Fractional Brownian Sheet Research Articles

Hölder continuity of stochastic heat equation with rough Gaussian noise

In this paper, we consider the stochastic heat equation ∂tu(t,x)=12Δu(t,x)+u(t,x)♢Ẇ(t,x), where Ẇ(t,x) is the formal derivative of a fractional Brownian sheet with H0>1/2 and 0<Hi<1fori=1,…,d. We obtain the Hölder continuity of the solution in time and space.

Spatial modeling and monitoring considering long-range dependence

Spatial modeling and monitoring are critical in geometric characterization and quality control of material/product surfaces. With advances in metrology technology, a long-range dependence (LRD) effect has recently been detected in spatial data over different fields. The spatial LRD refers to a type of dependence that decays slowly over the distance with heavy tails and non-summable autocovariances so that the correlation is high among surface measurements across long spatial distances. Physically, the spatial LRD effect can be caused by specific spatial patterns such as certain material textures, surface profiles, or manufacturing defects. In literature, although various Markovian and non-Markovian spatial models have been proposed to study material surfaces, none of them has yet considered the LRD effect, which can lead to inefficient surface characterization and inaccurate surface quality control. To overcome the challenge, in this article, we first propose a novel spatial model that can capture the spatial LRD on material surfaces. Both isotropic and anisotropic scenarios of the model are developed based on the Lévy fractional Brownian random field and the fractional Brownian sheet, respectively. Subsequently, based on the proposed spatial model we develop an LRD-integrated quality control framework to monitor surface quality via generalized likelihood ratio test. Comprehensive simulation studies and a real case study using images of wood surfaces are conducted to validate the proposed approach. Results show that the proposed model that integrates LRD significantly outperforms multiple existing models in anomaly detection, and traditional models mis-detect out-of-control surfaces when the spatial LRD actually presents.

Stability results for stochastic differential equations driven by an additive fractional Brownian sheet

Abstract The aim of the present paper is to establish some strong stability results for solutions of stochastic differential equations driven by a fractional Brownian sheet with Hurst parameters H , H ′ ∈ ( 0 , 1 ) {H,H^{\prime}\in(0,1)} for which pathwise uniqueness holds.

An Efficient Numerical Algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

An Efficient Numerical Algorithm for the Model Describing the Competition Between Super- and Sub-diffusions Driven by Fractional Brownian Sheet Noise

Aggregation of network traffic and anisotropic scaling of random fields

We discuss joint spatial-temporal scaling limits of sums A λ , γ A_{\lambda ,\gamma } (indexed by ( x , y ) ∈ R + 2 (x,y) \in \mathbb {R}^2_+ ) of large number O ( λ γ ) O(\lambda ^{\gamma }) of independent copies of integrated input process X = { X ( t ) , t ∈ R } X = \{X(t), t \in \mathbb {R}\} at time scale λ \lambda , for any given γ &gt; 0 \gamma &gt;0 . We consider two classes of inputs X X : (I) Poisson shot-noise with (random) pulse process, and (II) regenerative process with random pulse process and regeneration times following a heavy-tailed stationary renewal process. The above classes include several queueing and network traffic models for which joint spatial-temporal limits were previously discussed in the literature. In both cases (I) and (II) we find simple conditions on the input process in order that the normalized random fields A λ , γ A_{\lambda ,\gamma } tend to an α \alpha -stable Lévy sheet ( 1 &gt; α &gt; 2 ) (1&gt; \alpha &gt;2) if γ &gt; γ 0 \gamma &gt; \gamma _0 , and to a fractional Brownian sheet if γ &gt; γ 0 \gamma &gt; \gamma _0 , for some γ 0 &gt; 0 \gamma _0&gt;0 . We also prove an ‘intermediate’ limit for γ = γ 0 \gamma = \gamma _0 . Our results extend the previous works of R. Gaigalas and I. Kaj [Bernoulli 9 (2003), no. 4, 671–703] and T. Mikosch, S. Resnick, H. Rootzén and A. Stegeman [Ann. Appl. Probab. 12 (2002), no. 1, 23–68] and other papers to more general and new input processes.

Non-linear Young equations in the plane and pathwise regularization by noise for the stochastic wave equation

We study pathwise regularization by noise for equations on the plane in the spirit of the framework outlined by Catellier and Gubinelli (Stoch Process Appl 126(8):2323–2366, 2016). To this end, we extend the notion of non-linear Young equations to a two dimensional domain and prove existence and uniqueness of such equations. This concept is then used in order to prove regularization by noise for stochastic equations on the plane. The statement of regularization by noise is formulated in terms of the regularity of the local time associated to the perturbing stochastic field. For this, we provide two quantified example: a fractional Brownian sheet and the sum of two one-parameter fractional Brownian motions. As a further illustration of our regularization results, we also prove well-posedness of a 1D non-linear wave equation with a noisy boundary given by fractional Brownian motions. A discussion of open problems and further investigations is provided.

