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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
Pub Date : 2024-09-11 DOI: arxiv-2409.07029
Saloua Labed, Nacira Agram, Bernt Oksendal
In this paper, we study the probability distribution of solutions ofMcKean-Vlasov stochastic differential equations (SDEs) driven by fractionalBrownian motion. We prove the associated Fokker-Planck equation, which governsthe evolution of the probability distribution of the solution. For the casewhere the distribution is absolutely continuous, we present a more explicitform of this equation. To illustrate the result we use it to solve specificexamples, including the law of fractional Brownian motion and the geometricMcKean-Vlasov SDE, demonstrating the complex dynamics arising from theinterplay between fractional noise and mean-field interactions.
本文研究了由分数布朗运动驱动的麦克金-弗拉索夫随机微分方程(SDE)解的概率分布。我们证明了相关的福克-普朗克方程,该方程控制着解的概率分布的演化。对于分布绝对连续的情况,我们提出了该方程更明确的形式。为了说明这一结果,我们用它来求解具体的例子,包括分数布朗运动定律和几何麦克金-弗拉索夫 SDE,展示了分数噪声和均场相互作用之间相互作用所产生的复杂动力学。
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引用次数: 0
Jump Restore Light Transport 跳跃恢复光传输
Pub Date : 2024-09-11 DOI: arxiv-2409.07148
Sascha Holl, Gurprit Singh, Hans-Peter Seidel
Markov chain Monte Carlo (MCMC) algorithms come to rescue when sampling froma complex, high-dimensional distribution by a conventional method isintractable. Even though MCMC is a powerful tool, it is also hard to controland tune in practice. Simultaneously achieving both emph{local exploration} ofthe state space and emph{global discovery} of the target distribution is achallenging task. In this work, we present a MCMC formulation that subsumes allexisting MCMC samplers employed in rendering. We then present a novel frameworkfor emph{adjusting} an arbitrary Markov chain, making it exhibit invariancewith respect to a specified target distribution. To showcase the potential ofthe proposed framework, we focus on a first simple application in lighttransport simulation. As a by-product, we introduce continuous-time MCMCsampling to the computer graphics community. We show how any existingMCMC-based light transport algorithm can be embedded into our framework. Weempirically and theoretically prove that this embedding is superior to runningthe standalone algorithm. In fact, our approach will convert any existingalgorithm into a highly parallelizable variant with shorter running time,smaller error and less variance.
当用传统方法从复杂的高维分布中采样困难重重时,马尔可夫链蒙特卡洛(MCMC)算法就派上用场了。尽管 MCMC 是一种功能强大的工具,但在实际应用中也很难控制和调整。同时实现对状态空间的局部探索和对目标分布的全局发现是一项艰巨的任务。在这项工作中,我们提出了一种 MCMC 方案,它包含了渲染中使用的所有现有 MCMC 采样器。然后,我们提出了一个新颖的框架,用于渲染{调整}任意马尔可夫链,使其对指定的目标分布表现出不变性。为了展示所提框架的潜力,我们将重点放在光传输模拟中的第一个简单应用上。作为副产品,我们将连续时间 MCMC 采样引入计算机图形学领域。我们展示了如何将任何现有的基于 MCMC 的光传输算法嵌入到我们的框架中。我们从经验和理论上证明,这种嵌入优于运行独立算法。事实上,我们的方法可以将任何现有算法转化为高度可并行化的变体,其运行时间更短、误差更小、方差更小。
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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
Pub Date : 2024-09-11 DOI: arxiv-2409.07052
Yuyang Ye, Yunzhang Li, Shanjian Tang
In this paper, we study the Cauchy problem for backward stochastic partialdifferential equations (BSPDEs) involving fractional Laplacian operator.Firstly, by employing the martingale representation theorem and the fractionalheat kernel, we construct an explicit form of the solution for fractionalBSPDEs with space invariant coefficients, thereby demonstrating the existenceand uniqueness of strong solution. Then utilizing the freezing coefficientsmethod as well as the continuation method, we establish H"older estimates andwell-posedness for general fractional BSPDEs with coefficients dependent onspace-time variables. As an application, we use the fractional adjoint BSPDEsto investigate stochastic optimal control of the partially observed systemsdriven by $alpha$-stable L'evy processes.
