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Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion 分数布朗运动驱动的麦金-弗拉索夫 SDE 的福克-普朗克方程
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.07029
Saloua Labed, Nacira Agram, Bernt Oksendal
{"title":"Fokker-Planck equations for McKean-Vlasov SDEs driven by fractional Brownian motion","authors":"Saloua Labed, Nacira Agram, Bernt Oksendal","doi":"arxiv-2409.07029","DOIUrl":"https://doi.org/arxiv-2409.07029","url":null,"abstract":"In this paper, we study the probability distribution of solutions of\u0000McKean-Vlasov stochastic differential equations (SDEs) driven by fractional\u0000Brownian motion. We prove the associated Fokker-Planck equation, which governs\u0000the evolution of the probability distribution of the solution. For the case\u0000where the distribution is absolutely continuous, we present a more explicit\u0000form of this equation. To illustrate the result we use it to solve specific\u0000examples, including the law of fractional Brownian motion and the geometric\u0000McKean-Vlasov SDE, demonstrating the complex dynamics arising from the\u0000interplay between fractional noise and mean-field interactions.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"101 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212030","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}
引用次数: 0
Jump Restore Light Transport 跳跃恢复光传输
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.07148
Sascha Holl, Gurprit Singh, Hans-Peter Seidel
{"title":"Jump Restore Light Transport","authors":"Sascha Holl, Gurprit Singh, Hans-Peter Seidel","doi":"arxiv-2409.07148","DOIUrl":"https://doi.org/arxiv-2409.07148","url":null,"abstract":"Markov chain Monte Carlo (MCMC) algorithms come to rescue when sampling from\u0000a complex, high-dimensional distribution by a conventional method is\u0000intractable. Even though MCMC is a powerful tool, it is also hard to control\u0000and tune in practice. Simultaneously achieving both emph{local exploration} of\u0000the state space and emph{global discovery} of the target distribution is a\u0000challenging task. In this work, we present a MCMC formulation that subsumes all\u0000existing MCMC samplers employed in rendering. We then present a novel framework\u0000for emph{adjusting} an arbitrary Markov chain, making it exhibit invariance\u0000with respect to a specified target distribution. To showcase the potential of\u0000the proposed framework, we focus on a first simple application in light\u0000transport simulation. As a by-product, we introduce continuous-time MCMC\u0000sampling to the computer graphics community. We show how any existing\u0000MCMC-based light transport algorithm can be embedded into our framework. We\u0000empirically and theoretically prove that this embedding is superior to running\u0000the standalone algorithm. In fact, our approach will convert any existing\u0000algorithm into a highly parallelizable variant with shorter running time,\u0000smaller error and less variance.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212032","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}
引用次数: 0
Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes 分数后向随机偏微分方程及其在由勒维过程驱动的部分观测系统的随机优化控制中的应用
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.07052
Yuyang Ye, Yunzhang Li, Shanjian Tang
{"title":"Fractional Backward Stochastic Partial Differential Equations with Applications to Stochastic Optimal Control of Partially Observed Systems driven by Lévy Processes","authors":"Yuyang Ye, Yunzhang Li, Shanjian Tang","doi":"arxiv-2409.07052","DOIUrl":"https://doi.org/arxiv-2409.07052","url":null,"abstract":"In this paper, we study the Cauchy problem for backward stochastic partial\u0000differential equations (BSPDEs) involving fractional Laplacian operator.\u0000Firstly, by employing the martingale representation theorem and the fractional\u0000heat kernel, we construct an explicit form of the solution for fractional\u0000BSPDEs with space invariant coefficients, thereby demonstrating the existence\u0000and uniqueness of strong solution. Then utilizing the freezing coefficients\u0000method as well as the continuation method, we establish H\"older estimates and\u0000well-posedness for general fractional BSPDEs with coefficients dependent on\u0000space-time variables. As an application, we use the fractional adjoint BSPDEs\u0000to investigate stochastic optimal control of the partially observed systems\u0000driven by $alpha$-stable L'evy processes.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"288 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212033","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}
引用次数: 0
Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises 具有高斯彩色噪声的抛物线/超抛物线安德森模型的几乎确定的中心极限定理
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.07358
Panqiu Xia, Guangqu Zheng
{"title":"Almost sure central limit theorems for parabolic/hyperbolic Anderson models with Gaussian colored noises","authors":"Panqiu Xia, Guangqu Zheng","doi":"arxiv-2409.07358","DOIUrl":"https://doi.org/arxiv-2409.07358","url":null,"abstract":"This short note is devoted to establishing the almost sure central limit\u0000theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time\u0000Gaussian noises, completing recent results on quantitative central limit\u0000theorems for stochastic partial differential equations. We combine the\u0000second-order Gaussian Poincar'e inequality with Ibragimov and Lifshits' method\u0000of characteristic functions, effectively overcoming the challenge from the lack\u0000of It^o tools in this colored-in-time setting, and achieving results that are\u0000inaccessible with previous methods.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"30 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212028","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}
引用次数: 0
Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy 用于描述质子束放射治疗布拉格峰特征的跳跃式随机微分方程
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.06965
Alastair Crossley, Karen Habermann, Emma Horton, Jere Koskela, Andreas E. Kyprianou, Sarah Osman
{"title":"Jump stochastic differential equations for the characterisation of the Bragg peak in proton beam radiotherapy","authors":"Alastair Crossley, Karen Habermann, Emma Horton, Jere Koskela, Andreas E. Kyprianou, Sarah Osman","doi":"arxiv-2409.06965","DOIUrl":"https://doi.org/arxiv-2409.06965","url":null,"abstract":"Proton beam radiotherapy stands at the forefront of precision cancer\u0000treatment, leveraging the unique physical interactions of proton beams with\u0000human tissue to deliver minimal dose upon entry and deposit the therapeutic\u0000dose precisely at the so-called Bragg peak, with no residual dose beyond this\u0000point. The Bragg peak is the characteristic maximum that occurs when plotting\u0000the curve describing the rate of energy deposition along the length of the\u0000proton beam. Moreover, as a natural phenomenon, it is caused by an increase in\u0000the rate of nuclear interactions of protons as their energy decreases. From an\u0000analytical perspective, Bortfeld proposed a parametric family of curves that\u0000can be accurately calibrated to data replicating the Bragg peak in one\u0000dimension. We build, from first principles, the very first mathematical model\u0000describing the energy deposition of protons. Our approach uses stochastic\u0000differential equations and affords us the luxury of defining the natural\u0000analogue of the Bragg curve in two or three dimensions. This work is purely\u0000theoretical and provides a new mathematical framework which is capable of\u0000encompassing models built using Geant4 Monte Carlo, at one extreme, to pencil\u0000beam calculations with Bortfeld curves at the other.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"5 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212034","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}
引用次数: 0
Tempered space-time fractional negative binomial process 节制时空分数负二项过程
arXiv - MATH - Probability Pub Date : 2024-09-11 DOI: arxiv-2409.07044
Shilpa, Ashok Kumar Pathak, Aditya Maheshwari
{"title":"Tempered space-time fractional negative binomial process","authors":"Shilpa, Ashok Kumar Pathak, Aditya Maheshwari","doi":"arxiv-2409.07044","DOIUrl":"https://doi.org/arxiv-2409.07044","url":null,"abstract":"In this paper, we define a tempered space-time fractional negative binomial\u0000process (TSTFNBP) by subordinating the fractional Poisson process with an\u0000independent tempered Mittag-Leffler L'{e}vy subordinator. We study its\u0000distributional properties and its connection to partial differential equations.\u0000We derive the asymptotic behavior of its fractional order moments and\u0000long-range dependence property. It is shown that the TSTFNBP exhibits\u0000overdispersion. We also obtain some results related to the first-passage time.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212029","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}
引用次数: 0
Current fluctuations in the Dyson Gas 戴森气体中的电流波动
arXiv - MATH - Probability Pub Date : 2024-09-10 DOI: arxiv-2409.06881
Rahul Dandekar, P. L. Krapivsky, Kirone Mallick
