Dissipative Stochastic Evolution Equations Driven by General Gaussian and Non-Gaussian Noise

In this work, we establish a new concept of pseudo almost periodic processes in p-th mean sense using the measure theory. We use the μ-ergodic process to define the spaces of μ-pseudo almost periodic process in the p-th mean sense. We establish many interesting results on the functional space of such processes like completeness and composition theorems. The main objective of this paper is to use those results and some stochastic analysis approaches to study the existence, the uniqueness and the global attractiveness for a μ-pseudo almost periodic mild solution to a class of abstract stochastic evolution equations driven by fractional Brownian motion. We provide an example to illustrate our results.

Noise in an underwater acoustic (UWA) channel does not necessarily follow Gaussian statistics, especially in a shallow water environment which is dominated by impulsive noise sources. However most of the receivers are designed with the assumption that the channel noise is additive white Gaussian (AWGN). Such receivers may not be the optimum ones to deal with the non-Gaussian UWA noise. Several non-Gaussian statistics have been proposed in previous literature to model UWA noise among which the Generalized Gaussian (GG) model has been very popular due to its flexible parametric form. However, to the best of our knowledge, no analytical error analysis with the assumption of channel noise being generalized Gaussian has yet been reported. In this paper, we derive the analytical expression for probability of error considering the presence of GG noise in the UWA channel. We also try to study the performance of an AWGN receiver in the presence of non-Gaussian noise. We consider three different communication systems using BPSK, QPSK and M-ary PAM constellations, and observe, for all three of them, that if noise kurtosis is greater than the Gaussian kurtosis, the system performance degrades. Thus here is an initial phase of designing UWA system with GG noise assumptions instead of the traditional Gaussian receiver. It is still a matter of argument, due to all possible redesign complexities, whether it is worth taking all the redesign complexities or the performance degradation can be tolerated as per the requirements of system application.

We establish weak convergence rates for noise discretizations of a wide class of stochastic evolution equations with non-regularizing semigroups and additive or multiplicative noise. This class covers the nonlinear stochastic wave, HJMM, stochastic Schr\"odinger and linearized stochastic Korteweg-de Vries equation. For several important equations, including the stochastic wave equation, previous methods give only suboptimal rates, whereas our rates are essentially sharp.

A class of stochastic evolution equations with additive noise and weakly continuous drift is considered. First, regularity properties of the corresponding Ornstein-Uhlenbeck transition semigroupRt are obtained. We show thatRt is a compactC0-semigroup in all Sobolev spacesWn,p which are built on its invariant measure μ. Then we show the existence, uniqueness, compactness and smoothing properties of the transition semigroup for semilinear equations inLp(μ) spaces and spacesW1,p. As a consequence we prove the uniquencess of martingale solutions to the stochastic equation and the existence of a unique invariant measure equivalent to μ. It is shown also that the density of this measure with respect to μ is inLp(μ) for allp≧1.

The aim of this paper is to investigate the almost sure stability with a certain rate function λ(t) for a class of stochastic evolution equations in infinite dimensional spaces under various sufficient conditions. The results obtained here include exponential and polynomial stability as special cases. Much more refined sufficient conditions than the usual ones, for example, those described in [14], are obtained under our framework by the method of Liapunov functions. Two examples are given to illustrate our theory.

Stochastic evolution equations with general white noise disturbance

Upper and lower bounds on the capacity and minimum energy-per-bit for general additive white Gaussian noise (AWGN) and frequency-division AWGN (FD-AWGN) relay channel models are established. First, the max-flow min-cut bound and the generalized block-Markov coding scheme are used to derive upper and lower bounds on capacity. These bounds are never tight for the general AWGN model and are tight only under certain conditions for the FD-AWGN model. Two coding schemes that do not require the relay to decode any part of the message are then investigated. First, it is shown that the "side-information coding scheme" can outperform the block-Markov coding scheme. It is also shown that the achievable rate of the side-information coding scheme can be improved via time sharing. In the second scheme, the relaying functions are restricted to be linear. The problem is reduced to a "single-letter" nonconvex optimization problem for the FD-AWGN model. The paper also establishes a relationship between the minimum energy-per-bit and capacity of the AWGN relay channel. This relationship together with the lower and upper bounds on capacity are used to establish corresponding lower and upper bounds on the minimum energy-per-bit that do not differ by more than a factor of 1.45 for the FD-AWGN relay channel model and 1.7 for the general AWGN model.

