Sample Path Properties of Anisotropic Gaussian Random Fields

Xiao, Yimin

doi:10.1007/978-3-540-85994-9_5

Cited by 140 publications

(231 citation statements)

References 97 publications

Supporting

Mentioning

227

Contrasting

Unclassified

Order By: Relevance

“…The fractal dimension of its graph is still linked with H by the relation D = 2 − H a.s. (see [44] for instance).…”

mentioning

confidence: 99%

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

Biermé¹,

Moisan²,

Richard³

2015

Journal of Computational and Graphical Statistics

View full text Add to dashboard Cite

Abstract. In this paper, we propose a method for simulating realizations of twodimensional anisotropic fractional Brownian elds (AFBF) introduced by Bonami and Estrade (2003). The method is adapted from a generic simulation method called the turning-band method (TBM) due to Matheron (1973). The TBM reduces the problem of simulating a eld in two dimensions by combining independent processes simulated on oriented bands. In the AFBF context, the simulation elds are constructed by solving an integral equation arising from the application of the TBM to non-stationary anisotropic elds. This garantees the convergence of simulations as their precision is increased. The construction is followed by a theoretical study of the convergence rate. Another key feature of this work is the simulation of band processes. Using self-similarity properties, processes are simulated exactly on bands with a circulant embedding method, so that simulation errors are exclusively due to the eld approximation. Moreover, we design a Dynamic Programming algorithm that selects band orientations achieving the optimal trade-o between computational cost and precision. Finally, we conduct a numerical study showing that the approximation error does not signicantly depend on the regularity of the elds to be simulated, nor on their degree of anisotropy. Experiments also suggest that simulations preserve eld statistical properties. IntroductionIn this paper, we address the issue of simulating realizations of a generic class of Gaussian elds, known as Anisotropic Fractional Brownian Fields (AFBF) and introduced in [9]. Date: October 9, 2012. 1991 Mathematics Subject Classication. 60G60; 60G22; 60-04.

show abstract

“…The fractal dimension of its graph is still linked with H by the relation D = 2 − H a.s. (see [44] for instance).…”

mentioning

confidence: 99%

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

Biermé¹,

Moisan²,

Richard³

2015

Journal of Computational and Graphical Statistics

View full text Add to dashboard Cite

show abstract

“…From Theorem 3.3, we already know that the sample paths of the Gaussian process defined by (22) are Hölder continuous. However, under the standing assumptions, more can be said.…”

Section: Sample Paths Of the Process Vmentioning

confidence: 99%

“…Therefore, without loss of generality, we can reduce the analysis of the stochastic process given in (22) to the case where σ is the identity matrix in R d . By doing so, we are left to consider the Gaussian vector v(x) = (v i (x)) i with independent, identically distributed components defined by…”

Section: Gaussian Solutionsmentioning

confidence: 99%

“…Solutions to SPDEs, like the stochastic heat equation, belong to this class. In [22], different results relative to sample paths of anisotropic Gaussian random fields are presented, in particular on hitting probabilities, and a exhaustive number of references are given. The Gaussian process obtained from (6) with f = g = 0, provides yet another example of random field for which, to the best of our acknowledge, results on hitting probabilities have not been obtained before.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Systems of stochastic Poisson equations: Hitting probabilities

Sanz–Solé

Viles

2018

Stochastic Processes and their Applications

View full text Add to dashboard Cite

Abstract. We consider a d-dimensional random field u = (u(x), x ∈ D) that solves a system of elliptic stochastic equations on a bounded domain D ⊂ R k , with additive white noise and spatial dimension k = 1, 2, 3. Properties of u and its probability law are proved. For Gaussian solutions, using results from [9], we establish upper and lower bounds on hitting probabilities in terms of the Hausdorff measure and Bessel-Riesz capacity, respectively. This relies on precise estimates on the canonical distance of the process or, equivalently, on L 2 estimates of increments of the Green function of the Laplace equation.Keywords: Systems of stochastic Poisson equations; hitting probabilities, capacity; Hausdorff measure. AMS Subject Classification. Primary: 60H15, 60G15, 60J45; Secondary: 60G60, 60H07. IntroductionLet D be a bounded domain of R k , k = 1, 2, 3, for which the divergence theorem holds. Consider the following system of elliptic stochastic partial differential equations,,j≤d is a non-singular matrix with real-valued entries.The main motivation of this paper has been to find upper and lower bounds for the hitting probabilities P{u(I) ∩ A = ∅}, I ⊂ D, A ⊂ R d , in terms of the Hausdorff measure and the capacity of the set A, respectively. This is a fundamental problem in probabilistic potential theory that, in the context of stochastic partial differential equations, has been extensively studied for the stochastic heat and wave equations. We refer the reader to , for numerical approximations, among others. We observe that in (1), the stochastic forcing is an additive noise. Therefore, in the integral formulation of the system given in (6), the stochastic integral term contains a deterministic integrand and defines a Gaussian process. Since there is no time parameter in (1), considering a multiplicative noise would require a choice of anticipating stochastic integral in (6). For example, one could take the Skorohod integral. This would make the objective of this article difficult and rather speculative.The content of the paper is as follows. In Section 2, we prove the existence and uniqueness of a solution to (1), when the function f satisfies a monotonicity condition (see Theorem 2.2). This is a d-dimensional stochastic process indexed bȳ D, the closure of the domain D, with continuous sample paths and vanishing at the boundary of D, a.s. The proof applies standard methods of the theory of nonlinear monotone operators. In order to make the article self-contained, we include the details of the proof. In Section 3, we prove some properties of the solution to (1). With the a priori bound proved in Proposition 3.1, we prove that the solution lies in L p (Ω; R d ), uniformly in x ∈ D. Moreover, by using estimates of increments of the L 2 -norms of the Green function, we prove that the sample paths of the solution are Hölder continuous. Section 3 is devoted to study some aspects of the law of the solution. The integral formulation (6) suggests that the law of u is obtained from a Gaussian process by a non adapted ...

show abstract

“…The second metric has played an important role in the studying the anisotropic Gaussian fields and the selfsimilar random fields (see [14]). …”

Section: Definitionmentioning

confidence: 99%

Probability distributions of extremes of self-similar Gaussian random fields

Kozachenko¹,

Makogin²

2014

Journal of Classical Analysis

View full text Add to dashboard Cite

We have obtained some upper bounds for the probability distribution of extremes of a selfsimilar Gaussian random field with stationary rectangular increments that are defined on the compact spaces. The probability distributions of extremes for the normalized self-similar Gaussian random fields with stationary rectangular increments defined in R 2 + have been presented. In our work we have used the techniques developed for the self-similar fields and based on the classical series analysis of the maximal probability bounding from below for the Gaussian fields.

show abstract

Sample Path Properties of Anisotropic Gaussian Random Fields

Cited by 140 publications

References 97 publications

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

A Turning-Band Method for the Simulation of Anisotropic Fractional Brownian Fields

Systems of stochastic Poisson equations: Hitting probabilities

Probability distributions of extremes of self-similar Gaussian random fields

Contact Info

Product

Resources

About