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Spontaneous gas-bubble nucleation in weak gas–liquid solutions has been a challenging topic in theory, experimentation, and computer simulations. In analogy with recent advances in crystallization and droplet formation studies, the diffusive-shielding stabilization and thermodynamic irreversibility of bulk nanobubble (bNB) mechanisms are revisited and deployed to characterize nucleation processes in a stochastic framework of computer experiments using the large-scale atomic/molecular massively parallel simulator code. Theoretical bases, assumptions, and limitations underlying the irreversibility hypothesis of bNBs, and their computational counterparts, are extensively described and illustrated. In essence, it is established that the irreversibility hypothesis can be numerically investigated by converging the system volume (due to the finiteness of interatomic forces) and the initial dissolved-gas concentration in the solution (due to the single-bNB limitation). Helium nucleation in liquid Pb17Li alloy is selected as a representative case study, where it exhibits typical characteristics of noble-gas/liquid-metal systems. The proposed framework lays down the bases on which the stability of gas-bNBs in weak and supersaturated gas–liquid solutions can be inferred and explained from a novel perspective. In essence, it stochastically marches toward a unique irreversible state along out-of-equilibrium nucleation/growth trajectories. Moreover, it does not attempt to characterize the interface or any interface-related properties, neither theoretically nor computationally. It was concluded that bNBs of a few tens of He-atoms are irreversible when dissolved-He concentrations in the weak gas–liquid solution are at least ∼50 and ∼105 mol m−3 at 600 and 1000 K (and ∼ 80 MPa), respectively, whereas classical molecular dynamics -estimated solubilities are at least two orders of magnitude smaller. [ABSTRACT FROM AUTHOR]

Age and longevity are key parameters for demography and life-history evolution of organisms. In clonal species, a widespread life history among animals, plants, macroalgae and fungi, the sexually produced offspring (genet) grows indeterminately by producing iterative modules, or ramets, and so obscure their age. Here we present a novel molecular clock based on the accumulation of fixed somatic genetic variation that segregates among ramets. Using a stochastic model, we demonstrate that the accumulation of fixed somatic genetic variation will approach linearity after a lag phase, and is determined by the mitotic mutation rate, without direct dependence on asexual generation time. The lag phase decreased with lower stem cell population size, number of founder cells for the formation of new modules, and the ratio of symmetric versus asymmetric cell divisions. We calibrated the somatic genetic clock on cultivated eelgrass Zostera marina genets (4 and 17 years respectively). In a global data set of 20 eelgrass populations, genet ages were up to 1,403 years. The somatic genetic clock is applicable to any multicellular clonal species where the number of founder cells is small, opening novel research avenues to study longevity and, hence, demography and population dynamics of clonal species.

We consider a single-machine scheduling problem with due-date assignment and stochastic processing times, where only the mean and support (i.e. an interval bounded with lower and upper values) of processing times are known to the decision maker. The objective is to jointly determine a scheduling policy and a set of due dates for all jobs, so as to minimise the total expected individually weighted costs of earliness, tardiness and due-date assignment. By identifying an upper bound with the robust optimisation approach and a lower bound, and using a linear function of them to approximate the studied objective function, we establish an approximated problem. Then, a branch-and-bound algorithm is proposed to find an optimal solution for the approximated problem. Finally, a series of computational experiments are conducted to examine the performance of problem approximation and two developed heuristic algorithms. HIGHLIGHTS Study due-date assignment scheduling problem with stochastic processing times. Apply the mean and support (interval data) to model stochastic processing times. Use a linear function of identified lower and upper bounds to make an approximation. Derive optimal due-date assignment and develop branch-and-bound algorithms. Evaluate the efficiency of branch-and-bound algorithm and heuristic algorithms. [ABSTRACT FROM AUTHOR]

Cellular signaling, crucial for biological processes like immune response and homeostasis, relies on specificity and fidelity in signal transduction to accurately respond to stimuli amidst biological noise. Kinetic proofreading (KPR) is a key mechanism enhancing signaling specificity through time-delayed steps, although its effectiveness is debated due to intrinsic noise potentially reducing signal fidelity. In this study, we reformulate the theory of kinetic proofreading (KPR) by convolving multiple intermediate states into a single state and then define an overall processing time required to traverse these states. This simplification allows us to succinctly describe kinetic proofreading in terms of a single waiting time parameter, facilitating a more direct evaluation and comparison of KPR performance across different biological contexts such as DNA replication and T cell receptor (TCR) signaling. We find that loss of fidelity for longer proofreading steps relies on the specific strategy of information extraction and show that in the first-passage time (FPT) discrimination strategy, longer proofreading steps can exponentially improve the accuracy of KPR at the cost of speed. Thus, KPR can still be an effective discrimination mechanism in the high noise regime. However, in a product concentration-based discrimination strategy, longer proofreading steps do not necessarily lead to an increase in performance. However, by introducing activation thresholds on product concentrations, can we decompose the product-based strategy into a series of FPT-based strategies to better resolve the subtleties of KPR-mediated product discrimination. Our findings underscore the importance of understanding KPR in the context of how information is extracted and processed in the cell.

The aim of this study is to suggest a novel approach for estimating the intramolecular mobility of a charge carrier that migrates within a polymer chain and is involved in a pair reaction with a particle located on the same chain. The approach is based on the effect of an external electric field on the migration rate and, consequently, the kinetics of the reaction. As a first step, this problem is considered a stochastic one-step process with absorbing and reflecting boundaries, and an analytical solution is obtained in the case that the second reactant is immobile. With the use of computer simulations of stochastic migration, the effect of the mobility of both reactants and the influence of the Coulomb interaction between them are considered. It is found that the ratio of the pair reaction rates with and without an external field is relatively little dependent on these factors and that the analytical expressions derived can be applied to estimate the relative mobility of recombining particles with accuracy better than a factor of two in many realistic situations. [ABSTRACT FROM AUTHOR]

