In this paper we study the self-intersection of paths solving elliptic stochastic differential equations driven by fractional Brownian motion. We show that such a path has no self-intersection – except for paths forming a set of zero (r, q)-capacity in the sample space – provided the dimension d of the space and the Hurst parameter H satisfy the inequality d > rq + 2/H. This inequality is sharp in the case of brownian motion and fractional brownian motion according to existing results. Various results exist for the critical case where d = rq + 4 for Brownian motion.
{"title":"Quasi-sure non-self-intersection for rough differential equations driven by fractional Brownian motion","authors":"O. Cheng, William Roberson-Vickery","doi":"10.1214/22-ecp454","DOIUrl":"https://doi.org/10.1214/22-ecp454","url":null,"abstract":"In this paper we study the self-intersection of paths solving elliptic stochastic differential equations driven by fractional Brownian motion. We show that such a path has no self-intersection – except for paths forming a set of zero (r, q)-capacity in the sample space – provided the dimension d of the space and the Hurst parameter H satisfy the inequality d > rq + 2/H. This inequality is sharp in the case of brownian motion and fractional brownian motion according to existing results. Various results exist for the critical case where d = rq + 4 for Brownian motion.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43809384","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}
González Cázares and Ivanovs (2021) suggested a new method for “recovering” the Brownian motion component from the trajectory of a Lévy process that required sampling from an independent Brownian motion process. We show that such a procedure works equally well without any additional source of randomness if one uses normal quantiles instead of the ordered increments of the auxiliary Brownian motion process.
{"title":"A note on recovering the Brownian motion component from a Lévy process","authors":"K. Borovkov","doi":"10.1214/22-ecp477","DOIUrl":"https://doi.org/10.1214/22-ecp477","url":null,"abstract":"González Cázares and Ivanovs (2021) suggested a new method for “recovering” the Brownian motion component from the trajectory of a Lévy process that required sampling from an independent Brownian motion process. We show that such a procedure works equally well without any additional source of randomness if one uses normal quantiles instead of the ordered increments of the auxiliary Brownian motion process.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46442075","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 weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in [7].
{"title":"The weak functional representation of historical martingales","authors":"C. Mandler, L. Overbeck","doi":"10.1214/22-ecp492","DOIUrl":"https://doi.org/10.1214/22-ecp492","url":null,"abstract":"A weak extension of the Dupire derivative is derived, which turns out to be the adjoint operator of the integral with respect to the martingale measure associated with the historical Brownian motion a benchmark example of a measure valued process. This extension yields the explicit form of the martingale representation of historical functionals, which we compare to a classical result on the representation of historical functionals derived in [7].","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44525313","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}
E. Bernstein, Clare Hamblen, M. Junge, Lily Reeves
Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-$lambda$ Poisson process while being chased and consumed by blue particles according to a rate-$1$ Poisson process. Given a growing sequence of finite graphs, the critical rate $lambda_c$ is the largest value of $lambda$ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on $lambda_c$ for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.
{"title":"Chase-escape on the configuration model","authors":"E. Bernstein, Clare Hamblen, M. Junge, Lily Reeves","doi":"10.1214/22-ecp470","DOIUrl":"https://doi.org/10.1214/22-ecp470","url":null,"abstract":"Chase-escape is a competitive growth process in which red particles spread to adjacent empty sites according to a rate-$lambda$ Poisson process while being chased and consumed by blue particles according to a rate-$1$ Poisson process. Given a growing sequence of finite graphs, the critical rate $lambda_c$ is the largest value of $lambda$ for which red fails to reach a positive fraction of the vertices with high probability. We provide a conjecturally sharp lower bound and an implicit upper bound on $lambda_c$ for supercritical random graphs sampled from the configuration model with independent and identically distributed degrees with finite second moment. We additionally show that the expected number of sites occupied by red undergoes a phase transition and identify the location of this transition.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42606919","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}
Reinforced processes are known to provide a stochastic representation for the quasi-stationary distribution of a given killed Markov process – describing the killed Markov process at fixed time instants. In this paper we shall adapt the construction to provide a pathwise description. We also obtain a stochastic approximation for the quasi-limiting distributions of reducible killed Markov processes as a corollary.
