We consider two jointly stationary and ergodic random measures $xi$ and $eta$ on $mathbb{R}^d$ with equal finite intensities, assuming $xi$ to be diffuse. An allocation is a random mapping taking $mathbb{R}^d$ to $mathbb{R}^dcup{infty}$ in a translation invariant way. We construct allocations transporting the diffuse $xi$ to arbitrary $eta$, under the mild condition of existence of an `auxiliary' point process which is needed only in the case when $eta$ is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting $xi$ to $eta$ need not exist.
{"title":"Transportation of diffuse random measures on Rd","authors":"G. Last, H. Thorisson","doi":"10.30757/alea.v20-21","DOIUrl":"https://doi.org/10.30757/alea.v20-21","url":null,"abstract":"We consider two jointly stationary and ergodic random measures $xi$ and $eta$ on $mathbb{R}^d$ with equal finite intensities, assuming $xi$ to be diffuse. An allocation is a random mapping taking $mathbb{R}^d$ to $mathbb{R}^dcup{infty}$ in a translation invariant way. We construct allocations transporting the diffuse $xi$ to arbitrary $eta$, under the mild condition of existence of an `auxiliary' point process which is needed only in the case when $eta$ is diffuse. When that condition does not hold we show by a counterexample that an allocation transporting $xi$ to $eta$ need not exist.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42419635","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. Suppose X is a Markov process on the real line (or some interval). Do the distributions of its first passage times downwards (fptd) determine its law? In this paper we treat some special cases of this question. We prove that if the fptd process has the law of a subordinator, then necessarily X is a L´evy process with no negative jumps; specifying the law of the subordinator determines the law of X uniquely. We further show that, likewise, the classes of continuous-state branching processes and of self-similar processes without negative jumps are also respectively characterised by a certain structure of their fptd distributions; and each member of these classes separately is determined uniquely by the precise family of its fptd laws. The road to these results is paved by (i) the identification of Markov processes without negative jumps in terms of the nature of their fptd laws, and (ii) some general results concerning the identification of the fptd distributions for such processes.
{"title":"Some characterizations for Markov processes at first passage","authors":"M. Vidmar","doi":"10.30757/alea.v19-63","DOIUrl":"https://doi.org/10.30757/alea.v19-63","url":null,"abstract":". Suppose X is a Markov process on the real line (or some interval). Do the distributions of its first passage times downwards (fptd) determine its law? In this paper we treat some special cases of this question. We prove that if the fptd process has the law of a subordinator, then necessarily X is a L´evy process with no negative jumps; specifying the law of the subordinator determines the law of X uniquely. We further show that, likewise, the classes of continuous-state branching processes and of self-similar processes without negative jumps are also respectively characterised by a certain structure of their fptd distributions; and each member of these classes separately is determined uniquely by the precise family of its fptd laws. The road to these results is paved by (i) the identification of Markov processes without negative jumps in terms of the nature of their fptd laws, and (ii) some general results concerning the identification of the fptd distributions for such processes.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47912758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we investigate pointed $(mathbf{q}, g, n)$-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using the peeling exploration of arXiv:1809.02012 modified to this setting we show that its law in the non-generic critical phase can be coded in terms of a random walk confined to the positive integers by a new specific boundary condition. Under a technical assumption that we believe to be true, combining this observation with explicit quantities for the peeling law we derive the large deviations property for the distribution of the so-called nesting statistic and show that the exploration process possesses exactly the same scaling limit as in the rigid loop model on bipartite maps that is a specific self-similar Markov process introduced in arXiv:1809.02012. Besides, we conclude the equivalence of the admissible weight sequences related by the so-called fixed point equation by proving the missing direction in the argument of arXiv:1202.5521.
{"title":"The exploration process of critical Boltzmann planar maps decorated by a triangular O(n) loop model","authors":"Aleksandra Korzhenkova","doi":"10.30757/alea.v19-58","DOIUrl":"https://doi.org/10.30757/alea.v19-58","url":null,"abstract":"In this paper we investigate pointed $(mathbf{q}, g, n)$-Boltzmann loop-decorated maps with loops traversing only inner triangular faces. Using the peeling exploration of arXiv:1809.02012 modified to this setting we show that its law in the non-generic critical phase can be coded in terms of a random walk confined to the positive integers by a new specific boundary condition. Under a technical assumption that we believe to be true, combining this observation with explicit quantities for the peeling law we derive the large deviations property for the distribution of the so-called nesting statistic and show that the exploration process possesses exactly the same scaling limit as in the rigid loop model on bipartite maps that is a specific self-similar Markov process introduced in arXiv:1809.02012. Besides, we conclude the equivalence of the admissible weight sequences related by the so-called fixed point equation by proving the missing direction in the argument of arXiv:1202.5521.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47731585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A Markov process is constructed in an explicit way for an infinite system of entities arriving in and departing from a habitat X, which is a locally compact Polish space with a positive Radon measure χ. Along with its location x ∈ X, each particle is characterized by age α ≥ 0 – time since arriving. As the state space one takes the set of marked configurations Γ̂, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator L, expressed through the measure χ and a departure rate m(x, α) ≥ 0, and acting on bounded continuous functions F : Γ̂→ R. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.
