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Statistics for Spatial Data.

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

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Table Of Integrals Series And Products

Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

A Primer on Stochastic Partial Differential Equations- The Stochastic Wave Equation- Application of Malliavin Calculus to Stochastic Partial Differential Equations- Some Tools and Results for Parabolic Stochastic Partial Differential Equations- Sample Path Properties of Anisotropic Gaussian Random Fields

A Minicourse on Stochastic Partial Differential Equations

Anisotropic Gaussian random flelds arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operator-scaling Gaussian flelds with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random flelds in general. Let X = fX(t); t 2 R N g be a Gaussian random fleld with values in R d and with parameters H1;:::;HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly. Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random flelds. An important tool for our study is the various forms of strong local nondeterminism.

/pdf/sample-path-properties-of-anisotropic-gaussian-random-fields-1uxcz36kjw.pdf

Sample Path Properties of Anisotropic Gaussian Random Fields

Let Y(t) (t∈ℝ
 N
 ) be a real-valued, strongly locally nondeterministic Gaussian random field with stationary increments and Y(0)=0. Consider the (N,d) Gaussian random field defined by
 where X
1,…,X

 d
 are independent copies of Y. The local and global Holder conditions in the set variable for the local time of X(t) are established and the exact Hausdorff measure of the level set X
−1(x) is evaluated.

/pdf/holder-conditions-for-the-local-times-and-the-hausdorff-3yif1fb8td.pdf

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Let B H ={ B H (t), t ∈ R N } be an (N, d)-fractional Brownian sheet with index H = (H1 ,...,H N ) ∈ (0,1) N. The uniform and local asymptotic properties of B H are proved by using wavelet methods. The Hausdorff and packing dimensions of the range B H ((0,1) N ), the graph GrB H ((0,1) N ) and the level set are determined.

/pdf/asymptotic-properties-and-hausdorff-dimensions-of-fractional-4rmt8y7409.pdf

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

This is a survey on the sample path properties of Markov processes, especially fractal properties of the random sets and measures determined by their sample paths. The class of Markov processes considered in this paper include Levy processes in ℝ, diffusions on fractals and on ℝ, Feller processes determined by pseudo-differential operators and so on. We summarize recent results for Levy processes such as the Hausdorff and packing dimensions of their ranges, level sets, and multiple points; regularity properties of local times and self-intersection local times; multifractal analysis of the occupation measures and sample paths. Our emphasis is on general Markovian techniques that are applicable to other Markov processes.

/pdf/random-fractals-and-markov-processes-1yvfk9qzwu.pdf

Yimin Xiao

Papers

A Minicourse on Stochastic Partial Differential Equations

Sample Path Properties of Anisotropic Gaussian Random Fields

Hölder conditions for the local times and the Hausdorff measure of the level sets of Gaussian random fields

Asymptotic Properties and Hausdorff Dimensions of Fractional Brownian Sheets

Random Fractals and Markov Processes