We introduce a flexible methodology for modelling regular spatial point patterns using hyperuniform perturbed lattices. We show that, under suitable mixing conditions on the displacement field, lattices perturbed by stationary random fields are hyperuniform in arbitrary dimension. In particular, Gaussian perturbations with absolutely summable covariances yield class-I hyperuniform point processes. We further derive an explicit formula for the K-function of Gaussian models, which enables efficient parameter estimation via the minimum contrast method. The proposed framework provides a computationally tractable alternative to classical Gibbs models for repulsive data. The methodology is illustrated on three-dimensional data describing grain centers in a polycrystalline nickel-titanium alloy.
The non-parametric estimation of a non-linear reaction function in a reaction-diffusion stochastic partial differential equation (SPDE) is discussed. The estimation error can be bounded in terms of the diffusivity and the noise level, which obey a realistic coupling and tend to zero. The estimator achieves the minimax-optimal convergence rate and exhibits asymptotic normality due to the spatial ergodicity of the SPDE. Analysing the estimation error requires the control of spatial averages of non-linear transformations of the SPDE, and combines the Clark-Ocone formula from Malliavin calculus with the Markovianity of the SPDE. In contrast to previous results, the obtained variance bound is uniform in the Lipschitz-constant of the transformation.
This paper considers urn models, where balls are assigned to bins successively with non-linear feedback. The regime of super-linear feedback is known to lead to strong monopoly, where all but finitely many balls are assigned to one bin. We derive the tail distribution of balls in losing bins and the time to monopoly, extending previous results for two bins to an arbitrary number with general feedback. We use a novel approach based on the tail distribution of explosive birth processes and cover also dependencies among losers. Perhaps surprisingly, losing bins with weak feedback are more likely to win many steps than those with strong feedback.
We study a class of reaction-diffusion systems based on the symmetric partial exclusion process on a one-dimensional discrete torus, where each site can accommodate up to two particles. For this system, we rigorously derive both the hydrodynamic limit and the hydrostatic limit, and provide quantitative estimates on the law of large numbers under the non-equilibrium steady state. By employing Yau’s relative entropy method and related inequalities, we further demonstrate that the density fluctuations around the hydrostatic limit are described by an infinite-dimensional Ornstein-Uhlenbeck process. This result establishes a central limit theorem for the density of particles under the non-equilibrium stationary states, characterizing the asymptotic behavior of fluctuations in the macroscopic regime.
This paper investigates near-optimal controls for a class of fully-coupled stochastic functional differential equations (SFDEs) with two-time scales, in which all coefficients depend on the segment processes of both the fast and slow components. The underlying problem is to minimize a cost functional subject to the SFDEs mentioned above. Our primary tools are probabilistic methods, in particular, weak convergence methods. The main challenge lies in the complete coupling of the fast and slow processes through their segment processes along with the resulting effects on the tightness of the segment process of the slow component. To address these challenges, the boundedness and H & ouml;lder continuity for such segment process are established in a continuous function space. In addition, it is also shown that the segment process of a fixed-x SFDE is uniformly bounded, exponentially ergodic, and continuously dependent on the parameter x. By using the relaxed control representation and the martingale problem formulation, it is proved that the slow process and the corresponding value function in the original problem converge to that of a limit problem, where the coefficients of the limit problem are obtained by averaging coefficients of the original problem with respect to the invariant measure of the fixed-x equation. Finally, by solving the optimal control problem for the limit problem, a practically useful control for the original system is constructed. Such constructed controls are shown to be nearly optimal for the original problem.
This paper investigates the stochastic heat equation driven by Gaussian noise d/dt (u(t, x)) = Au(1, x) + u(t, x) W(t, x), where W(t,x) is the fractional Brownian sheet with temporal Hurst parameter Ho & euro; (1/4,1/2] and spatial Hurst parameters (HH) (0,1), and W(t, x) = W(t,x). The symbol "o" denotes the Wick product. We establish distinct suf-ficient conditions and necessary conditions for the existence and uniqueness of the solution. We also derive spatial H & ouml;lder continuity and spatial asymptotic upper bound of the solution.