ON THE QHASI CLASS AND ITS EXTENSION TO SOME GAUSSIAN SHEETS

Introduced in 2018 the generalized bifractional Brownian motion is considered as an element of the quasi-helix with approximately stationary increment class of real centered Gaussian processes conditioning by parameters. This paper proves that the generalized bifractional Brownian motion is an element of the above mentioned class with no condition on parameters. The quasi-helix with approximately stationary increment class of real centered Gaussian processes is extended to two-dimensional processes as the fractional Brownian sheet, the sub-fractional Brownian sheet, and the bifractional Brownian sheet. This generalized presentation of the class of stochastic processes is used to augment the training samples for generative adversarial networks in computer vision problem.

Optimal Hölder continuity and hitting probabilities for SPDEs with rough fractional noises

We investigate the optimal Hölder continuity and hitting probabilities for systems of stochastic heat equations and stochastic wave equations driven by an additive fractional Brownian sheet with temporal index 12 and spatial index H≤1/2. Using stochastic calculus for fractional Brownian motion, we prove that these systems are well-posed and the solutions are Hölder continuous. Furthermore, the optimal Hölder exponents are obtained, which is the first result, as far as we knew, on the optimal Hölder continuity of SHEs and SWEs driven by fractional Brownian sheet that is rough in space. Based on this sharp regularity, we obtain lower and upper bounds of hitting probabilities of the solutions in terms of Bessel–Riesz capacity and Hausdorff measure, respectively.

The Hausdorff measure of the range and level sets of Gaussian random fields with sectorial local nondeterminism

We determine the exact Hausdorff measure functions for the range and level sets of a class of Gaussian random fields satisfying sectorial local nondeterminism and other assumptions. We also establish a Chung-type law of the iterated logarithm. The results can be applied to the Brownian sheet, fractional Brownian sheets whose Hurst indices are the same in all directions, and systems of linear stochastic wave equations in one spatial dimension driven by space–time white noise or colored noise.

The Persistence Exponents of Gaussian Random Fields Connected by the Lamperti Transform

The (fractional) Brownian sheet is the simplest example of a Gaussian random field $$X$$ whose covariance is the tensor product of a finite number (d) of nonnegative correlation functions of self- similar Gaussian processes. We consider the homogeneous Gaussian field $$Y$$ obtained by applying to X the exponential change of time (more precisely, the Lamperti transform). Under some assumptions, we prove the existence of the persistence exponents for both fields, $$X$$ and $$Y$$ , and find the relation between them. The exponent for any random function $$Z$$ is $$\psi (T)$$ if lim $$\ln P(Z({\mathbf{t}}) \le ,{\mathbf{t}} \in TG)/\psi (T) = - 1$$ , $$T > > {1}$$ , where $$G$$ is a d-dimensional region containing 0, and $$T$$ is a similarity coefficient. The considered problem was raised by Li and Shao [Ann. Probab. 32:1, 2004] and originally concerned the Brownian sheet.

Reflecting time-Space Gaussian random field on compact Riemannian manifold and excursion probability

Motivated from measuring the performance of quantum computing and storage systems together with nanorheology over different shapes of devices, we introduce a reflecting time-space Gaussian random field (RGRF) on a general -parameter compact Riemannian manifold to model the quantum particle movement dynamics. The main task in studying the RGRF is to estimate an asymptotic upper bound of its excursion probability, which asymptotically converges to zero. In doing so, we establish a relationship between the RGRF and its netput Gaussian random field (GRF) via a Skorohod mapping. Then, as an intermediate step, we approximate the excursion probability of the GRF by constructing a chart oriented homeomorphism mapping between a flat manifold and a general compact Riemannian manifold. The netput GRF can be either isotropic or anisotropic. Case studies in terms of anisotropic GRFs over a flat manifold such as normalized fractional Brownian sheets (FBSs) and those over a sphere such as normalized anisotropic fractional spherical Brownian motions (FSBMs) are also presented, where an α-FSBM is introduced and its existence is proved for some constant .

Recent advances on eigenvalues of matrix-valued stochastic processes

Since the introduction of Dyson’s Brownian motion in early 1960s, there have been a lot of developments in the investigation of stochastic processes on the space of Hermitian matrices. Their properties, especially, the properties of their eigenvalues have been studied in great detail. In particular, the limiting behaviours of the eigenvalues are found when the dimension of the matrix space tends to infinity, which connects with random matrix theory. This survey reviews a selection of results on the eigenvalues of stochastic processes from the literature of the past three decades. For most recent variations of such processes, such as matrix-valued processes driven by fractional Brownian motion or Brownian sheet, the eigenvalues of them are also discussed in this survey. In the end, some open problems in the area are also proposed.

