In this paper, we study the probability distribution of solutions of McKean-Vlasov stochastic differential equations (SDEs) driven by fractional Brownian motion. We prove the associated Fokker-Planck equation, which governs the evolution of the probability distribution of the solution. For the case where the distribution is absolutely continuous, we present a more explicit form of this equation. To illustrate the result we use it to solve specific examples, including the law of fractional Brownian motion and the geometric McKean-Vlasov SDE, demonstrating the complex dynamics arising from the interplay between fractional noise and mean-field interactions.
{"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":null,"pages":null},"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}
Markov chain Monte Carlo (MCMC) algorithms come to rescue when sampling from a complex, high-dimensional distribution by a conventional method is intractable. Even though MCMC is a powerful tool, it is also hard to control and tune in practice. Simultaneously achieving both emph{local exploration} of the state space and emph{global discovery} of the target distribution is a challenging task. In this work, we present a MCMC formulation that subsumes all existing MCMC samplers employed in rendering. We then present a novel framework for emph{adjusting} an arbitrary Markov chain, making it exhibit invariance with respect to a specified target distribution. To showcase the potential of the proposed framework, we focus on a first simple application in light transport simulation. As a by-product, we introduce continuous-time MCMC sampling to the computer graphics community. We show how any existing MCMC-based light transport algorithm can be embedded into our framework. We empirically and theoretically prove that this embedding is superior to running the standalone algorithm. In fact, our approach will convert any existing algorithm into a highly parallelizable variant with shorter running time, smaller error and less variance.
{"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":null,"pages":null},"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}
In this paper, we study the Cauchy problem for backward stochastic partial differential equations (BSPDEs) involving fractional Laplacian operator. Firstly, by employing the martingale representation theorem and the fractional heat kernel, we construct an explicit form of the solution for fractional BSPDEs with space invariant coefficients, thereby demonstrating the existence and uniqueness of strong solution. Then utilizing the freezing coefficients method as well as the continuation method, we establish H"older estimates and well-posedness for general fractional BSPDEs with coefficients dependent on space-time variables. As an application, we use the fractional adjoint BSPDEs to investigate stochastic optimal control of the partially observed systems driven by $alpha$-stable L'evy processes.
{"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":null,"pages":null},"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}
This short note is devoted to establishing the almost sure central limit theorem for the parabolic/hyperbolic Anderson models driven by colored-in-time Gaussian noises, completing recent results on quantitative central limit theorems for stochastic partial differential equations. We combine the second-order Gaussian Poincar'e inequality with Ibragimov and Lifshits' method of characteristic functions, effectively overcoming the challenge from the lack of It^o tools in this colored-in-time setting, and achieving results that are inaccessible with previous methods.
{"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":null,"pages":null},"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}
Alastair Crossley, Karen Habermann, Emma Horton, Jere Koskela, Andreas E. Kyprianou, Sarah Osman
Proton beam radiotherapy stands at the forefront of precision cancer treatment, leveraging the unique physical interactions of proton beams with human tissue to deliver minimal dose upon entry and deposit the therapeutic dose precisely at the so-called Bragg peak, with no residual dose beyond this point. The Bragg peak is the characteristic maximum that occurs when plotting the curve describing the rate of energy deposition along the length of the proton beam. Moreover, as a natural phenomenon, it is caused by an increase in the rate of nuclear interactions of protons as their energy decreases. From an analytical perspective, Bortfeld proposed a parametric family of curves that can be accurately calibrated to data replicating the Bragg peak in one dimension. We build, from first principles, the very first mathematical model describing the energy deposition of protons. Our approach uses stochastic differential equations and affords us the luxury of defining the natural analogue of the Bragg curve in two or three dimensions. This work is purely theoretical and provides a new mathematical framework which is capable of encompassing models built using Geant4 Monte Carlo, at one extreme, to pencil beam calculations with Bortfeld curves at the other.
{"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":null,"pages":null},"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}
In this paper, we define a tempered space-time fractional negative binomial process (TSTFNBP) by subordinating the fractional Poisson process with an independent tempered Mittag-Leffler L'{e}vy subordinator. We study its distributional properties and its connection to partial differential equations. We derive the asymptotic behavior of its fractional order moments and long-range dependence property. It is shown that the TSTFNBP exhibits overdispersion. We also obtain some results related to the first-passage time.
{"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":null,"pages":null},"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}
We study large fluctuations of the current in a Dyson gas, a 1D system of particles interacting through a logarithmic potential and subjected to random noise. We adapt the macroscopic fluctuation theory to the Dyson gas and derive two coupled partial differential equations describing the evolution of the density and momentum. These equations are nonlinear and non-local, and the `boundary' conditions are mixed: some at the initial time and others at the final time. If the initial condition can fluctuate (annealed setting), this boundary-value problem is tractable. We compute the cumulant generating function encoding all the cumulants of the current.
{"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":null,"pages":null},"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}
The marked Hawkes risk process is a compound point process for which the occurrence and amplitude of past events impact the future. Thanks to its autoregressive properties, it found applications in various fields such as neuosciences, social networks and insurance.Since data in real life is acquired over a discrete time grid, we propose a strong discrete-time approximation of the continuous-time Hawkes risk process obtained be embedding from the same Poisson measure. We then prove trajectorial convergence results both in some fractional Sobolev spaces and in the Skorokhod space, hence extending the theorems proven in the literature. We also provide upper bounds on the convergence speed with explicit dependence on the size of the discretisation step, the time horizon and the regularity of the kernel.
{"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":null,"pages":null},"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}
Random functions $mu(x)$, generated by values of stochastic measures are considered. The Besov regularity of the continuous paths of $mu(x)$, $xin[0,1]^d$ is proved. Fourier series expansion of $mu(x)$, $xin[0,2pi]$ is obtained. These results are proved under weaker conditions than similar results in previous papers.
{"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":null,"pages":null},"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}
Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon
When the memory parameter of the elephant random walk is above a critical threshold, the process becomes superdiffusive and, once suitably normalised, converges to a non-Gaussian random variable. In a recent paper by the three first authors, it was shown that this limit variable has a density and that the associated moments satisfy a nonlinear recurrence relation. In this work, we exploit this recurrence to derive an asymptotic expansion of the moments and the asymptotic behaviour of the density at infinity. In particular, we show that an asymmetry in the distribution of the first step of the random walk leads to an asymmetry of the tails of the limit variable. These results follow from a new, explicit expression of the Stieltjes transformation of the moments in terms of special functions such as hypergeometric series and incomplete beta integrals. We also obtain other results about the random variable, such as unimodality and, for certain values of the memory parameter, log-concavity.
{"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":null,"pages":null},"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}
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