{"title":"Current fluctuations in the Dyson Gas","authors":"Rahul Dandekar, P. L. Krapivsky, Kirone Mallick","doi":"arxiv-2409.06881","DOIUrl":"https://doi.org/arxiv-2409.06881","url":null,"abstract":"We study large fluctuations of the current in a Dyson gas, a 1D system of\u0000particles interacting through a logarithmic potential and subjected to random\u0000noise. We adapt the macroscopic fluctuation theory to the Dyson gas and derive\u0000two coupled partial differential equations describing the evolution of the\u0000density and momentum. These equations are nonlinear and non-local, and the\u0000`boundary' conditions are mixed: some at the initial time and others at the\u0000final time. If the initial condition can fluctuate (annealed setting), this\u0000boundary-value problem is tractable. We compute the cumulant generating\u0000function encoding all the cumulants of the current.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"178 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212035","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}
引用次数: 0
Functional approximation of the marked Hawkes risk process 标记霍克斯风险过程的函数近似值
arXiv - MATH - Probability Pub Date : 2024-09-10 DOI: arxiv-2409.06276
Laure CoutinIMT, Mahmoud Khabou
{"title":"Functional approximation of the marked Hawkes risk process","authors":"Laure CoutinIMT, Mahmoud Khabou","doi":"arxiv-2409.06276","DOIUrl":"https://doi.org/arxiv-2409.06276","url":null,"abstract":"The marked Hawkes risk process is a compound point process for which the\u0000occurrence and amplitude of past events impact the future. Thanks to its\u0000autoregressive properties, it found applications in various fields such as\u0000neuosciences, social networks and insurance.Since data in real life is acquired\u0000over a discrete time grid, we propose a strong discrete-time approximation of\u0000the continuous-time Hawkes risk process obtained be embedding from the same\u0000Poisson measure. We then prove trajectorial convergence results both in some\u0000fractional Sobolev spaces and in the Skorokhod space, hence extending the\u0000theorems proven in the literature. We also provide upper bounds on the\u0000convergence speed with explicit dependence on the size of the discretisation\u0000step, the time horizon and the regularity of the kernel.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212040","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}
引用次数: 0
Regularity of paths of stochastic measures 随机测量路径的规律性
arXiv - MATH - Probability Pub Date : 2024-09-10 DOI: arxiv-2409.06497
Vadym Radchenko
{"title":"Regularity of paths of stochastic measures","authors":"Vadym Radchenko","doi":"arxiv-2409.06497","DOIUrl":"https://doi.org/arxiv-2409.06497","url":null,"abstract":"Random functions $mu(x)$, generated by values of stochastic measures are\u0000considered. The Besov regularity of the continuous paths of $mu(x)$,\u0000$xin[0,1]^d$ is proved. Fourier series expansion of $mu(x)$, $xin[0,2pi]$\u0000is obtained. These results are proved under weaker conditions than similar\u0000results in previous papers.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212039","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}
引用次数: 0
On the limit law of the superdiffusive elephant random walk 论超扩散大象随机漫步的极限规律
arXiv - MATH - Probability Pub Date : 2024-09-10 DOI: arxiv-2409.06836
Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon
{"title":"On the limit law of the superdiffusive elephant random walk","authors":"Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon","doi":"arxiv-2409.06836","DOIUrl":"https://doi.org/arxiv-2409.06836","url":null,"abstract":"When the memory parameter of the elephant random walk is above a critical\u0000threshold, the process becomes superdiffusive and, once suitably normalised,\u0000converges to a non-Gaussian random variable. In a recent paper by the three\u0000first authors, it was shown that this limit variable has a density and that the\u0000associated moments satisfy a nonlinear recurrence relation. In this work, we\u0000exploit this recurrence to derive an asymptotic expansion of the moments and\u0000the asymptotic behaviour of the density at infinity. In particular, we show\u0000that an asymmetry in the distribution of the first step of the random walk\u0000leads to an asymmetry of the tails of the limit variable. These results follow\u0000from a new, explicit expression of the Stieltjes transformation of the moments\u0000in terms of special functions such as hypergeometric series and incomplete beta\u0000integrals. We also obtain other results about the random variable, such as\u0000unimodality and, for certain values of the memory parameter, log-concavity.","PeriodicalId":501245,"journal":{"name":"arXiv - MATH - Probability","volume":"28 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142212031","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}
引用次数: 0
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