Spectrum sensing is an indispensable core part of cognitive radio dynamic spectrum access (DSA) and a key approach to alleviating spectrum scarcity in the Internet of Things (IoT). The key issue in practical IoT networks is robust sensing under the coexistence of low signal-to-noise ratios (SNRs) and non-Gaussian impulsive noise, where observations may be distorted differently across feature modalities, making conventional fusion unstable and degrading detection reliability. To address this challenge, the generalized Gaussian distribution (GGD) is adopted as the noise model, and a multimodal fusion framework termed BCAM-Net (bidirectional cross-attention multimodal network) is proposed. BCAM-Net adopts a parallel dual-branch architecture: a time-frequency branch that leverages the continuous wavelet transform (CWT) to extract time-frequency representations, and a temporal branch that learns long-range dependencies from raw signals. BCAM-Net utilizes a bidirectional cross-attention mechanism to achieve deep alignment and mutual calibration of temporal and time-frequency features, generating a fused representation that is highly robust to complex noise. Simulation results show that, under GGD noise with shape parameter β=0.5, BCAM-Net achieves high detection probabilities in the low-SNR regime and outperforms representative baselines. At a false alarm probability Pf=0.1 and SNR of −14 dB, it attains a detection probability of 0.9020, exceeding the CNN-Transformer, WT-ResNet, TFCFN, and conventional CNN benchmarks by 5.75%, 6.98%, 33.3%, and 21.1%, respectively. These results indicate that BCAM-Net can effectively improve spectrum sensing performance in low-SNR impulsive-noise scenarios, and provides a lightweight, high-performance solution for practical cognitive radio spectrum sensing.

We obtain necessary and sufficient conditions for equivalence of law for linear stochastic evolution equations driven by a general Gaussian noise by identifying the suitable space of controls for the corresponding deterministic control problem. This result is applied to semilinear (reaction-diffusion) equations driven by a fractional Brownian motion. We establish the equivalence of continuous dependence of laws of solutions to semilinear equations on the initial datum in the topology of pointwise convergence of measures and null controllability for the corresponding deterministic control problem.

Hybrid Taylor and block-pulse functions operational matrix algorithm and its application to obtain the approximate solution of stochastic evolution equation driven by fractional Brownian motion

Much attention has been paid to the arbitrage opportunities in the Black–Scholes model when it is driven by fractional Brownian motions. It is natural to ask whether there exists arbitrage or not when we focus on other fractional processes, such as the Hermite process. We set forth an approximation of the Hermite Black–Scholes model by random walks in the Skorokhod topology, and apply the Donsker type approximation to the Hermite process as the Hurst index is greater than 12. We find that the binary model approximation of the Black–Scholes model driven by Hermite processes also admits arbitrage opportunities. Several numerical examples of the Hermite binomial model are presented as demonstration. Moreover, we provide an option pricing model when the geometric Hermite processes drives the price fluctuations of the underlying asset.

Asymptotic behavior of the quadratic variation of the sum of two Hermite processes of consecutive orders

This paper considers a problem of communication over an additive noise channel where the noise is distributed according to a Generalized Gaussian (GG) distribution. In the first part of the paper, a number of properties of the family of GG distributions are derived which are of independent interest. For example, considerable attention is given to the properties of the characteristic function of the GG distribution. In the second part of the paper, the capacity of an additive noise channel with GG noise is considered under p-th absolute moment constraints. It is shown that, even though Shannon's upper bound is achievable in some instances, in general such achievability is not possible. Moreover, it is shown that discrete inputs can achieve capacity within a constant gap or full degree of freedom for any p-th absolute moment constraint. Following the seminal work of Smith, the paper also gives a condition under which discrete inputs are exactly optimal.

We address the issue of detecting hidden periodicity when the signal exhibits periodic correlation, but is additionally affected by non-Gaussian noise with unknown characteristics. This scenario is common in various applications. The conventional approach for identifying periodically correlated (PC) behavior involves the frequency domain-based analysis. In our investigation, we also employ such an approach; however, we use a robust version of the discrete Fourier transform incorporating the Huber function-based M-estimation, unlike the classical algorithm. Building upon this approach, we propose robust coherent and incoherent statistics originally designed to identify hidden periodicity in pure PC models. The novelty of this paper lies in introducing robust coherent and incoherent statistics through the application of the robust discrete Fourier transform in classical algorithms and proposing a new technique for period estimation based on the proposed methodology. We explore two types of PC models and two types of additive noise, resulting in PC signals disturbed by non-Gaussian additive noise. Detecting hidden periodicity in such cases proves to be significantly more challenging than in classical scenarios. Through Monte Carlo simulations, we demonstrate the effectiveness of the proposed robust approaches and their superiority over classical. To further substantiate our findings, we analyze three datasets in which hidden periodicity had previously been confirmed in the literature. Among them, two datasets correspond to the condition monitoring area, being a main motivation of our research.

In this paper, we study the asymptotic behavior of the maximum queue length in a fluid queue driven by a self-similar Gaussian process with non-stationary increments. We derive almost sure and in-probability convergence of the normalized maximum queue length, described by a function [Formula: see text], where [Formula: see text] is the Hurst parameter. We obtain sharp tail bounds for the distribution of the queue length, using Malliavin calculus, which explicitly depend on [Formula: see text] and the drift parameter [Formula: see text]. Our results generalize classical results on fractional Brownian motion (fBm) to broader classes such as sub-fractional Brownian motion (sub-fBm) and bi-fractional Brownian motion (bi-fBm). The key techniques introduced in this framework include Gaussian supremum localization, path-penalty large deviations, and metric entropy. Systems that exhibit long-range dependence and self-similarity are developed based on these results.