Generative Adversarial Networks (GANs) have shown immense potential in fields such as text and image generation. Only very recently attempts to exploit GANs to statistical-mechanics models have been reported. Here we quantitatively test this approach by applying it to a prototypical stochastic process on a lattice. By suitably adding noise to the original data we succeed in bringing both the Generator and the Discriminator loss functions close to their ideal value. Importantly, the discreteness of the model is retained despite the noise. As typical for adversarial approaches, oscillations around the convergence limit persist also at large epochs. This undermines model selection and the quality of the generated trajectories. We demonstrate that a simple multi-model procedure where stochastic trajectories are advanced at each step upon randomly selecting a Generator leads to a remarkable increase in accuracy. This is illustrated by quantitative analysis of both the predicted equilibrium probability distribution and of the escape-time distribution. Based on the reported findings, we believe that GANs are a promising tool to tackle complex statistical dynamics by machine learning techniques. [ABSTRACT FROM AUTHOR]

The brain produces diverse functions, from perceiving sounds to producing arm reaches, through the collective activity of populations of many neurons. Determining if and how the features of these exogenous variables (e.g., sound frequency, reach angle) are reflected in population neural activity is important for understanding how the brain operates. Often, high-dimensional neural population activity is confined to low-dimensional latent spaces. However, many current methods fail to extract latent spaces that are clearly structured by exogenous variables. This has contributed to a debate about whether or not brains should be thought of as dynamical systems or representational systems. Here, we developed a new latent process Bayesian regression framework, the orthogonal stochastic linear mixing model (OSLMM) which introduces an orthogonality constraint amongst time-varying mixture coefficients, and provide Markov chain Monte Carlo inference procedures. We demonstrate superior performance of OSLMM on latent trajectory recovery in synthetic experiments and show superior computational efficiency and prediction performance on several real-world benchmark data sets. We primarily focus on demonstrating the utility of OSLMM in two neural data sets: μECoG recordings from rat auditory cortex during presentation of pure tones and multi-single unit recordings form monkey motor cortex during complex arm reaching. We show that OSLMM achieves superior or comparable predictive accuracy of neural data and decoding of external variables (e.g., reach velocity). Most importantly, in both experimental contexts, we demonstrate that OSLMM latent trajectories directly reflect features of the sounds and reaches, demonstrating that neural dynamics are structured by neural representations. Together, these results demonstrate that OSLMM will be useful for the analysis of diverse, large-scale biological time-series datasets.

"Gating" is a widely observed phenomenon in biochemistry that describes the transition between the activated (or open) and deactivated (or closed) states of an ion-channel, which makes transport through that channel highly selective. In general, gating is a mechanism that imposes an additional restriction on a transport, as the process ends only when the "gate" is open and continues otherwise. When diffusion occurs in the presence of a constant bias to a gated target, i.e., to a target that switches between an open and a closed state, the dynamics essentially slow down compared to ungated drift-diffusion, resulting in an increase in the mean completion time, ⟨TG⟩ > ⟨T⟩, where T denotes the random time of transport and G indicates gating. In this work, we utilize stochastic resetting as an external protocol to counterbalance the delay due to gating. We consider a particle in the positive semi-infinite space that undergoes drift-diffusion in the presence of a stochastically gated target at the origin and is moreover subjected to rate-limiting resetting dynamics. Calculating the minimal mean completion time ⟨ T r ⋆ G ⟩ rendered by an optimal resetting rate r⋆ for this exactly solvable system, we construct a phase diagram that owns three distinct phases: (i) where resetting can make gated drift-diffusion faster even compared to the original ungated process, ⟨ T r ⋆ G ⟩ < ⟨ T ⟩ < ⟨ T G ⟩ , (ii) where resetting still expedites gated drift-diffusion but not beyond the original ungated process, ⟨ T ⟩ ≤ ⟨ T r ⋆ G ⟩ < ⟨ T G ⟩ , and (iii) where resetting fails to expedite gated drift-diffusion, ⟨ T ⟩ < ⟨ T G ⟩ ≤ ⟨ T r ⋆ G ⟩. We also highlight various non-trivial behaviors of the completion time as the resetting rate, gating parameters, and geometry of the set-up are carefully ramified. Gated drift-diffusion aptly models various stochastic processes such as chemical reactions that exclusively take place in certain activated states of the reactants. Our work predicts the conditions under which stochastic resetting can act as a useful strategy to enhance the rate of such processes without compromising their selectivity. [ABSTRACT FROM AUTHOR]

In this paper, we derive a condition-based maintenance strategy for a degradation-shock dependence system subject to two causes of failure, degradation and random shocks. We develop a reliability model for a system under warranty for failure in which we consider the dependence between the degradation process and random shocks and do a warranty cost analysis based on the suggested model. We consider a degradation-shock dependence system that is under warranty with three separate services: repair, replacement, and preventive maintenance. These are triggered when a product degradation crosses one of three predetermined degradation thresholds. We also investigate the dependence between random shocks and degradation modelled by a time-scaled covariate factor. We study the random shock considering fatal shocks that require replacement and nonfatal shocks that require only repair and the degradation caused by nonfatal shocks. The condition-based maintenance strategy for a degradation-shock dependence system is a novel approach to analysing system degradation, with a goal of optimising warranty costs; we also minimise the expected total system cost to find an optimal warranty length. We present several numeric examples and real-world applications to illustrate our proposed approach. [ABSTRACT FROM AUTHOR]

This article presents a robust security‐based model predictive control (MPC) framework for a class of discrete‐time nonlinear cyber‐physical systems (CPSs) suffering random deception attacks and persistent actuator saturation. In CPSs, the attacks can significantly degrade the control performance. Naturally, a larger degree of control is required, prompting the consideration of actuator saturation in CPS's structure. In our work, the cyberattacks on the controller‐actuator (C‐A) channel are described as a stochastic process with specific probability and bounded energy. The security‐related constraint is introduced to improve the resistance and robustness against attacks. Mean‐square quadratic boundedness is derived to estimate the robustly invariant set. By placing the saturated input into a convex hull, a robust security‐based MPC optimization problem is addressed less conservatively with linear matrix inequality technique. Furthermore, the feasibility of the proposed method is proved recursively, and the stability and security of the closed‐loop system are established in mean‐square sense. In the end, the laboratory tank and reactor‐separator process are applied to validate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The present paper develops a new procedure for finding the most unfavorable long-period excitation for isolated building structures based on the probabilistic critical excitation method. The upper bound of earthquake input energy per unit mass is considered the problem's constraint, and a modified power spectral density (PSD) function is proposed to conform to the new constraint. It is shown that the new constraint can be used as a criterion to estimate the possible range of input energy of similar earthquakes. In addition, the new PSD function can control the excitation's intensity. Finally, three base-isolated shear buildings modeled as multi degree of freedom systems with nonproportional damping are considered, and the critical excitations for each model are found based on the proposed method. To examine the reliability of the results, three actual accelerograms with a considerably high level of total input energy are selected as benchmarks and linear dynamic analyses are conducted using these accelerograms along with the generated critical excitations. The results show that the generated long-period excitations can better estimate the structural behavior (i.e., maximum displacement, drift, and absolute acceleration of stories) than the actual benchmark accelerograms. [ABSTRACT FROM AUTHOR]

