suitable variable within the model is chosen to be represented by a SP. Spectral analysis of ice core time series shows a red noise character of the underlying fluctuations. Fitting Ornstein Uhlenbeck processes to the observed spectrum defines the parameters for the stochastic dynamic model (SDM). Numerical simulations for different sets of ice core data lead to three sets of strongly differing systems. Pathwise, statistical and spectral analysis of these models show the importance of carefully choosing suitable stochastic terms in order to get a physically meaningful SDM.
We study a two-species interacting particle model on a subset of Z with open boundaries. The two ... more We study a two-species interacting particle model on a subset of Z with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an "order book" on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation. In the literature a slightly different model is often studies where particles of different type annihilate with rate one; this leads to the existence of a "coexistence region" which in our model is excluded.
Schrodinger operators with potentials generated by primitive substitutions are simple models for ... more Schrodinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically verifiable sufficient condition for their spectrum to be singular continuous and supported on a Cantor set of zero Lebesgue measure. Applications to specific examples are discussed. We consider one dimensional Schrodinger operators H defined by H/ n = / n+1 + / nGamma1 + V n / n ; / 2 l 2 (ZZ); (1) where (V n ) n2ZZ is an aperiodic sequence generated by a substitution. A substitution is a map ¸ from a finite alphabet A to the set A of words on A, which can be naturally extended to a map from A to A and then to a map from A IN to A IN . We also define the free group b A , extension of A obtained by addition of the formal inverses of the letters in A as generators. ¸ is said primitive if 9k 2 IN s.t. 8(ff; fi) 2 A 2 , ¸ k (ff) cont...
We consider the random variable $\xi(\beta)= \sum^\infty_{n=0}\varepsilon_n\beta^n$, where the $\... more We consider the random variable $\xi(\beta)= \sum^\infty_{n=0}\varepsilon_n\beta^n$, where the $\varepsilon_n$ are i.i.d. Bernoulli random variables defined on a probability space $(\Omega,\Sigma,P)$, $P(\varepsilon_n=\pm1)=1/2$. If $\beta<1$, this series converges and $\xi(\beta)$ is a well-defined random variable. The probability distribution of this random variable $\xi(\beta)$ has been the object of considerable interest in the last half century. Results by {\it B. Jessen} and {\it A. Wintner} [Trans. Am. Math. Soc. 38, 48-88 (1935; Zbl 0014.15401)], {\it A. Wintner} [Am. J. Math. 57, 821-826 (1935; Zbl 0013.25603) and ibid. 57, 827-838 (1935; Zbl 0013.25701)], {\it R. Kershner} and {\it A. Wintner} [ibid. 57, 541-548 (1935; Zbl 0012.06302)] show that this distribution is always continuous and pure, that is to say either absolutely continuous or singularly continuous but not a mixture of the two. They also show that for $\beta<1/2$ this distribution is always singularly co...
International Journal of Theoretical and Applied Finance
We propose a class of Markovian agent based models for the time evolution of a share price in an ... more We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
We study a model for Darwinian evolution in an asexual population with a large but non-constant p... more We study a model for Darwinian evolution in an asexual population with a large but non-constant populations size. The model incorporates the basic evolutionary mechanisms, namely natural birth, density-depending logistic death due to age and competition and a probability $u$ of mutation at each birth event. The difference between mothers and mutants trait will be random, but of order $\sigma$. In the present paper, we study the long-term behavior of the systems in the limit of large population ($K\to \infty$) size, rare mutations ($u\to 0$), and small mutational effects ($\sigma\to 0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to, e.g. [Champagnat and M\'el\'eard, 2011], we study the three limits simultaneously, i.e. $u=u_K$ and $\sigma=\sigma_K$, tend to zero with $K$, subject to conditions that ensure that the time-scales of birth and death events remains separated from that of successful mutational events. The fact that $\sigm...
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2015
In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (tim... more In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.
Mathematical Aspects of Spin Glasses and Neural Networks, 1996
We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a... more We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of "small α"; a particularly satisfactory result concerns a non-trivial regime of parameters in which we prove 1) the convergence of the local "mean fields" to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) "propagation of chaos", i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the "replica symmetric solution" of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques. #
Genome-wide assessment of protein-DNA interaction by chromatin immunoprecipitation followed by ma... more Genome-wide assessment of protein-DNA interaction by chromatin immunoprecipitation followed by massive parallel sequencing (ChIP-seq) is a key technology for studying transcription factor (TF) localization and regulation of gene expression. Signal-to-noise-ratio and signal specificity in ChIP-seq studies depend on many variables, including antibody affinity and specificity. Thus far, efforts to improve antibody reagents for ChIP-seq experiments have focused mainly on generating higher quality antibodies. Here we introduce KOIN (knockout implemented normalization) as a novel strategy to increase signal specificity and reduce noise by using TF knockout mice as a critical control for ChIP-seq data experiments. Additionally, KOIN can identify 'hyper ChIPable regions' as another source of false-positive signals. As the use of the KOIN algorithm reduces false-positive results and thereby prevents misinterpretation of ChIP-seq data, it should be considered as the gold standard for ...