{"title":"A Markov process for an infinite age-structured population","authors":"Dominika Jasińska, Y. Kozitsky","doi":"10.30757/alea.v19-18","DOIUrl":"https://doi.org/10.30757/alea.v19-18","url":null,"abstract":"A Markov process is constructed in an explicit way for an infinite system of entities arriving in and departing from a habitat X, which is a locally compact Polish space with a positive Radon measure χ. Along with its location x ∈ X, each particle is characterized by age α ≥ 0 – time since arriving. As the state space one takes the set of marked configurations Γ̂, equipped with a metric that makes it a complete and separable metric space. The stochastic evolution of the system is described by a Kolmogorov operator L, expressed through the measure χ and a departure rate m(x, α) ≥ 0, and acting on bounded continuous functions F : Γ̂→ R. For this operator, we pose the martingale problem and show that it has a unique solution, explicitly constructed in the paper. We also prove that the corresponding process has a unique stationary state and is temporarily egrodic if the rate of departure is separated away from zero.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46301197","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distribution. We show that after a sufficiently long time, the fitness distribution from the lowest to the highest fitness levels approximately evolves as a traveling wave with a profile which is asymptotically related the the Airy function. Our work complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.
{"title":"Particle configurations for branching Brownian motion with an inhomogeneous branching rate","authors":"Jiaqi Liu, Jason Schweinsberg","doi":"10.30757/alea.v20-28","DOIUrl":"https://doi.org/10.30757/alea.v20-28","url":null,"abstract":"Aiming to understand the distribution of fitness levels of individuals in a large population undergoing selection, we study the particle configurations of branching Brownian motion where each particle independently moves as Brownian motion with negative drift, particles can die or undergo dyadic fission, and the difference between the birth rate and the death rate is proportional to the particle's location. Under some assumptions, we obtain the limit in probability of the number of particles in any given interval and an explicit formula for the asymptotic empirical density of the fitness distribution. We show that after a sufficiently long time, the fitness distribution from the lowest to the highest fitness levels approximately evolves as a traveling wave with a profile which is asymptotically related the the Airy function. Our work complements the results in Roberts and Schweinsberg (2021), giving a fuller picture of the fitness distribution.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46527834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we study the stochastic wave equation on the entire space $mathbb{R}^d$, driven by a space-time L'evy white noise with possibly infinite variance (such as the $alpha$-stable L'evy noise). In this equation, the noise is multiplied by a Lipschitz function $sigma(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the L'evy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a c`adl`ag modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.
在本文中，我们研究了整个空间$mathbb{R}^d$上的随机波动方程，该方程是由一个可能具有无限方差的时空l郁闷白噪声(如$alpha$ -稳定l郁闷噪声)驱动的。在这个方程中，噪声乘以解的Lipschitz函数$sigma(u)$。我们假设空间维度为$d=1$或$d=2$。在噪声的lsamvy测度的一般条件下，证明了该解的存在性，并证明了该解作为一个函数值过程，在$r<1/4$ if $d=1$阶的局部分数Sobolev空间中具有càdlàg修正，分别为$r<-1$ if $d=2$。
{"title":"Stochastic wave equation with Lévy white noise","authors":"R. Balan","doi":"10.30757/ALEA.v20-16","DOIUrl":"https://doi.org/10.30757/ALEA.v20-16","url":null,"abstract":"In this article, we study the stochastic wave equation on the entire space $mathbb{R}^d$, driven by a space-time L'evy white noise with possibly infinite variance (such as the $alpha$-stable L'evy noise). In this equation, the noise is multiplied by a Lipschitz function $sigma(u)$ of the solution. We assume that the spatial dimension is $d=1$ or $d=2$. Under general conditions on the L'evy measure of the noise, we prove the existence of the solution, and we show that, as a function-valued process, the solution has a c`adl`ag modification in the local fractional Sobolev space of order $r<1/4$ if $d=1$, respectively $r<-1$ if $d=2$.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46547892","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We prove that a system of locally interacting diffusions carrying discrete masses, subject to an environmental noise and undergoing mass coagulation, converges to a system of Stochastic Partial Differential Equations (SPDEs) with Smoluchowski-type nonlinearity. Existence, uniqueness and regularity of the SPDEs are also proven.