Threshold exceedances of stochastic processes in space and time often appear to be more localized the more extreme they are. While classical regularly varying stochastic processes cannot model this effect, we introduce an adapted version of regular variation, where a suitable domain-scaling can be incorporated to accommodate this behaviour. Our theory is inspired by the triangular array convergence of domain-scaled maxima of Gaussian processes to a Brown-Resnick process and turns out to be natural in this context. We study key properties of the resulting tail approximating process and demonstrate its ability to approximate conditional exceedance probabilities of Gaussian processes. Mathematical convenience arises from the recently rediscovered concept of vague convergence based on boundedness.
We study a multilayer SIR model with two levels of mixing, namely a global level which is uniformly mixing, and a local level with two layers distinguishing household and workplace contacts, respectively. We establish the large population convergence of the corresponding stochastic process. For this purpose, we use an individual-based model whose state space specifies the remaining infectious period length for each infected. This allows to deal with the natural correlation of the epidemic states of individuals whose household and workplace share a common infected. In a general setting where a non-exponential distribution of infectious periods may be considered, convergence to the unique deterministic solution of a measure-valued equation is obtained. In the particular case of exponentially distributed infectious periods, we show that it is possible to further reduce the obtained deterministic limit, leading to a closed, finite dimensional dynamical system capturing the epidemic dynamics. This model reduction subsequently is studied from a numerical point of view. We illustrate that the dynamical system derived from the large population approximation is a pertinent model reduction when compared to simulations of the stochastic process or to an alternative edge-based compartmental model, both in terms of accuracy and computational cost.
In this paper, we consider a stochastic version of a nonlocal Cahn-Hilliard-Navier-Stokes model with a singular potential in a bounded domain of R-d, d= 2,3. The system describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under random influences. We prove the existence of a global weak martingale solution. The proof relies on a splitting up method, which is a numerical scheme based on the method of fractional steps. In the two dimensional case, we also prove the pathwise uniqueness of the solution.
In this paper, we study the Dirichlet problem for the following integro-differential operator on bounded (not necessarily connected) open sets in & Ropf;(d) : Lu = 1 2 div(A(x)del(x)) + b(x) & sdot; del u(x) + 1/2 lim (epsilon -> 0) integral(d)({y is an element of & Ropf;) & ratio;|y-x|>epsilon} (u(y) - u(x))J(x, y)dy, where A(x) = (a(ij)(x))(1 <= i,j <= d) is a measurable d & times; d matrix-valued function on & Ropf;(d) that is uniformly elliptic and bounded, b is an & Ropf;(d) -valued function so that |b| (2) is in the Kato class Kd , and J(x, y) >= 0 is a measurable symmetric non-trivial kernel on & Ropf;(d) & times; & Ropf;(d) bounded above by c max{|x - y| -(d+alpha) , |x - y| -(d+beta)} for some 0 < beta <= alpha < 2 and c > 0. We show that there is a Feller process X on & Ropf;(d) having strong Feller property associated with the non-local operator L. We further show that for any bounded open set D in & Ropf;(d) that is regular with respect to the Feller process X and for every bounded function phi on D-c that is continuous on partial derivative D, the Dirichlet problem for L on D has a unique weak solution on & Ropf;(d) that is continuous on D. Moreover, the solution can be represented in terms of the associated Feller process
We investigate sparse and low-rank estimation methods for the intensity functions of multivariate Hawkes processes. While univariate Hawkes processes are approximated by integer-valued autoregressive models in the weak topology, we extend this approximation result to the multivariate setting. Building on this foundation, we develop new sparse and higher-order reduced-rank estimation procedures for multivariate Hawkes processes by leveraging methodology for multivariate integer-valued time series. In addition, we study a sparse weighted least squares estimator and establish error bounds for all proposed estimators. All theoretical results are derived by verifying novel moment and mixing conditions that ensure the applicability of concentration inequalities to weakly dependent data. Finally, we conduct an empirical study to assess the finite-sample performance of all estimation methods.