We derive consistency and asymptotic normality results for quasi‐maximum likelihood methods for drift parameters of ergodic stochastic processes observed in discrete time in an underlying continuous‐time setting. The special feature of our analysis is that the stochastic integral part is unobserved and nonparametric. Additionally, the drift may depend on the (unknown and unobserved) stochastic integrand. Our results hold for ergodic semi‐parametric diffusions and backward SDEs. Simulation studies confirm that the methods proposed yield good convergence results. [ABSTRACT FROM AUTHOR]

Flow lines are often used to perform assembly operations in multi-stage processes. During these assembly operations, components that are relatively small, compared to the work pieces travelling down the flow line, are mounted to the work pieces at a given stage. Those components, or more generally, any kind of auxiliary material, are provisioned to the corresponding production stage in a repetitive but not necessarily deterministic manner using a certain delivery frequency, each time filling the local storage up to a predetermined order-up-to level. Just like random processing times, machine failures, and repairs, the randomness of the provisioning process can impact the long-term throughput of such a flow line. In this paper, we develop a fast and accurate analytical performance evaluation method to estimate the long-term throughput of a Markovian flow line of this type for the practically important case of limited buffer capacities between the production stages. We first give an exact characterization of a two-machine line of that type and show how to determine system state probabilities and aggregate performance measures. Furthermore, we show how to use this two-machine model as the building block of an approximate decomposition approach for longer flow lines. As opposed to previous decomposition approaches, even the state space of the two-machine lines can become so large that an exact solution of the Markov chains can become impractical. We hence show how to set up, train, and use an artificial neural network to replace the Markov chain solver embedded in the decomposition approach, which then leads to an accurate and extremely fast flow line evaluation tool. The proposed methodology is evaluated by a comparison with simulation results and used to characterize the structural patterns describing the behaviour of flow lines of this type. The method can be used to systematically consider the combined impact of the delivery frequency and the local order-up-to levels for the auxiliary material when designing a flow line of this type. [ABSTRACT FROM AUTHOR]

Tasks of different nature and difficulty levels are a part of people's lives. In this context, there is a scientific interest in the relationship between the difficulty of the task and the persistence need to accomplish it. Despite the generality of this problem, some tasks can be simulated in the form of games. In this way, we employ data from a large online platform, called Steam, to analyze games and the performance of their players. More specifically, we investigated persistence in completing tasks based on the proportion of players who accomplished game achievements. Overall, we present five major findings. First, the probability distribution for the number of achievements is log-normal distribution. Second, the distribution of game players also follows a log-normal. Third, most games require neither a very high degree of persistence nor a very low one. Fourth, players also prefer games that demand a certain intermediate persistence. Fifth, the proportion of players as a function of the number of achievements declines approximately exponentially. As both the log-normal and the exponential functions are memoryless, they are mathematical forms that describe random effects arising from the nature of the system. Therefore our first two findings describe random processes of fragmenting achievements and players while the last three provide a quantitative measure of the human preference in the pursuit of challenging, achievable and justifiable tasks. [ABSTRACT FROM AUTHOR]

Aerobic anoxygenic phototrophic bacteria (AAPB) are significant bacterial groups in aquatic ecosystems, known for their rapid growth and photoheterotrophic characteristics. However, the distribution and ecological assembly process of AAPB in low irradiation freshwater basins remain unclear, warranting further investigation. In this study, we present the diversity, abundance, spatial variations, ecological process, and community interaction of AAPB in sediment of Three Gorges Reservoir (TGR) under low irradiation. Our findings demonstrate the dominant genera of AAPB community that exist in the TGR area also are appeared in different waters, with some regional preference. Moreover, the concentration of pufM gene, an indicator for AAPB, maintains a consistently high numerical level ranging from (2.21 ± 0.44) × 104 to (9.98 ± 0.30) × 107 gene copies/g. Although solar irradiation is suggested as the major factor affecting AAPB, it remains unclear whether and how AAPB differ between regions due to varying solar irradiation levels. Our results show spatial differences between total bacteria and AAPB communities, with significant differences observed only in AAPB. Geographical and environmental factor contributed less than 10% to the spatial difference of community, with sediment type and environmental factors being the key factors influencing microbial community structure. The stochastic process plays a dominant role in the aggregation and replacement of AAPB communities, among which the most contribution is dispersal limitation. For AAPB network, Yoonia and Gemmobacter are the hubs for modules. Those results valuable insights into the AAPB communities in TGR with low irradiation. [Display omitted] [ABSTRACT FROM AUTHOR]

This paper gives a new property for stochastic processes, called square-mean μ - pseudo-S-asymptotically Bloch-type periodicity. We show how this property is preserved under some operations, such as composition and convolution, for stochastic processes. Our main results extend the classical results on S-asymptotically Bloch-type periodic functions. We also apply our results to some problems involving semilinear stochastic integrodifferential equations in abstract spaces [ABSTRACT FROM AUTHOR]