本文研究了涉及分数拉普拉斯算子的后向随机偏微分方程(BSPDEs)的Cauchy问题。首先,利用马丁格尔表示定理和分数热核,构建了具有空间不变系数的分数BSPDEs解的显式,从而证明了强解的存在性和唯一性。然后,利用冻结系数法和延续法,我们为系数依赖于时空变量的一般分数 BSPDE 建立了 "老 "估计和好求解性。作为应用,我们利用分数邻接 BSPDEst 来研究由 $alpha$ 稳定 L'evy 过程驱动的部分观测系统的弹性最优控制。
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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
Pub Date : 2024-09-11 DOI: arxiv-2409.07358
Panqiu Xia, Guangqu Zheng
This short note is devoted to establishing the almost sure central limittheorem for the parabolic/hyperbolic Anderson models driven by colored-in-timeGaussian noises, completing recent results on quantitative central limittheorems for stochastic partial differential equations. We combine thesecond-order Gaussian Poincar'e inequality with Ibragimov and Lifshits' methodof characteristic functions, effectively overcoming the challenge from the lackof It^o tools in this colored-in-time setting, and achieving results that areinaccessible with previous methods.
这篇短文致力于建立由有色高斯噪声驱动的抛物/超抛物安德森模型的几乎确定的中心极限定理,完成了最近关于随机偏微分方程定量中心极限定理的成果。我们将这些二阶高斯 Poincar'e 不等式与 Ibragimov 和 Lifshits 的特征函数方法结合起来,有效地克服了在这种有色-时间设置中缺乏 It^o 工具所带来的挑战,并获得了以前的方法无法获得的结果。
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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
Pub Date : 2024-09-11 DOI: arxiv-2409.06965
Alastair Crossley, Karen Habermann, Emma Horton, Jere Koskela, Andreas E. Kyprianou, Sarah Osman
Proton beam radiotherapy stands at the forefront of precision cancertreatment, leveraging the unique physical interactions of proton beams withhuman tissue to deliver minimal dose upon entry and deposit the therapeuticdose precisely at the so-called Bragg peak, with no residual dose beyond thispoint. The Bragg peak is the characteristic maximum that occurs when plottingthe curve describing the rate of energy deposition along the length of theproton beam. Moreover, as a natural phenomenon, it is caused by an increase inthe rate of nuclear interactions of protons as their energy decreases. From ananalytical perspective, Bortfeld proposed a parametric family of curves thatcan be accurately calibrated to data replicating the Bragg peak in onedimension. We build, from first principles, the very first mathematical modeldescribing the energy deposition of protons. Our approach uses stochasticdifferential equations and affords us the luxury of defining the naturalanalogue of the Bragg curve in two or three dimensions. This work is purelytheoretical and provides a new mathematical framework which is capable ofencompassing models built using Geant4 Monte Carlo, at one extreme, to pencilbeam calculations with Bortfeld curves at the other.
质子束放射治疗是精准癌症治疗的前沿技术,它利用质子束与人体组织之间独特的物理相互作用,在进入人体组织时将剂量降至最低,并将治疗剂量精确地沉积在所谓的布拉格峰上,在此点之外没有任何残余剂量。布拉格峰是沿着质子束长度绘制能量沉积率曲线时出现的特征性最大值。此外,作为一种自然现象,布拉格峰是由质子能量下降时核相互作用速率增加引起的。从分析的角度来看,波特菲尔德提出了一个曲线参数族,可以精确地校准复制布拉格峰的一维数据。我们从第一原理出发,建立了第一个描述质子能量沉积的数学模型。我们的方法使用随机微分方程,使我们能够在二维或三维空间中定义布拉格曲线的自然类似物。这项工作纯粹是理论性的,它提供了一个新的数学框架,能够涵盖从使用 Geant4 蒙特卡洛建立的模型到使用波特菲尔德曲线进行的铅笔光束计算。
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引用次数: 0
Tempered space-time fractional negative binomial process 节制时空分数负二项过程
Pub Date : 2024-09-11 DOI: arxiv-2409.07044
Shilpa, Ashok Kumar Pathak, Aditya Maheshwari
In this paper, we define a tempered space-time fractional negative binomialprocess (TSTFNBP) by subordinating the fractional Poisson process with anindependent tempered Mittag-Leffler L'{e}vy subordinator. We study itsdistributional properties and its connection to partial differential equations.We derive the asymptotic behavior of its fractional order moments andlong-range dependence property. It is shown that the TSTFNBP exhibitsoverdispersion. We also obtain some results related to the first-passage time.
本文通过将分数泊松过程与独立的回火米塔格-勒夫勒 L'{e}vy 附属器进行附属,定义了回火时空分数负二项式过程(TSTFNBP)。我们研究了它的分布特性及其与偏微分方程的联系,并推导出其分数阶矩的渐近行为和长程依赖特性。结果表明,TSTFNBP 表现出过度离散性。我们还得到了一些与首过时间相关的结果。
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引用次数: 0
Current fluctuations in the Dyson Gas 戴森气体中的电流波动
Pub Date : 2024-09-10 DOI: arxiv-2409.06881
Rahul Dandekar, P. L. Krapivsky, Kirone Mallick
We study large fluctuations of the current in a Dyson gas, a 1D system ofparticles interacting through a logarithmic potential and subjected to randomnoise. We adapt the macroscopic fluctuation theory to the Dyson gas and derivetwo coupled partial differential equations describing the evolution of thedensity and momentum. These equations are nonlinear and non-local, and the`boundary' conditions are mixed: some at the initial time and others at thefinal time. If the initial condition can fluctuate (annealed setting), thisboundary-value problem is tractable. We compute the cumulant generatingfunction encoding all the cumulants of the current.