{"title":"Stochastic approximation of the paths of killed Markov processes conditioned on survival","authors":"Oliver Tough","doi":"10.1214/22-ecp475","DOIUrl":"https://doi.org/10.1214/22-ecp475","url":null,"abstract":"Reinforced processes are known to provide a stochastic representation for the quasi-stationary distribution of a given killed Markov process – describing the killed Markov process at fixed time instants. In this paper we shall adapt the construction to provide a pathwise description. We also obtain a stochastic approximation for the quasi-limiting distributions of reducible killed Markov processes as a corollary.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41932513","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}
Occupancy processes are a broad class of discrete time Markov chains on ${0,1}^{n}$ encompassing models from diverse areas. This model is compared to a collection of $n$ independent Markov chains on ${0,1}$, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.}
{"title":"Dominating occupancy processes by the independent site approximation","authors":"R. McVinish","doi":"10.1214/22-ecp499","DOIUrl":"https://doi.org/10.1214/22-ecp499","url":null,"abstract":"Occupancy processes are a broad class of discrete time Markov chains on ${0,1}^{n}$ encompassing models from diverse areas. This model is compared to a collection of $n$ independent Markov chains on ${0,1}$, which we call the independent site model. We establish conditions under which an occupancy process is smaller in the lower orthant order than the independent site model. An analogous result for spin systems follows by a limiting argument.}","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46252478","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 prove the weak convergence, in a high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions under a higher temperature condition. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.
{"title":"Marginals of a spherical spin glass model with correlated disorder","authors":"Jean Barbier, M. S'aenz","doi":"10.1214/22-ecp489","DOIUrl":"https://doi.org/10.1214/22-ecp489","url":null,"abstract":"In this paper we prove the weak convergence, in a high-temperature phase, of the finite marginals of the Gibbs measure associated to a symmetric spherical spin glass model with correlated couplings towards an explicit asymptotic decoupled measure. We also provide upper bounds for the rate of convergence in terms of the one of the energy per variable. Furthermore, we establish a concentration inequality for bounded functions under a higher temperature condition. These results are exemplified by analysing the asymptotic behaviour of the empirical mean of coordinate-wise functions of samples from the Gibbs measure of the model.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45623919","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 consider the elephant random walk with general step distribution. We cal-culate the first four moments of the limiting distribution of the position rescaled by n α in the superdiffusive regime where α is the memory parameter. This extends the results obtained by Bercu in [Ber17].
{"title":"Moments of the superdiffusive elephant random walk with general step distribution","authors":"J. Kiss, B'alint VetHo","doi":"10.1214/22-ecp485","DOIUrl":"https://doi.org/10.1214/22-ecp485","url":null,"abstract":"We consider the elephant random walk with general step distribution. We cal-culate the first four moments of the limiting distribution of the position rescaled by n α in the superdiffusive regime where α is the memory parameter. This extends the results obtained by Bercu in [Ber17].","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46680065","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}
Gideon Amir, Omer Angel, Rangel Baldasso, R. Peretz
We study how the consensus opinion of the voter model on finite graphs varies in light of noise sensitivity with respect to both the initial opinions and the dynamics. We first prove that the final opinion is stable with respect to small perturbations of the initial configuration. Different effects are observed when a perturbation is introduced in the dynamics governing the evolution of the process, and the final opinion is noise sensitive in this case. Our proofs rely on the relationship between the voter model and coalescing random walks, and on a precise description of this evolution when we have coupled dynamics.
{"title":"Dynamical noise sensitivity for the voter model","authors":"Gideon Amir, Omer Angel, Rangel Baldasso, R. Peretz","doi":"10.1214/22-ECP483","DOIUrl":"https://doi.org/10.1214/22-ECP483","url":null,"abstract":"We study how the consensus opinion of the voter model on finite graphs varies in light of noise sensitivity with respect to both the initial opinions and the dynamics. We first prove that the final opinion is stable with respect to small perturbations of the initial configuration. Different effects are observed when a perturbation is introduced in the dynamics governing the evolution of the process, and the final opinion is noise sensitive in this case. Our proofs rely on the relationship between the voter model and coalescing random walks, and on a precise description of this evolution when we have coupled dynamics.","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47034968","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 investigate eigenvector statistics of the Truncated Unitary ensemble TUE( N, M ) in the weakly non-unitary case M = 1 , that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when M ≥ N , suggesting possible extensions to TUE( N, M ) for all values of M .
{"title":"Explicit formulas concerning eigenvectors of weakly non-unitary matrices","authors":"Guillaume Dubach","doi":"10.1214/22-ECP507","DOIUrl":"https://doi.org/10.1214/22-ECP507","url":null,"abstract":". We investigate eigenvector statistics of the Truncated Unitary ensemble TUE( N, M ) in the weakly non-unitary case M = 1 , that is when only one row and column are removed. We provide an explicit description of generalized overlaps as deterministic functions of the eigenvalues, as well as a method to derive an exact formula for the expectation of diagonal overlaps (or squared eigenvalue condition numbers), conditionally on one eigenvalue. This complements recent results obtained in the opposite regime when M ≥ N , suggesting possible extensions to TUE( N, M ) for all values of M .","PeriodicalId":50543,"journal":{"name":"Electronic Communications in Probability","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2021-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49345961","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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