This study proposes a novel velocity divergence‐generalized adaptive probability density evolution method (VD‐GAPDEM) for calculating the probability density function of the stochastic response process of stochastic structures under stochastic dynamic loads. First, based on the principle of probability conservation, the velocity divergence‐generalized adaptive probability density evolution equation (VD‐GAPDEE) is derived for a stochastic system that can effectively consider the shape and location changes of the joint transitional probability density of representative points (RPs) in the stochastic response process. Second, a novel VD‐GAPDEM is proposed to solve the VD‐GAPDEE directly using the point selection technique based on the generalized F discrepancy and the second‐order Runge–Kutta method with a smoothing kernel method (Runge–Kutta‐SKFAM). Furthermore, the differences and connections between VD‐GAPDEM and the existing probability density evolution method are analyzed. Additionally, the high computational efficiency and accuracy of the proposed VD‐GAPDEM are demonstrated through three typical examples of stochastic response analysis, involving stochastic systems subjected to stochastic dynamic loads. [ABSTRACT FROM AUTHOR]

In [Trans. Am. Math. Soc. 375 (2022), no. 1, 627–668], Kahn gave the strongest possible, affirmative, answer to Shamir's problem, which had been open since the late 1970s: Let r⩾3$$ r\geqslant 3 $$ and let n$$ n $$ be divisible by r$$ r $$. Then, in the random r$$ r $$‐uniform hypergraph process on n$$ n $$ vertices, as soon as the last isolated vertex disappears, a perfect matching emerges. In the present work, we prove the analogue of this result for clique factors in the random graph process: at the time that the last vertex joins a copy of the complete graph Kr$$ {K}_r $$, the random graph process contains a Kr$$ {K}_r $$‐factor. Our proof draws on a novel sequence of couplings which embeds the random hypergraph process into the cliques of the random graph process. An analogous result is proved for clique factors in the s$$ s $$‐uniform hypergraph process (s⩾3$$ s\geqslant 3 $$). [ABSTRACT FROM AUTHOR]

Geometric properties of exceedance regions above a given quantile level provide meaningful theoretical and statistical characterizations for stochastic processes defined on Euclidean domains. Many theoretical results have been obtained for excursions of Gaussian processes and include expected values of the so-called Lipschitz–Killing curvatures (LKCs), such as the area, perimeter and Euler characteristic in two-dimensional Euclidean space. In this paper, we derive novel results for the expected LKCs of excursion sets of more general processes whose construction is based on location or scale mixtures of a Gaussian process, which means that the mean or the standard deviation, respectively, of a stationary Gaussian process is a random variable. We first present exact formulas for peaks-over-threshold-stable limit processes (so-called Pareto processes) arising from the use of Gaussian or log-Gaussian spectral functions in the spectral construction of max-stable processes. These peaks-over-threshold limits are known to arise for Gaussian location or scale mixtures if the mixing distributions satisfies certain regular-variation properties. As a second important result, we show that expected LKCs of excursion sets of such general mixture processes converge to the corresponding expressions of their Pareto process limits. We further provide exact subasymptotic formulas of expected LKCs for various specific choices of the distribution of the mixing variable. Finally, we discuss consistent empirical estimation of LKCs of exceedance regions and implement numerical experiments to illustrate the rate of convergence towards asymptotic expressions. An application to daily temperature data simulated by climate models for the period 1951–2005 over a regular pixel grid covering continental France showcases the practical utility of the new results. [ABSTRACT FROM AUTHOR]

For a 24/7 database system, backups should be implemented right after a large volume of data has been updated, putting their backup windows in non-busy states with user's convenience.From this viewpoint, this paper studies periodic and random incremental backup policies, in which, incremental backup is implemented right after data update and full backup is performed at periodic times KT , or at a number N of data updates, respectively. We firstly describe the stochastic processes of data update and database failure, and then model the expected cost rates for data backup and data restoration.Respective K ∗ , N ∗ , K f ∗ , and N f ∗ are obtained to minimize their expected cost rates in analytical ways, respectively. Finally, numerical examples are given to illustrate the optimum policies. [ABSTRACT FROM AUTHOR]

Plant communities are complex systems shaped by a combination of deterministic and stochastic ecological processes. Bryophytes are an essential component of plant diversity in natural ecosystems, yet our understanding of their community ecology needs to catch up to that of other organisms. The unique geological history, alpine climatic conditions, and high habitat heterogeneity of Tibet provide suitable areas for bryophytes to survive in the alpine regions. Therefore, field surveys were conducted across 184 plots in forest, thicket, and herbaceous vegetation of Tibet to investigate the role of deterministic processes such as biological interactions and abiotic effects, along with stochastic processes, in shaping the distribution of bryophyte assemblages. We employed various analytical methods, including mixed effects models, partial least squares path modeling, null model analysis, and neutral community models. The study showed that bryophyte richness was highest in forests. Bryophyte assemblages showed greater segregation in forest and thicket environments compared to herbaceous vegetation. As the influence of stochastic processes increased, that of deterministic processes decreased from forests through thickets to herbaceous vegetation. Deterministic processes were the main driving forces for the bryophyte assemblage pattern. Soil properties and climatic factors, particularly pH played a key role in determining bryophyte patterns in Tibet. This study has deepened our comprehension of how deterministic and stochastic ecological processes interplay and shape bryophyte distribution patterns in Tibet. [ABSTRACT FROM AUTHOR]

In the analysis of clustered failure time data, Cox frailty models have been extensively studied by incorporating frailty with a prespecified distribution to address potential correlation of data within clusters. In this paper, we propose a frailty proportional mean residual life regression model to analyze such data. A novel conditional quasi‐likelihood inference procedure is developed, utilizing a stochastic process and the inverse probability of censoring weighting (IPCW) to form estimating equations for regression parameters. Our proposal employs conditional inference based on a penalized quasi‐likelihood to address within‐cluster correlation without need to specify the frailty distribution, bringing the method closer to what suffices for real‐world applications. By adopting the Buckley–James estimator in the IPCW, the method further allows for dependent censoring. We establish asymptotic properties of the proposed estimator and evaluate its finite sample performance via simulation studies. An application to the data from a multi‐institutional breast cancer study is presented for illustration. [ABSTRACT FROM AUTHOR]