{"title":"Coagulation dynamics under environmental noise: Scaling limit to SPDE","authors":"F. Flandoli, Ruojun Huang","doi":"10.30757/alea.v19-51","DOIUrl":"https://doi.org/10.30757/alea.v19-51","url":null,"abstract":"We prove that a system of locally interacting diffusions carrying discrete masses, subject to an environmental noise and undergoing mass coagulation, converges to a system of Stochastic Partial Differential Equations (SPDEs) with Smoluchowski-type nonlinearity. Existence, uniqueness and regularity of the SPDEs are also proven.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45830033","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We study the non-linear supersymmetric hyperbolic sigma model H 2 | 2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2 | 2 -model; its size is logarithmic in the size of the original model. The model deals with spin variables taking values in the H with two Grassmann components over the hyperboloid , 0 . The model has supersymmetries, which extend the generators of the Lorentz group acting on H 2 . In Disertori et al. (2010), Disertori, Spencer, and Zirnbauer examine this model over boxes in Z d , d ≥ 3 . For sufficiently small temperature, they derive moment estimates and conclude that the spins are aligned with high probability. For high temperature in any dimension d , Disertori and Spencer (2010) show exponential decay
{"title":"The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions","authors":"M. Disertori, F. Merkl, S. Rolles","doi":"10.30757/alea.v19-62","DOIUrl":"https://doi.org/10.30757/alea.v19-62","url":null,"abstract":". We study the non-linear supersymmetric hyperbolic sigma model H 2 | 2 on a complete graph with hierarchical interactions. For interactions which do not decrease too fast in the hierarchical distance, we prove tightness of certain spin variables in horospherical coordinates, uniformly in the pinning and in the size of the graph. The proof relies on a reduction to an effective H 2 | 2 -model; its size is logarithmic in the size of the original model. The model deals with spin variables taking values in the H with two Grassmann components over the hyperboloid , 0 . The model has supersymmetries, which extend the generators of the Lorentz group acting on H 2 . In Disertori et al. (2010), Disertori, Spencer, and Zirnbauer examine this model over boxes in Z d , d ≥ 3 . For sufficiently small temperature, they derive moment estimates and conclude that the spins are aligned with high probability. For high temperature in any dimension d , Disertori and Spencer (2010) show exponential decay","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46904248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Louis H. Y. Chen, Arturo Jaramillo, Xiaochuan Yang
We present an assessment of the distance in total variation of textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]={1,dots, n}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and collection of"small primes". As applications, we show a generalized version of the celebrated Erd"os Kac theorem for not necessarily uniform samples of numbers.
{"title":"A generalized Kubilius-Barban-Vinogradov bound for prime multiplicities","authors":"Louis H. Y. Chen, Arturo Jaramillo, Xiaochuan Yang","doi":"10.30757/alea.v20-27","DOIUrl":"https://doi.org/10.30757/alea.v20-27","url":null,"abstract":"We present an assessment of the distance in total variation of textit{arbitrary} collection of prime factor multiplicities of a random number in $[n]={1,dots, n}$ and a collection of independent geometric random variables. More precisely, we impose mild conditions on the probability law of the random sample and the aforementioned collection of prime multiplicities, for which a fast decaying bound on the distance towards a tuple of geometric variables holds. Our results generalize and complement those from Kubilius et al. which consider the particular case of uniform samples in $[n]$ and collection of\"small primes\". As applications, we show a generalized version of the celebrated Erd\"os Kac theorem for not necessarily uniform samples of numbers.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43744545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. In this paper we investigate the local limit theorem for additive functionals of nonstationary Markov chains that converge in distribution. We consider both the lattice and the non-lattice cases. The results are also new in the stationary setting and lead to local limit theorems linked to convergence to stable distributions. The conditions are imposed to individual summands and are expressed in terms of lower psi-mixing coefficients.
{"title":"On the local limit theorems for lower psi-mixing Markov chains","authors":"F. Merlevède, M. Peligrad, C. Peligrad","doi":"10.30757/alea.v19-45","DOIUrl":"https://doi.org/10.30757/alea.v19-45","url":null,"abstract":". In this paper we investigate the local limit theorem for additive functionals of nonstationary Markov chains that converge in distribution. We consider both the lattice and the non-lattice cases. The results are also new in the stationary setting and lead to local limit theorems linked to convergence to stable distributions. The conditions are imposed to individual summands and are expressed in terms of lower psi-mixing coefficients.","PeriodicalId":49244,"journal":{"name":"Alea-Latin American Journal of Probability and Mathematical Statistics","volume":" ","pages":""},"PeriodicalIF":0.7,"publicationDate":"2021-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43686211","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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