We study a class of Markov chains that describe reversible stochastic dynamics of a large class o... more We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of "admissible transitions". For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
We study the large deviation behaviour of S n = n j=1 W j Z j , where (W j ) j∈N and (Z j ) j∈N a... more We study the large deviation behaviour of S n = n j=1 W j Z j , where (W j ) j∈N and (Z j ) j∈N are sequences of real-valued, independent and identically distributed random variables satisfying certain moment conditions, independent of each other. More precisely, we prove a conditional strong large deviation result and describe the fluctuations of the random rate function through a functional central limit theorem.
We study a Hopfield model whose number of patterns M grows to infinity with the system size N, in... more We study a Hopfield model whose number of patterns M grows to infinity with the system size N, in such a way that M N 2 log M N /N tends to zero. In this model the unbiased Gibbs state in volume N can essentially be decomposed into M N pairs of disjoint measures. We investigate the distributions of the corresponding weights, and show, in particular, that these weights concentrate for any given N very closely to one of the pairs, with probability tending to 1. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.
A method to describe unresolved processes in meteorological models by physically based stochastic... more A method to describe unresolved processes in meteorological models by physically based stochastic processes (SP) is proposed by the example of an energy budget model (EBM). Contrary to the common approach using additive white noise, a suitable variable within the model is chosen to be represented by a SP. Spectral analysis of ice core time series shows a red noise character of the underlying fluctuations. Fitting Ornstein Uhlenbeck processes to the observed spectrum defines the parameters for the stochastic dynamic model (SDM). Numerical simulations for different sets of ice core data lead to three sets of strongly differing systems. Pathwise, statistical and spectral analysis of these models show the importance of carefully choosing suitable stochastic terms in order to get a physically meaningful SDM.
We study a two-species interacting particle model on a subset of Z with open boundaries. The two ... more We study a two-species interacting particle model on a subset of Z with open boundaries. The two species are injected with time dependent rate on the left, resp. right boundary. Particles of different species annihilate when they try to occupy the same site. This model has been proposed as a simple model for the dynamics of an "order book" on a stock market. We consider the hydrodynamic scaling limit for the empirical process and prove a large deviation principle that implies convergence to the solution of a non-linear parabolic equation. In the literature a slightly different model is often studies where particles of different type annihilate with rate one; this leads to the existence of a "coexistence region" which in our model is excluded.
Schrodinger operators with potentials generated by primitive substitutions are simple models for ... more Schrodinger operators with potentials generated by primitive substitutions are simple models for one dimensional quasi-crystals. We review recent results on their spectral properties. These include in particular an algorithmically verifiable sufficient condition for their spectrum to be singular continuous and supported on a Cantor set of zero Lebesgue measure. Applications to specific examples are discussed. We consider one dimensional Schrodinger operators H defined by H/ n = / n+1 + / nGamma1 + V n / n ; / 2 l 2 (ZZ); (1) where (V n ) n2ZZ is an aperiodic sequence generated by a substitution. A substitution is a map ¸ from a finite alphabet A to the set A of words on A, which can be naturally extended to a map from A to A and then to a map from A IN to A IN . We also define the free group b A , extension of A obtained by addition of the formal inverses of the letters in A as generators. ¸ is said primitive if 9k 2 IN s.t. 8(ff; fi) 2 A 2 , ¸ k (ff) cont...
We consider the random variable $\xi(\beta)= \sum^\infty_{n=0}\varepsilon_n\beta^n$, where the $\... more We consider the random variable $\xi(\beta)= \sum^\infty_{n=0}\varepsilon_n\beta^n$, where the $\varepsilon_n$ are i.i.d. Bernoulli random variables defined on a probability space $(\Omega,\Sigma,P)$, $P(\varepsilon_n=\pm1)=1/2$. If $\beta<1$, this series converges and $\xi(\beta)$ is a well-defined random variable. The probability distribution of this random variable $\xi(\beta)$ has been the object of considerable interest in the last half century. Results by {\it B. Jessen} and {\it A. Wintner} [Trans. Am. Math. Soc. 38, 48-88 (1935; Zbl 0014.15401)], {\it A. Wintner} [Am. J. Math. 57, 821-826 (1935; Zbl 0013.25603) and ibid. 57, 827-838 (1935; Zbl 0013.25701)], {\it R. Kershner} and {\it A. Wintner} [ibid. 57, 541-548 (1935; Zbl 0012.06302)] show that this distribution is always continuous and pure, that is to say either absolutely continuous or singularly continuous but not a mixture of the two. They also show that for $\beta<1/2$ this distribution is always singularly co...
