Dynamical Correlations among Vicious Random Walkers

P.F. Stadler: Landscapes and Their Correlation Functions
Most of the knowledge about landscapes has so far been derived using statistical methods, considering random models of landscapes rather than a single landscape. The distribution of local optima and the statistical characteristics of down-hill walks have been computed for the uncorrelated landscape of the random energy model 25, 26, 27], Furthermore, two one-parameter families of tunably rugged landscapes have been studied extensively: the Nk model and its variants 28, 29, 30, 16] and the p-spin models 31, 32, 33, 34]. Local optima of 2-spin models are considered in 35, 36, 37]. While the statistical approach is the natural one, e.g., in the physics of spin glasses, it seems to be rather contrived in evolutionary biology because it is by no means clear what a reasonable statistical model should look like, even if there is a computational procedure to model the landscapes of, say, RNA free energies. A theory of landscapes is based on three ingredients: we are given a nite, but very large set V of \con gurations" and a \ tness function" f : V ! IR. The third ingredient is a notion of neighborhood between the con gurations, which allows us to interpret V as the vertex set of a graph ?. We will refer to ? as the con guration space of the landscape f . Let us brie y discuss two examples here: Consider the set of RNA molecules of given chain length n. A particular molecule x can be represented as a string of length n taken from the alphabet fG, C, A, Ug; molecular biologists call this string the sequence of the RNA. The \ tness" function f is, for instance, the free energy of folding x into its secondary structure 15]. In silico the folding is done by an algorithm containing a large number of experimentally determined parameters 38]. In nature as well as in in vitro experiments variation is introduced by mutations, predominantly point-mutations. Neighboring sequences are thus those that di er in only a single position. The resulting graph is known as the sequence space 39, 40]. A very di erent example is the Travelling Salesman Problem. A salesman starts from his home city and visits exactly once each of the n cities on a given list, then he returns home. The con gurations are the possible tours, i.e., all permutations of the cities on the salesman's list. The numerical value assigned to a particular tour {2{

2021,41A (1):227-236数学物理学报http: // a ct a 混合型随机微分方程的传输不等式徐丽平李治**收稿日期：2019-12-24;修订日期：2020-10-25E-mail: ****************基金项目：国家自然科学基金(11901058, 62076039)Supported by the NSFC(11901058, 62076039)*通讯作者(长江大学信息与数学学院湖北荆州434023)摘要：该文探讨一类由Wiener 过程和Hurst 参数1/2 < H < 1分数布朗运动驱动的混合 型随机微分方程.通过使用一些变换技巧和逼近方法，这类方程的强解在d 2度量和一致度量 d *下的二次传输不等式被建立.关键词：传输不等式；混合型随机微分方程；分数布朗运动；轨道积分.MR(2010)主题分类：60H15; 60H05 中图分类号：0221 文献标识码：A文章编号：1003-3998(2021)01-227-101引言考虑如下的混合型随机微分方程X (t ) = X o + / a (s,X (s ))d s + / b (s,X (s ))d W (s )Jo Joc (s,X (s ))d B H (s ),t G [0,T ],(1.1)这里a : [0,T ] x 及b,c : [0,T ] x t R d x k 是一些适当的函数，我们将在后面详细讨论他们；W 是标准的Wiener 过程，B H 是Hurs t 参数1/2 <H < 1的分数布朗运动； 随机积分f (0 c (s,X (s ))d B H (s )是样本轨道意义下的Riemann-Stieltjes 积分，这就是说这种随 机积分是Young [22〕和Zahle [23-24〕在分数Sobolev 型空间中定义的积分.考虑这种混合型随机微分方程的动机来自于金融数学.在模拟金融市场的随机性时，有 必要区分两种不同的主要随机性来源.噪音的第一个来源是证券交易所的数千个经纪人，这 类随机噪声可以被认为为白噪声，可以使用Wiener 过程来模拟.第二类噪声源于常常具有 长相依性和自相似性的财政和经济状况，这类噪声能够使用Hurst 参数1/2 < H < 1的分 数布朗运动来模拟.因此，利用Wiener 过程和分数布朗运动驱动的混合型随机微分方程描 述证券的价格行为更加合理.最近，混合型随机微分方程吸引了广泛的关注•例如，Guerra 和Nualart 间及Kubilius [6] 考虑了混合型随机微分方程(1.1)强解的存在唯一性；Mishura 和Shevchenko [12]研究了方程228数学物理学报Vol.41A(1.1)解的存在唯一性及收敛性；Mishura等【回调查了混合时滞微分方程解的Euler逼近的 收敛性；Liu和Luo®〕使用修正的Euler方法讨论了方程(1.1)解的均方收敛率；Melnikov 等［11］研究了方程(1.1)的随机可行性和比较定理.另一方面，关于各种随机过程的Talagrand-类型的传输不等式已有大量的研究结果.文献［2,20-21］研究了底上的扩散过程，文献［17-18］研究了Riemannian流形上的扩散过程，文献［14］研究了多维半鞅，文献［19］研究了纯跳过程驱动的随机微分方程，文献［16］研究了分数布朗运动驱动的随机微分方程，文献［8］研究了Hurst参数1/2<H<1分数布朗运动驱动的随机时滞发展方程以及文献［1］研究了Hurst参数0<H<1/2分数布朗运动驱动的中立性随机时滞发展方程.TCIs是处理测量现象的集中问题的有效工具，TCIs在Tsirel'son-类型和Hoeffding-类型不等式t2'20-21!以及经验测度集中度血」9］等中的广泛应用使得它吸引了越来越多的关注.假设(E,d)是。

P.Sollich@
Peter Sollich
Abstract
We present a new method for obtaining the response function G and its average G from which most of the properties of learning and generalization in linear perceptrons can be derived. We rst rederive the known results for the `thermodynamic limit' of in nite perceptron size N and show explicitly that G is self-averaging in this limit. We then discuss extensions of our method to more general learning scenarios with anisotropic teacher space priors, input distributions, and weight decay terms. Finally, we use our method to calculate the nite N corrections of order 1=N to G and discuss the corresponding nite size e ects on generalization and learning dynamics. An important spin-o is the observation that results obtained in the thermodynamic limit are often directly relevant to systems of fairly modest, `real-world' sizes.

• Math/Theory
• Application/Optics • Data Analysis
2 texts: ‘Light scattering by Small Particles’ by van de Hulst ‘Dynamic Light Scattering with applications to Chemistry, Biology and Physics’ by Berne and Pecora
G1( ! ) )
1 T 10 E( t )E +t ( ! ,d! T
Constructive interference
G1(t) decays as and exponential with a decay constant 345for a system undergoing Brownian motion
Dynamic Light Scattering (aka QLS, PCS)
Oriented particles create interference patterns, each bright spot being a speckle. The speckle pattern moves as the particle move, creating flickering. All the motions and measurements are described by correlations functions • G2(!)- intensity correlation function describes particle motion • G1(!)- electric field correlation function describes measured fluctuations Which are related to connect the measurement and motion