Abstract We provide a connection between Brownian motion and a classical Newton mechanical system in dimension d ≥ 3 {dgeq 3} . This paper is an extension of [S. Liang, A mechanical model of Brownian motion for one massive particle including slow light particles, J. Stat. Phys. 170 2018, 2, 286–350]. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random Newton mechanical principles, via interaction potentials, without any assumption requiring that the initial energies of the environmental particles should be restricted to be “high enough”. We prove that, as in the high-dimensional case, the position/velocity process of the massive particle converges to a diffusion process when the mass of the environmental particles converges to 0, while the density and the velocities of them go to infinity.
{"title":"A mechanical model of Brownian motion for one massive particle including low energy light particles in dimension d ≥ 3","authors":"Song Liang","doi":"10.1515/rose-2021-2062","DOIUrl":"https://doi.org/10.1515/rose-2021-2062","url":null,"abstract":"Abstract We provide a connection between Brownian motion and a classical Newton mechanical system in dimension d ≥ 3 {dgeq 3} . This paper is an extension of [S. Liang, A mechanical model of Brownian motion for one massive particle including slow light particles, J. Stat. Phys. 170 2018, 2, 286–350]. Precisely, we consider a system of one massive particle interacting with an ideal gas, evolved according to non-random Newton mechanical principles, via interaction potentials, without any assumption requiring that the initial energies of the environmental particles should be restricted to be “high enough”. We prove that, as in the high-dimensional case, the position/velocity process of the massive particle converges to a diffusion process when the mass of the environmental particles converges to 0, while the density and the velocities of them go to infinity.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2062","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44008223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {Hin(frac{1}{2},1)} in a Hilbert space.
{"title":"Global attracting sets of neutral stochastic functional integro-differential equations driven by a fractional Brownian motion","authors":"A. Bakka, S. Hajji, D. Kiouach","doi":"10.1515/rose-2021-2058","DOIUrl":"https://doi.org/10.1515/rose-2021-2058","url":null,"abstract":"Abstract By means of the Banach fixed point principle, we establish some sufficient conditions ensuring the existence of the global attracting sets of neutral stochastic functional integrodifferential equations with finite delay driven by a fractional Brownian motion (fBm) with Hurst parameter H ∈ ( 1 2 , 1 ) {Hin(frac{1}{2},1)} in a Hilbert space.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2058","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42173364","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.
摘要:对n。V。利用可测函数的L d {L_{d}}范数求解随机积分分布的Krylov方法是众所周知的，并广泛应用于随机微分方程理论和受控扩散过程。我们对可选半鞅推广了这类估计，然后应用这些估计证明了Sobolev空间W d 2 {W^{2}_{d}}中一类一般函数的变量变换公式。我们还展示了如何使用这些估计来研究可选SDE解的l2 {L^{2}}收敛性。
{"title":"On Krylov’s estimates for optional semimartingales","authors":"M. Abdelghani, A. Melnikov, A. Pak","doi":"10.1515/rose-2021-2059","DOIUrl":"https://doi.org/10.1515/rose-2021-2059","url":null,"abstract":"Abstract The estimates of N. V. Krylov for distributions of stochastic integrals by means of the L d {L_{d}} -norm of a measurable function are well-known and are widely used in the theory of stochastic differential equations and controlled diffusion processes. We generalize estimates of this type for optional semimartingales, then apply these estimates to prove the change of variables formula for a general class of functions from the Sobolev space W d 2 {W^{2}_{d}} . We also show how to use these estimates for the investigation of L 2 {L^{2}} -convergence of solutions of optional SDE’s.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2059","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42447998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.
{"title":"Reflected generalized BSDEs with discontinuous barriers driven by a Lévy process","authors":"M. El Otmani","doi":"10.1515/rose-2021-2060","DOIUrl":"https://doi.org/10.1515/rose-2021-2060","url":null,"abstract":"Abstract This article deals with the reflected and doubly reflected generalized backward stochastic differential equations when the noise is given by Brownian motion and Teugels martingales associated with an independent pure jump Lévy process. We prove the existence and the uniqueness of the solution for these equations with monotone generators and right continuous left limited obstacles.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44478536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov–Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class of processes is provided to point that such estimates considering several transition steps may be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.