International Journal of Theoretical and Applied Finance
We propose a class of Markovian agent based models for the time evolution of a share price in an ... more We propose a class of Markovian agent based models for the time evolution of a share price in an interactive market. The models rely on a microscopic description of a market of buyers and sellers who change their opinion about the stock value in a stochastic way. The actual price is determined in realistic way by matching (clearing) offers until no further transactions can be performed. Some analytic results for a non-interacting model are presented. We also propose basic interaction mechanisms and show in simulations that these already reproduce certain particular features of prices in real stock markets.
We study a model for Darwinian evolution in an asexual population with a large but non-constant p... more We study a model for Darwinian evolution in an asexual population with a large but non-constant populations size. The model incorporates the basic evolutionary mechanisms, namely natural birth, density-depending logistic death due to age and competition and a probability $u$ of mutation at each birth event. The difference between mothers and mutants trait will be random, but of order $\sigma$. In the present paper, we study the long-term behavior of the systems in the limit of large population ($K\to \infty$) size, rare mutations ($u\to 0$), and small mutational effects ($\sigma\to 0$), proving convergence to the canonical equation of adaptive dynamics (CEAD). In contrast to, e.g. [Champagnat and M\'el\'eard, 2011], we study the three limits simultaneously, i.e. $u=u_K$ and $\sigma=\sigma_K$, tend to zero with $K$, subject to conditions that ensure that the time-scales of birth and death events remains separated from that of successful mutational events. The fact that $\sigm...
Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2015
In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (tim... more In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process.
Mathematical Aspects of Spin Glasses and Neural Networks, 1996
We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a... more We give a comprehensive self-contained review on the rigorous analysis of the thermodynamics of a class of random spin systems of mean field type whose most prominent example is the Hopfield model. We focus on the low temperature phase and the analysis of the Gibbs measures with large deviation techniques. There is a very detailed and complete picture in the regime of "small α"; a particularly satisfactory result concerns a non-trivial regime of parameters in which we prove 1) the convergence of the local "mean fields" to gaussian random variables with constant variance and random mean; the random means are from site to site independent gaussians themselves; 2) "propagation of chaos", i.e. factorization of the extremal infinite volume Gibbs measures, and 3) the correctness of the "replica symmetric solution" of Amit, Gutfreund and Sompolinsky [AGS]. This last result was first proven by M. Talagrand [T4], using different techniques. #
Genome-wide assessment of protein-DNA interaction by chromatin immunoprecipitation followed by ma... more Genome-wide assessment of protein-DNA interaction by chromatin immunoprecipitation followed by massive parallel sequencing (ChIP-seq) is a key technology for studying transcription factor (TF) localization and regulation of gene expression. Signal-to-noise-ratio and signal specificity in ChIP-seq studies depend on many variables, including antibody affinity and specificity. Thus far, efforts to improve antibody reagents for ChIP-seq experiments have focused mainly on generating higher quality antibodies. Here we introduce KOIN (knockout implemented normalization) as a novel strategy to increase signal specificity and reduce noise by using TF knockout mice as a critical control for ChIP-seq data experiments. Additionally, KOIN can identify 'hyper ChIPable regions' as another source of false-positive signals. As the use of the KOIN algorithm reduces false-positive results and thereby prevents misinterpretation of ChIP-seq data, it should be considered as the gold standard for ...
We study a class of Markov chains that describe reversible stochastic dynamics of a large class o... more We study a class of Markov chains that describe reversible stochastic dynamics of a large class of disordered mean field models at low temperatures. Our main purpose is to give a precise relation between the metastable time scales in the problem to the properties of the rate functions of the corresponding Gibbs measures. We derive the analog of the Wentzell-Freidlin theory in this case, showing that any transition can be decomposed, with probability exponentially close to one, into a deterministic sequence of "admissible transitions". For these admissible transitions we give upper and lower bounds on the expected transition times that differ only by a constant factor. The distribution rescaled transition times are shown to converge to the exponential distribution. We exemplify our results in the context of the random field Curie-Weiss model.
We study the large deviation behaviour of S n = n j=1 W j Z j , where (W j ) j∈N and (Z j ) j∈N a... more We study the large deviation behaviour of S n = n j=1 W j Z j , where (W j ) j∈N and (Z j ) j∈N are sequences of real-valued, independent and identically distributed random variables satisfying certain moment conditions, independent of each other. More precisely, we prove a conditional strong large deviation result and describe the fluctuations of the random rate function through a functional central limit theorem.
We study a Hopfield model whose number of patterns M grows to infinity with the system size N, in... more We study a Hopfield model whose number of patterns M grows to infinity with the system size N, in such a way that M N 2 log M N /N tends to zero. In this model the unbiased Gibbs state in volume N can essentially be decomposed into M N pairs of disjoint measures. We investigate the distributions of the corresponding weights, and show, in particular, that these weights concentrate for any given N very closely to one of the pairs, with probability tending to 1. Our analysis is based upon a new result on the asymptotic distribution of order statistics of certain correlated exchangeable random variables.
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Papers by Anton Bovier