Continuous time Bayesian networks (CTBNs) represent a class of stochastic processes, which can be used to model complex phenomena, for instance, they can describe interactions occurring in living processes, social science models or medicine. The literature on this topic is usually focused on a case when a dependence structure of a system is known and we are to determine conditional transition intensities (parameters of a network). In the paper, we study a structure learning problem, which is a more challenging task and the existing research on this topic is limited. The approach, which we propose, is based on a penalized likelihood method. We prove that our algorithm, under mild regularity conditions, recognizes a dependence structure of a graph with high probability. We also investigate properties of the procedure in numerical studies. [ABSTRACT FROM AUTHOR]

This research explores the reliability of deep learning, specifically Long Short‐Term Memory (LSTM) networks, for estimating the Hurst parameter in fractional stochastic processes. The study focuses on three types of processes: fractional Brownian motion (fBm), fractional Ornstein–Uhlenbeck (fOU) process, and linear fractional stable motions (lfsm). The work involves a fast generation of extensive datasets for fBm and fOU to train the LSTM network on a large volume of data in a feasible time. The study analyses the accuracy of the LSTM network's Hurst parameter estimation regarding various performance measures like root mean squared error (RMSE), mean absolute error (MAE), mean relative error (MRE), and quantiles of the absolute and relative errors. It finds that LSTM outperforms the traditional statistical methods in the case of fBm and fOU processes; however, it has limited accuracy on lfsm processes. The research also delves into the implications of training length and valuation sequence length on the LSTM's performance. The methodology is applied to estimating the Hurst parameter in li‐ion battery degradation data and obtaining confidence bounds for the estimation. The study concludes that while deep learning methods show promise in parameter estimation of fractional processes, their effectiveness is contingent on the process type and the quality of training data. [ABSTRACT FROM AUTHOR]

Time‐varying group formation tracking control problems for multi‐agent systems are investigated based on distributed multi‐sensor multi‐target filtering with intermittent observations. First, in order to estimate the states of multiple targets under the phenomenon of intermittent observations accurately, a distributed multi‐sensor multi‐target filtering algorithm is proposed based on cubature Kalman filtering. Second, a time‐varying group formation tracking protocol is designed for multi‐agent systems by using the state estimations obtained from the filtering algorithm and the neighboring interaction. The protocol enables multi‐agent systems to form time‐varying subformations and track multiple targets in the same subgroups, respectively. Third, the boundedness of the error covariance matrices is proved under the condition that the observation probability is higher than the minimum threshold. Then the estimation errors of the filtering algorithm are proved to be stochastically bounded by introducing a stochastic process. Furthermore, the boundedness of the group formation tracking errors is proved. Finally, a numerical example is used to verify the performance of the proposed algorithm and protocol. [ABSTRACT FROM AUTHOR]

Subsampling is a practical strategy for analyzing vast survival data, which are progressively encountered across diverse research domains. While the optimal subsampling method has been applied to inferences for Cox models and parametric accelerated failure time (AFT) models, its application to semi‐parametric AFT models with rank‐based estimation have received limited attention. The challenges arise from the non‐smooth estimating function for regression coefficients and the seemingly zero contribution from censored observations in estimating functions in the commonly seen form. To address these challenges, we develop optimal subsampling probabilities for both event and censored observations by expressing the estimating functions through a well‐defined stochastic process. Meanwhile, we apply an induced smoothing procedure to the non‐smooth estimating functions. As the optimal subsampling probabilities depend on the unknown regression coefficients, we employ a two‐step procedure to obtain a feasible estimation method. An additional benefit of the method is its ability to resolve the issue of underestimation of the variance when the subsample size approaches the full sample size. We validate the performance of our estimators through a simulation study and apply the methods to analyze the survival time of lymphoma patients in the surveillance, epidemiology, and end results program. [ABSTRACT FROM AUTHOR]

Microorganisms are important members of seagrass bed ecosystems and play a crucial role in maintaining the health of seagrasses and the ecological functions of the ecosystem. In this study, we systematically quantified the assembly processes of microbial communities in fragmented seagrass beds and examined their correlation with environmental factors. Concurrently, we explored the relative contributions of species replacement and richness differences to the taxonomic and functional β-diversity of microbial communities, investigated the potential interrelation between these components, and assessed the explanatory power of environmental factors. The results suggest that stochastic processes dominate community assembly. Taxonomic β-diversity differences are governed by species replacement, while for functional β-diversity, the contribution of richness differences slightly outweighs that of replacement processes. A weak but significant correlation (p < 0.05) exists between the two components of bdiversity in taxonomy and functionality, with almost no observed significant correlation with environmental factors. This implies significant differences in taxonomy, but functional convergence and redundancy within microbial communities. Environmental factors are insufficient to explain the β-diversity differences. In conclusion, the assembly of microbial communities in fragmented seagrass beds is governed by stochastic processes. The patterns of taxonomic and functional β-diversity provide new insights and evidence for a better understanding of these stochastic assembly rules. This has important implications for the conservation and management of fragmented seagrass beds. [ABSTRACT FROM AUTHOR]

A network evolution with predicted tail and extremal indices of PageRank and the Max-Linear Model used as node influence indices in random graphs is considered. The tail index shows a heaviness of the distribution tail. The extremal index is a measure of clustering (or local dependence) of the stochastic process. The cluster implies a set of consecutive exceedances of the process over a sufficiently high threshold. Our recent results concerning sums and maxima of non-stationary random length sequences of regularly varying random variables are extended to random graphs. Starting with a set of connected stationary seed communities as a hot spot and ranking them with regard to their tail indices, the tail and extremal indices of new nodes that are appended to the network may be determined. This procedure allows us to predict a temporal network evolution in terms of tail and extremal indices. The extremal index determines limiting distributions of a maximum of the PageRank and the Max-Linear Model of newly attached nodes. The exposition is provided by algorithms and examples. To validate our theoretical results, our simulation and real data study concerning a linear preferential attachment as a tool for network growth are provided. [ABSTRACT FROM AUTHOR]