Multiple points of Gaussian random fields

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.

Harnack-type inequality for linear fractional stochastic equations

Abstract Using the coupling method and Girsanov theorem, we prove a Harnack-type inequality for a stochastic differential equation with non-Lipschitz drift and driven by a fractional Brownian motion with Hurst parameter H &lt; 1 2 {H&lt;\frac{1}{2}} . We also investigate this inequality for a stochastic differential equation driven by an additive fractional Brownian sheet.

Sample Path Properties of Generalized Random Sheets with Operator Scaling

We consider operator scaling alpha -stable random sheets, which were introduced in Hoffmann (Operator scaling stable random sheets with application to binary mixtures. Dissertation Universität Siegen, 2011). The idea behind such fields is to combine the properties of operator scaling alpha -stable random fields introduced in Biermé et al. (Stoch Proc Appl 117(3):312–332, 2007) and fractional Brownian sheets introduced in Kamont (Probab Math Stat 16:85–98, 1996). We establish a general uniform modulus of continuity of such fields in terms of the polar coordinates introduced in Biermé et al. (2007). Based on this, we determine the box-counting dimension and the Hausdorff dimension of the graph of a trajectory over a non-degenerate cube I subset {mathbb {R}}^d.

Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet

We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space ℝ 1+1 with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.

On a mixed fractional Burgers type equation with polynomial nonlinearity and perturbed by fractional Brownian sheet

In this paper, we discuss a class of stochastic nonlinear partial differential equation of Burgers type driven by pseudo differential operator for and fractional Brownian sheet. The existence and uniqueness of an Lp-valued (local) solution is established for the initial valued problem to the equation.

Pointwise Rectangular Lipschitz Regularities for Fractional Brownian Sheets and Some Sierpinski Selfsimilar Functions

We consider pointwise rectangular Lipschitz regularity and pointwise level coordinate axes Lipschitz regularities for continuous functions f on the unit cube I 2 in R 2 . Firstly, we provide characterizations by simple estimates on the decay rate of the coefficients (resp. leaders) of the expansion of f in the rectangular Schauder system, near the point considered. We deduce that pointwise rectangular Lipschitz regularity yields pointwise level coordinate axes Lipschitz regularities. As an application, we refine earlier results in Ayache et al. (Drap brownien fractionnaire. Potential Anal. 2002, 17, 31–43) and Kamont (On the fractional anisotropic Wiener field. Probab. Math. Statist. 1996, 16, 85–98), where uniform rectangular Lipschitz regularity of the trajectories of the fractional Brownian sheet over the total I 2 (or any cube) was considered. Actually, we prove that fractional Brownian sheets are pointwise rectangular and level coordinate axes monofractal. On the opposite, we construct a class of Sierpinski selfsimilar functions that are pointwise rectangular and level coordinate axes multifractal.

Multi-scale invariant fields: estimation and prediction

Extending the concept of multi-selfsimilar random field, we study multi-scale invariant (MSI) fields with component-wise discrete scale invariant properties. Assuming scale parameters as λ i > 1, i = 1, …, d and the parameter space as (1, ∞) d , the first scale rectangle is referred to the rectangle (1, λ 1) ×⋯× (1, λ d ). We show that the covariance functions of the sampled Markov MSI field are characterized by the variances and covariances of samples inside the first scale rectangle. As an example of an MSI field, a two-dimensional simple fractional Brownian sheet (sfBss) is demonstrated. In addition, real data for precipitation in the area of Brisbane, Australia, covering a period of two days (25 and 26 January 2013), are examined. We show that precipitation in this area demonstrates MSI properties, and estimate it as a simple MSI field with stationary increments inside scale intervals. This structure enables us to predict the precipitation in terms of surface and time. We apply the mean absolute percentage error as a measure for the accuracy of the predictions.

Asymptotic behavior for quadratic variations of non-Gaussian multiparameter Hermite random fields

Let Zt q,H t∈[0,1]d denote a d-parameter Hermite random field of order q ≥ 1 and self-similarity parameter H = H₁, . . . ,Hd ∈ ½, 1d. This process is H-self-similar, has stationary increments and exhibits long-range dependence. Particular examples include fractional Brownian motion q = 1, d = 1, fractional Brownian sheet q = 1, d ≥ 2, the Rosenblatt process q = 2, d = 1 as well as the Rosenblatt sheet q = 2, d ≥ 2. For any q ≥ 2, d ≥ 1 and H ∈ ½, 1d we show in this paper that a proper renormalization of the quadratic variation of Zq,H converges in L2Ω to a standard d-parameter Rosenblatt random variable with self-similarity index H' = 1 + 2H − 2/q.