{"title":"A new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains","authors":"A. Shchegolev","doi":"10.1515/rose-2022-2084","DOIUrl":"https://doi.org/10.1515/rose-2022-2084","url":null,"abstract":"Abstract In the paper, we study a new rate of convergence estimate for homogeneous discrete-time nonlinear Markov chains based on the Markov–Dobrushin condition. This result generalizes the convergence estimates for any positive number of transition steps. An example of a class of processes is provided to point that such estimates considering several transition steps may be applicable when one transition can not guarantee any convergence. Moreover, a better estimate can be obtained for a higher number of transitions steps. A law of large numbers is presented for a class of ergodic nonlinear Markov chains with finite state space that may serve as a basis for nonparametric estimation and other statistics.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44232847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract A one-dimensional Fisher–Wright diffusion process on the interval ( 0 , 1 ) {(0,1)} with mutations is considered. This is a widely known model in population genetics. The goal of this paper is an exponential recurrence of the process, which also implies an exponential rate of convergence towards the invariant measure.
{"title":"On recurrent properties of Fisher--Wright's diffusion on (0,1) with mutation","authors":"Roman Sineokiy, A. Veretennikov","doi":"10.1515/rose-2021-2061","DOIUrl":"https://doi.org/10.1515/rose-2021-2061","url":null,"abstract":"Abstract A one-dimensional Fisher–Wright diffusion process on the interval ( 0 , 1 ) {(0,1)} with mutations is considered. This is a widely known model in population genetics. The goal of this paper is an exponential recurrence of the process, which also implies an exponential rate of convergence towards the invariant measure.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45686856","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The G-pencil law under the G-Lindeberg condition for a random matrix is proven.
摘要证明了随机矩阵在G-Lindeberg条件下的G-铅笔定律。
{"title":"VICTORIA transform, RESPECT and REFORM methods for the proof of the G-permanent pencil law under G-Lindeberg condition for some random matrices from G-elliptic ensemble","authors":"V. Girko","doi":"10.1515/rose-2021-2057","DOIUrl":"https://doi.org/10.1515/rose-2021-2057","url":null,"abstract":"Abstract The G-pencil law under the G-Lindeberg condition for a random matrix is proven.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2057","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47675710","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract In the present work, we consider a stochastic predator-prey model with disease in prey and distributed delay. Firstly, we establish sufficient conditions for the extinction of the disease and also permanence of healthy prey and predator. Besides, we obtain the condition for the existence of an ergodic stationary distribution through the stochastic Lyapunov function. Finally, we provide some numerical simulations to validate our theoretical findings.
{"title":"Analysis on stochastic predator-prey model with distributed delay","authors":"C. Gokila, M. Sambath","doi":"10.1515/rose-2021-2056","DOIUrl":"https://doi.org/10.1515/rose-2021-2056","url":null,"abstract":"Abstract In the present work, we consider a stochastic predator-prey model with disease in prey and distributed delay. Firstly, we establish sufficient conditions for the extinction of the disease and also permanence of healthy prey and predator. Besides, we obtain the condition for the existence of an ergodic stationary distribution through the stochastic Lyapunov function. Finally, we provide some numerical simulations to validate our theoretical findings.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2056","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45434103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.
{"title":"Stability of stochastic differential equations driven by multifractional Brownian motion","authors":"Oussama El Barrimi, Y. Ouknine","doi":"10.1515/rose-2021-2055","DOIUrl":"https://doi.org/10.1515/rose-2021-2055","url":null,"abstract":"Abstract Our aim in this paper is to establish some strong stability results for solutions of stochastic differential equations driven by a Riemann–Liouville multifractional Brownian motion. The latter is defined as a Gaussian non-stationary process with a Hurst parameter as a function of time. The results are obtained assuming that the pathwise uniqueness property holds and using Skorokhod’s selection theorem.","PeriodicalId":43421,"journal":{"name":"Random Operators and Stochastic Equations","volume":null,"pages":null},"PeriodicalIF":0.4,"publicationDate":"2021-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1515/rose-2021-2055","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41492280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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