AbstractFunctional Time Series (FTS) are sequences of dependent random elements taking values on some functional space. Most of the research on this domain focuses on producing a predictor able to forecast the next function, having observed a part of the sequence. For this, the Autoregressive Hilbertian process is a suitable framework. Here, we address the problem of constructing simultaneous predictive confidence bands for a stationary FTS. The method is based on an entropy measure for stochastic processes. To construct predictive bands, we use a Reproducing Kernel Hilbert Space (RKHS) to represent the functions and a functional bootstrap procedure that allows us to estimate the prediction law and a Reproducing Kernel Hilbert Spaces (RKHS) to represent the functions, considering then the basis associated to the reproducing kernel. We then classify the points on the projected space according to those that belong to the minimum entropy set (MES) and those that do not. We map the minimum entropy set back to the functional space and construct a band using the regularity property of the RKHS. The proposed methodology is illustrated through artificial and real data sets. [ABSTRACT FROM AUTHOR]

A cruise was conducted in the summer of 2023 from the Pearl River Estuary (PRE) to the adjacent waters of the Xisha Islands in the northern South China Sea (NSCS) to investigate the distribution, community structure, and assembly patterns of eukaryotic and prokaryotic phytoplankton using high-throughput sequencing (HTS) and microscopic observation. Dinophyta were the most abundant phylum in the eukaryotic phytoplankton community based on HTS, accounting for 92.17% of the total amplicon sequence variants (ASVs). Syndiniales was the most abundant order among eukaryotic phytoplankton, whereas Prochlorococcus was the most abundant genus within cyanobacteria. The alpha diversity showed the lowest values in the PRE area and decreased gradually with depth, while cyanobacteria exhibited higher alpha diversity indices in the PRE and at depths ranging from 75 m to 750 m. The morphological results were different from the data based on HTS. Diatoms (37 species) dominated the phytoplankton community, with an average abundance of 3.01 × 104 cells L-1, but only six species of dinoflagellate were observed. Spearman correlation analysis and redundancy analysis (RDA) showed that the distribution and community structure of phytoplankton were largely influenced by geographical location and environmental parameters in the NSCS. The neutral community model (NCM) and null model indicated that deterministic processes played a significant role in the assembly of eukaryotic phytoplankton, with heterogeneous selection and homogeneous selection accounting for 47.27 and 29.95%, respectively. However, stochastic processes (over 60%) dominated the assembly of cyanobacteria and undominated processes accounted for 63.44%. In summary, the formation of eukaryotic phytoplankton was mainly influenced by environmental factors and geographic location, but the assembly of cyanobacteria was shaped by both stochastic processes, which accounted for over 60%, and environmental selection in the NSCS. [ABSTRACT FROM AUTHOR]

This paper establishes a new fractional frSIS model utilizing a continuous time random walk method. There are two main innovations in this paper. On the one hand, the model is analyzed from a mathematical perspective. First, unlike the classic SIS infectious disease model, this model presents the infection rate and cure rate in a fractional order. Then, we proved the basic regeneration number R0 of the model and studied the influence of orders a and b on R0. Second, we found that frSIS has a disease-free equilibrium point E0 and an endemic equilibrium point E∗. Moreover, we proved frSIS global stability of the model using R0. If R0<1, the model of E0 is globally asymptotically stable. If R0>1, the model of E∗ is globally asymptotically stable. On the other hand, from the perspective of infectious diseases, we discovered that appropriately increasing a and decreasing b are beneficial for controlling the spread of diseases and ultimately leading to their disappearance. This can help us provide some dynamic adjustments in prevention and control measures based on changes in the disease. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with a stochastic semilinear differential equations driven by both Brownian motion and fractional Brownian motion. Firstly, we establish an inequality for the distance between finite‐dimensional distributions of a random process at two different moments. Then, using the properties of stochastic integrals, fixed point theorems, and based on this inequality, we establish the existence and uniqueness of Besicovich almost automorphic solutions in finite‐dimensional distributions for this type of semilinear equation. Finally, we provide an example to demonstrate the effectiveness of our results. Our results are new to stochastic differential equations driven by Brownian motion or stochastic differential equations driven by fractional Brownian motion. [ABSTRACT FROM AUTHOR]

Properties of the intervals between zero-crossings of non-Gaussian stable random processes are investigated. The classes of process considered are divided into those having short- and long-term memories, characterized by a coherence function that imbues dependence between a random variable measured at different times, which replaces the auto-correlation function, which exists only for the Gaussian case. For exponential coherence functions, all moments of probability densities for the intervals exist and a persistence parameter is determined that characterizes the rate of decay of the exponential tail of the probability densities, together with the full form of the density functions for selected values of the stability index. Results are verified with numerical simulation of a Cauchy process using a method independent of the theoretical development. For power-law coherence functions, the existence of moments of the distribution is conditional upon the indices characterizing the stable process and coherence function. The probability densities of the intervals have power-law tails depending on the product of these indices, with a logarithmic correction for specific combinations of these. An outer-scale or cut-off to power-law coherence functions regains a persistence parameter to the densities and the existence of all moments of the intervals. [ABSTRACT FROM AUTHOR]

AbstractMultistate Markov models are a canonical parametric approach for data modeling of observed or latent stochastic processes supported on a finite state space. Continuous-time Markov processes describe data that are observed irregularly over time, as is often the case in longitudinal medical data, for example. Assuming that a continuous-time Markov process is time-homogeneous, a closed-form likelihood function can be derived from the Kolmogorov forward equations – a system of differential equations with a well-known matrix-exponential solution. Unfortunately, however, the forward equations do not admit an analytical solution for continuous-time, time-inhomogeneous Markov processes, and so researchers and practitioners often make the simplifying assumption that the process is piecewise time-homogeneous. In this paper, we provide intuitions and illustrations of the potential biases for parameter estimation that may ensue in the more realistic scenario that the piecewise-homogeneous assumption is violated, and we advocate for a solution for likelihood computation in a truly time-inhomogeneous fashion. Particular focus is afforded to the context of multistate Markov models that allow for state label misclassifications, which applies more broadly to hidden Markov models (HMMs), and Bayesian computations bypass the necessity for computationally demanding numerical gradient approximations for obtaining maximum likelihood estimates (MLEs). Supplemental materials are available online. [ABSTRACT FROM AUTHOR]

