We prove a sufficient stochastic maximum principle for continuous-state branching processes with immigration, so-called CBI processes, which possess the self-exciting property, meaning that jump clustering effects can be captured by this class of stochastic processes. In our setup, the control process is contained in the drift of the state equation, while the stochastic drivers of the state process are given by Gaussian and Poisson random fields. We apply the result to several stochastic control problems stemming from finance (optimal consumption rates), epidemiology (optimal control of infection numbers), and the medical sciences (optimal drug dosing).

In this work, we revisit the theory of considering diffusion processes as 1-currents defined by line integrals along their trajectories. Specifically, we will study diffusion processes defined by solutions to stochastic differential equations, with a particular focus on foliated Brownian motion. We will explore the connection between this theory, the Godbillon-Vey class, and the entropy of foliations on compact Riemannian manifolds.

In this article, we introduce the Itô integral and arbitrary time scales, and we deduce some of its properties. The Itô product formula and Itô formula are derived on arbitrary time scales. As their applications, linear first-order stochastic dynamic equations on time scales are investigated.

We study the two-dimensional electroconvection model driven by transport noise with particular emphasis on its long-term dynamics. Through the framework of Lyons’ rough path analysis, we establish that pathwise solutions exist for this system and that it produces a continuous stochastic flow in the controlled path space. In the absence of solution uniqueness, we employ semiflow selection techniques to identify a solution system where each initial condition corresponds to a unique trajectory while preserving the conventional semiflow characteristics under proper rough path translations. Moreover, when the driving rough path satisfies appropriate regularity conditions, we prove that this electroconvection system forms a well-defined measurable random dynamical system.

We introduce the concept of dissipative measure-valued martingale solutions for the stochastic compressible Navier–Stokes equations. These solutions are probabilistically weak, as they incorporate both the underlying Wiener process and the probability space as intrinsic components of the formulation. For the stochastic compressible Navier–Stokes system, we further derive a relative energy inequality, which serves as the foundation for establishing a path-wise weak–strong uniqueness principle. We also look at the inviscid–incompressible limit of the underlying system of equations using the relative energy inequality.

In this article, we are concerned with distribution-dependent stochastic differential equations driven by G-Brownian motion (in short form, distribution-dependent G-SDEs). We propose a non-Lipschitz condition for the coefficients under which we can establish existence and uniqueness of solutions to the distribution-dependent G-SDEs by utilizing Picard’s iteration. Furthermore, we derive moment estimates for solutions of the distribution- dependent G-SDEs. Finally, we prove the Euler-Maruyama convergence theorem under this non-Lipschitz condition.

Different methods are used to evaluate products and conditions with antibiofilm or biofilm detachment properties. Using the super biofilm producer Cobetia marina strain MM1IDA2H-1 as a model, our aim is to provide statistical support for biological hypotheses on biofilm evaluation across wide ranges of temperatures in which bacteria can grow. The analytical biofilm formation was defined as the ratio of bacterial adhesion to an organic surface and the amount of bacterial biomass used in the assay (absorbance at 540 nm/ absorbance at 600 nm) as our dependent random variable. Biofilms were measured across a wide range of growth temperatures during the exponential growth phase. A non-standard ANOVA-like approach to assessing heterogeneity across temperatures is developed to explain biological behaviors related to biofilm formation, biomass, and temperatures. We developed an exponential family-based ANOVA-like analysis with a novel statistical approach that has been developed to provide deep biological insights.

. An implicit Euler-Maruyama (EM) scheme for Caputo stochastic fractional delay differential equations (CSFDDE) is considered. Under standard regularity conditions on the coefficients, this scheme converges at a certain rate in L p -norm for p≥2. The scheme also strongly converges under local Lipschitz and linear growth assumptions, along with bounded p-th moments of the exact and approximate solutions. The same results are obtained as a direct corollary when applied to Caputo stochastic fractional differential equations (CSFDE) with less restrictive hypotheses. Numerical simulations are given to support our theoretical results. In addition, it is shown that solutions of CSFDDE have well-posedness and regularity.

In this article, we study the asymptotic behavior of stochastic 3D Navier-Stokes-Voigt equations with infinite delay. We focus on the stability properties of stationary solutions. We first prove the local stability of stationary solutions for general delay terms by using a direct method and then apply the obtained results to two kinds of infinite delay. Then, the exponential stability of stationary solutions is established in the case of unbounded distributed delay. Moreover, we investigate the polynomial stability of stationary solutions in the case of proportional delay. Finally, the almost sure exponential/polynomial stability of the stationary solutions is also studied.

Background : As digital data in multiple languages increases, analyzing multilingual legal texts has become increasingly important. However, existing models often struggle with the complexity of legal language and cross-lingual understanding, making accurate classification and interpretation challenging. Objective : The study aims to develop a natural language processing multilingual legal data model (NMLDM) to address domain-specific legal terminology, adversarial and prompt injection attacks, data sparsity, and the interpretability and transparency of legal text classification. Methodology : The proposed NMLDM introduces XRXNet, a hybrid model combining XLM-R and XLNet transformer-based language models for robust multilingual legal text analysis. NMLDM integrates data collection, executes preprocessing with an improvised mBERT tokenizer, and performs feature extraction using legal-specific lexicons and linguistic features. XRXNet combines cross-lingual understanding and long-range dependency modeling to classify legal documents accurately while mitigating adversarial and prompt injection attacks. Results : The proposed framework shows exceptional performance across several metrics after evaluating NMLDM with existing models. The simulation results show that NMLDM attained an accuracy of 99.66%, highlighting its competence. Thus, the proposed model shows promise for analyzing multilingual legal documents.