戴森气体是一个通过对数势能相互作用并受到随机噪声影响的一维粒子系统,我们研究了戴森气体中电流的大波动。我们将宏观波动理论应用于戴森气体,并推导出描述密度和动量演化的两个耦合偏微分方程。这些方程是非线性和非局部的,而且 "边界 "条件是混合的:一些在初始时,另一些在最终时。如果初始条件可以波动(退火设置),那么这个边界值问题就很容易解决。我们计算的累积生成函数编码了当前的所有累积。
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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
Pub Date : 2024-09-10 DOI: arxiv-2409.06276
Laure CoutinIMT, Mahmoud Khabou
The marked Hawkes risk process is a compound point process for which theoccurrence and amplitude of past events impact the future. Thanks to itsautoregressive properties, it found applications in various fields such asneuosciences, social networks and insurance.Since data in real life is acquiredover a discrete time grid, we propose a strong discrete-time approximation ofthe continuous-time Hawkes risk process obtained be embedding from the samePoisson measure. We then prove trajectorial convergence results both in somefractional Sobolev spaces and in the Skorokhod space, hence extending thetheorems proven in the literature. We also provide upper bounds on theconvergence speed with explicit dependence on the size of the discretisationstep, the time horizon and the regularity of the kernel.
标记霍克斯风险过程是一个复合点过程,过去事件的发生和振幅会对未来产生影响。由于现实生活中的数据是在离散时间网格上获取的,因此我们提出了连续时间霍克斯风险过程的强离散时间近似值,该近似值通过嵌入相同的泊松量度获得。然后,我们证明了某些分数 Sobolev 空间和 Skorokhod 空间中的轨迹收敛结果,从而扩展了文献中证明的定理。我们还提供了收敛速度的上限,并明确依赖于离散化步骤的大小、时间跨度和核的正则性。
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引用次数: 0
Regularity of paths of stochastic measures 随机测量路径的规律性
Pub Date : 2024-09-10 DOI: arxiv-2409.06497
Vadym Radchenko
Random functions $mu(x)$, generated by values of stochastic measures areconsidered. 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 similarresults in previous papers.
考虑了由随机测量值生成的随机函数 $mu(x)$。证明了 $mu(x)$, $xin[0,1]^d$ 的连续路径的贝索夫正则性。得到了 $mu(x)$, $xin[0,2pi]$ 的傅里叶级数展开。这些结果是在比以前论文中类似结果更弱的条件下证明的。
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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
Pub Date : 2024-09-10 DOI: arxiv-2409.06836
Hélène Guérin, Lucile Laulin, Kilian Raschel, Thomas Simon
When the memory parameter of the elephant random walk is above a criticalthreshold, the process becomes superdiffusive and, once suitably normalised,converges to a non-Gaussian random variable. In a recent paper by the threefirst authors, it was shown that this limit variable has a density and that theassociated moments satisfy a nonlinear recurrence relation. In this work, weexploit this recurrence to derive an asymptotic expansion of the moments andthe asymptotic behaviour of the density at infinity. In particular, we showthat an asymmetry in the distribution of the first step of the random walkleads to an asymmetry of the tails of the limit variable. These results followfrom a new, explicit expression of the Stieltjes transformation of the momentsin terms of special functions such as hypergeometric series and incomplete betaintegrals. We also obtain other results about the random variable, such asunimodality and, for certain values of the memory parameter, log-concavity.
当大象随机游走的记忆参数超过临界阈值时,过程就会变得超扩散,一旦适当归一化,就会收敛为非高斯随机变量。在三位第一作者最近发表的一篇论文中,证明了这种极限变量具有密度,而且相关矩满足非线性递推关系。在本文中,我们利用这一递推关系推导出了矩的渐近展开和密度在无穷大时的渐近行为。特别是,我们证明了随机游走第一步分布的不对称性导致了极限变量尾部的不对称性。这些结果源于用特殊函数(如超几何级数和不完全贝特积分)对矩的斯蒂尔杰斯变换的新的明确表达。我们还得到了有关随机变量的其他结果,如单调性,以及在记忆参数的某些值下的对数凹性。
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引用次数: 0
期刊
arXiv - MATH - Probability
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