In this paper, we consider a stochastic version of a nonlocal Cahn-Hilliard-Navier-Stokes model with a singular potential in a bounded domain of Rd, d=2,3. The system describes the motion of an incompressible isothermal mixture of two (partially) immiscible fluids under random influences. We prove the existence of a global weak martingale solution. The proof relies on a splitting up method, which is a numerical scheme based on the method of fractional steps. In the two dimensional case, we also prove the pathwise uniqueness of the solution.
We establish a Perron-Frobenius type theorem for products of stationary and ergodic nonnegative random matrices, which provides a precise asymptotic description of their coefficients, and reveals a close connection with the ergodic theory for stationary and ergodic sequences of real random variables. As applications, we improve upon existing laws of large numbers for products of nonnegative random matrices by relaxing moment conditions. We then derive limit theorems for a multi-type branching process (Z(n)) in a stationary and ergodic environment, offering a precise description of the growth rate of the population size. In particular, we establish a Kesten-Stigum type theorem for the scalar product < Z(n) ,y > with any nonnegative vector y, revealing a key link between the branching process and products of random matrices. We also prove the convergence of the direction Z(n)/vertical bar vertical bar Z(n vertical bar vertical bar), a complement to the Kesten-Stigum type theorem, and establish new laws of large numbers and central limit theorems for < Z(n) ,y >.
We provide supplementary explanations for certain confusing and insufficiently detailed expressions. We also correct a number of confusing typographical errors.
This paper provides a bound for the supremum of sample averages over a class of functions for a general class of mixing stochastic processes with arbitrary mixing rates. Regardless of the speed of mixing, the bound is comprised of a concentration rate and a novel measure of complexity. The speed of mixing, however, affects the former quantity implying a phase transition. Fast mixing leads to the standard root-n concentration rate, while slow mixing leads to a slower concentration rate whose speed depends on the mixing structure. Our findings are applied to obtain new Glivenko-Cantelli type results.
We consider the problem of nonasymptotic mean estimation for heavy-tailed random variables taking values in a smooth Banach space. In particular, we extend and improve a simple truncation-based mean estimator by Catoni and Giulini. We show that this estimator enjoys finite-sample, high probability deviation bounds that hold uniformly over the class of distributions with a pth central moment bounded above by a known constant v > 0 for p is an element of (1, 2]. Our main contributions exploit connections between truncation-based mean estimation and the concentration of martingales with bounded increments in smooth Banach spaces. We prove two types of time-uniform bounds on the distance between the estimator and unknown mean: line-crossing inequalities, which can be optimized for a fixed sample size n, and iterated logarithm inequalities, which match the tightness of line-crossing inequalities at all points in time up to a doubly logarithmic factor in n. Our results do not depend on the dimension of the Banach space, hold under martingale dependence, and all employed constants are known and small.
In this paper, we study the solvability of multidimensional anticipated backward stochastic differential equations with quadratic growth. We establish three new results concerning both local and global solutions. More precisely, for the local solution with bounded terminal values, the generator (, , , +, + ) is of general growth in and +. For the global solution with bounded terminal values, the generator (,, , + , + ) is of skew sub-quadratic but is also "strictly and diagonally" quadratic growth in . For the global solution with unbounded terminal values, the generator (, , , +) is of diagonal quadratic growth in in the first case; the generator (, ) + L[(, , , +, +)] is of diagonal quadratic growth in and linear growth in + in the second case.
Spectrally correlated processes are harmonizable processes with spectral measures concentrated on a countable union of curves. Our study focuses on the spectral analysis of spectrally correlated processes, particularly in scenarios where these support curves take the form of lines with possible non-unit slopes. This subclass of spectrally correlated processes finds practical application, for example, in the problem of locating moving sources like aircraft, rockets, or other hostile jamming emitters emitting communication signals. Our research is focused on a frequency-smoothed periodogram along the support line as the spectral density function estimator. We derive its asymptotic distribution and based on that, we discuss the asymptotic properties of the introduced coherence estimator. Additionally, we formulate a consistent subsampling technique tailored for spectral analysis of spectrally correlated processes. We construct subsampling-based confidence intervals for spectral characteristics. Finally, we provide a numerical example illustrating a model relevant to applications in acoustics and communications.