We deal with the problem of optimal estimation of linear functionals constructed from the missed values of a continuous time stochastic process ξ ⁢ ( t ) {\xi(t)} with periodically stationary increments at points t ∈ [ 0 ; ( N + 1 ) ⁢ T ] t\in[0;(N+1)T] based on observations of this process with periodically stationary noise. To solve the problem, a sequence of stochastic functions { ξ j ( d ) ⁢ ( u ) = ξ j ( d ) ⁢ ( u + j ⁢ T , τ ) , u ∈ [ 0 , T ) , j ∈ ℤ } {\{\xi^{(d)}_{j}(u)=\xi^{(d)}_{j}(u+jT,\tau),u\in[0,T),\,j\in\mathbb{Z}\}} is constructed. It forms an L 2 ⁢ ( [ 0 , T ) ; H ) {L_{2}([0,T);H)} -valued stationary increment sequence { ξ j ( d ) , j ∈ ℤ } {\{\xi^{(d)}_{j},j\in\mathbb{Z}\}} or corresponding to it an (infinite-dimensional) vector stationary increment sequence { ξ → j ( d ) = ( ξ k ⁢ j ( d ) , k = 1 , 2 , … ) ⊤ , j ∈ ℤ } {\{\vec{\xi}^{(d)}_{j}=(\xi^{(d)}_{kj},k=1,2,\dots)^{\top},\,j\in\mathbb{Z}\}} . In the case of a known spectral density, we obtain formulas for calculating values of the mean square errors and the spectral characteristics of the optimal estimates of the functionals. Formulas determining the least favorable spectral densities and the minimax (robust) spectral characteristics of the optimal linear estimates of functionals are derived in the case where the sets of admissible spectral densities are given. [ABSTRACT FROM AUTHOR]

Fluorescence microscopic imaging is essentially a convolution process distorted by random noise, limiting critical parameters such as imaging speed, duration, and resolution. Though algorithmic compensation has shown great potential to enhance these pivotal aspects, its fidelity remains questioned. Here we develop a physics-rooted computational resolution extension and denoising method with ensured fidelity. Our approach employs a multi-resolution analysis (MRA) framework to extract the two main characteristics of fluorescence images against noise: across-edge contrast, and along-edge continuity. By constraining the two features in a model-solution framework using framelet and curvelet, we develop MRA deconvolution algorithms, which improve the signal-to-noise ratio (SNR) up to 10 dB higher than spatial derivative based penalties, and can provide up to two-fold fidelity-ensured resolution improvement rather than the artifact-prone Richardson-Lucy inference. We demonstrate our methods can improve the performance of various diffraction-limited and super-resolution microscopies with ensured fidelity, enabling accomplishments of more challenging imaging tasks. [ABSTRACT FROM AUTHOR]

In this study, we explore optimal service allocation within the Chinese hierarchical healthcare system with green channels, providing valuable insights for practitioners to understand how optimal service allocation is affected by various realistic factors. These green channels are designed to streamline referrals from community healthcare centers to comprehensive hospitals. We aim to determine the optimal capacity allocation for these green channels within comprehensive hospitals. Our research employs techniques from queuing theory and stochastic processes, e.g., diffusion analysis, to develop a mathematical model that approximates the optimal allocation of resources. We uncover both closed-form and numerical solutions for this optimal capacity allocation. By analyzing the impact of various cost factors, we find that an increase in costs within the green channel results in a lower optimal service rate. Additionally, patient preferences for specific treatments influence allocation, reducing the optimal share of services provided by general hospitals. The optimal solution is also affected by the proportions of different patient types. Through extensive simulations, we validate the accuracy of our model approximations under heavy traffic conditions, further examining sources of error to ensure robustness. Our findings provide valuable insights into optimizing resource allocation in hierarchical healthcare systems, ensuring efficient and cost-effective patient care. [ABSTRACT FROM AUTHOR]

We consider local probabilities of lower deviations for branching process Zn = Xn,1 + ⋯ + Xn,Zn−1 in random environment η. We assume that η is a sequence of independent identically distributed variables and for fixed η the variables Xi,j are independent and have geometric distributions. We suppose that steps ξi of the associated random walk Sn = ξ1 + ⋯ + ξn has positive mean and satisfies left-side Cramér condition: E exp(hξi) < ∞ if h− < h < 0 for some h− < − 1. Under these assumptions we find the asymptotic of the local probabilities P(Zn = ⌊exp(θn)⌋), n → ∞, for θ ∈ (max(m−, 0); m(− 1)) and for θ in a neighbourhood of m(− 1), where m− and m(− 1) are some constants. [ABSTRACT FROM AUTHOR]

An integro‐differential equation for the probability density of the generalized stochastic Ornstein–Uhlenbeck process with jump diffusion is considered for a special case of the Laplacian distribution of jumps. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly, as a finite sum. Alternatively, the fundamental solution can be represented as converging power series. The properties of this solution are investigated. The fundamental solution makes it possible to obtain explicit formulas for the density at each instant of time, which is important, for example, for testing numerical methods. [ABSTRACT FROM AUTHOR]

Let (Zn) be a supercritical branching process with immigration in an independent and identically distributed random environment. Under necessary moment conditions, we show the exact convergence rate in the central limit theorem on log Zn and establish the corresponding local limit theorem by using the moments of the natural submartingale and the convergence rates of its logarithm. By similar approach and with the help of a change of measure, we also present the so-called integrolocal theorem and integral large deviation theorem to characterize the precise asymptotics of the upper large deviations. [ABSTRACT FROM AUTHOR]

River water quality often follows a long-memory stochastic process with power-type autocorrelation decay, which can only be reproduced using appropriate mathematical models. The selection of a stochastic process model, particularly its memory structure, is often subject to misspecifications owing to low data quality and quantity. Therefore, environmental risk assessment should account for model misspecification through mathematically rigorous and efficiently implementable approaches; however, such approaches have been still rare. We address this issue by first modeling water quality dynamics through the superposition of an affine diffusion process that is stationary and has a long memory. Second, the worst-case upper deviation of the water quality value from a prescribed threshold value under model misspecifications is evaluated using either the divergence risk or Wasserstein risk measure. The divergence risk measure can consistently deal with the misspecification of the memory structure to the worst-case upper deviation. The Wasserstein risk measure is more flexible but fails in this regard, as it does not directly consider the memory structure information. We theoretically compare both approaches to demonstrate that their assumed uncertainties differed substantially. From the application to the 30-year water quality data of a river in Japan, we categorized the water quality indices to be those with truly long memory (Total nitrogen, NO3-N, NH4-N, and SO 4 2 - ), those with moderate power-type memory (NO2-N, PO4-P, and Total Organic Carbon), and those with almost exponential memory (Total phosphorus and Chemical Oxygen demand). The risk measures are successfully computed numerically considering the seasonal variations of the water quality indices. [ABSTRACT FROM AUTHOR]

Background and aims: Afforestation can alter belowground microbial diversity and affect soil‒plant feedback, which is crucial for understanding nutrient cycles and ecosystem productivity. Most studies have focused on the carbon and nitrogen cycles in forest ecosystems; however, there are few studies on the effect of afforestation in soil microbial diversity on phosphorus (P) nutrients. In this study, we explored the effect of soil microbial communities on soil‒plant P feedback under long-term afforestation. Methods: Typical poplar plantations were selected from the Lhasa River Basin. We assessed the P content, activity of extracellular enzyme related to P acquisition, rhizosphere soil microbes, and mechanism of microbial community. Results: Increased bacterial richness and evenness were beneficial to soil and plant P nutrition; however, the community differentiation of bacteria and fungi had no positive effect on the P content. Ecological stochasticity drove for the assembly of the rhizosphere soil microbial community after afforestation on the Tibetan Plateau. Homogenizing dispersal dominated bacterial and fungal assemblies. Ecological drift was another stochastic process that affected the microbial assembly. Conclusion: Bacterial richness and evenness had positive effects on soil‒plant P feedback, whereas microbial differentiation did not. Homogenizing dispersal and ecological drift are intrinsic factors that regulate the effects of microbial community diversity on soil‒plant P feedback processes. This study provides the evidence on the intrinsic mechanism of belowground microbial diversity influencing soil‒plant P feedback. It also provides new insights into the selection of plantation species on the Tibetan Plateau from the perspective of enhancement of P nutrition by microbes. [ABSTRACT FROM AUTHOR]

The Lindley distribution is useful in a wide variety of fields, such as biology and astronomy. Many generalizations of the Lindley distribution have been introduced in the literature, with various motivations. Inspired by the concept of superstatistics in nonequilibrium statistical mechanics, we introduce here a novel generalization of the Lindley distribution, by regarding its shape parameter as a stochastic variable. This procedure is particularly well motivated in astronomy applications, since this parameter may fluctuate over a large spatiotemporal scale. By appealing to the central limit theorem, two families of distributions are constructed, associated with additive and multiplicative random processes. We discuss in detail these distributions, we study their asymptotic behavior, and compute their moments. We briefly discuss their possible use in astronomy, biology, and elsewhere. [ABSTRACT FROM AUTHOR]

As stated in the no-free-lunch (NFL) theory, there is not any optimizer suitable for all problems. This circumstance becomes the motivation of introducing a new swarm-based metaheuristic called best couple algorithm (BCA). BCA is constructed as a swarm-based metaheuristic where the swarm is split into two sub-swarms. There are two types of splitting. The first split is dividing the swarm into the first half and second half of swarms. The second split is dividing the swarm into the odd indexed swarm members and even indexed swarm members. There is a sub swarm leader representing the highest quality swarm member in every sub swarm. There are two sequential searches for every split: the motion toward the middle between two sub swarm leaders and the motion relative to the middle between two randomly picked sub swarm members. In the benchmark assessment, BCA is compared with total interaction algorithm (TIA), coati optimization algorithm (COA), language education algorithm (LEO), osprey optimization algorithm (OOA), and walrus optimization algorithm (WaOA). The result shows that BCA is superior to these five contenders as it is better than TIA, COA, LEO, OOA, and WaOA in 18, 18, 16, 18, and 18 functions respectively out of 23 functions. [ABSTRACT FROM AUTHOR]

In his famous letter to A. N. Prior dated 3 September 1958, Saul Kripke suggested the use of branching time in temporal logic. In this paper, however, it is argued that Prior worked with an idea close to the notion of branching time ('the big Y') already the year before he received Kripke's letter. It is likely that Prior's findings based on this early study can explain why Prior so quickly accepted the idea of branching time when he received Kripke's letter. [ABSTRACT FROM AUTHOR]

In this article, convolution-type fractional derivatives generated by Dickman subordinator and inverse Dickman subordinator are discussed. The Dickman subordinator and its inverse are generalizations of stable and inverse stable subordinators, respectively. The series representations of densities of the Dickman subordinator and inverse Dickman subordinator are also obtained, which could be helpful for computational purposes. Moreover, the space and time-fractional Poisson-Dickman processes, space-fractional Skellam Dickman process and non-homogenous Poisson-Dickman process are introduced and their main properties are studied. [ABSTRACT FROM AUTHOR]

The coalescent is a stochastic process representing ancestral lineages in a population undergoing neutral genetic drift. Originally defined for a well-mixed population, the coalescent has been adapted in various ways to accommodate spatial, age, and class structure, along with other features of real-world populations. To further extend the range of population structures to which coalescent theory applies, we formulate a coalescent process for a broad class of neutral drift models with arbitrary – but fixed – spatial, age, sex, and class structure, haploid or diploid genetics, and any fixed mating pattern. Here, the coalescent is represented as a random sequence of mappings C = C t t = 0 ∞ from a finite set G to itself. The set G represents the "sites" (in individuals, in particular locations and/or classes) at which these alleles can live. The state of the coalescent, C t : G → G , maps each site g ∈ G to the site containing g 's ancestor, t time-steps into the past. Using this representation, we define and analyze coalescence time, coalescence branch length, mutations prior to coalescence, and stationary probabilities of identity-by-descent and identity-by-state. For low mutation, we provide a recipe for computing identity-by-descent and identity-by-state probabilities via the coalescent. Applying our results to a diploid population with arbitrary sex ratio r , we find that measures of genetic dissimilarity, among any set of sites, are scaled by 4 r (1 − r) relative to the even sex ratio case. [ABSTRACT FROM AUTHOR]

The introduction of the spatial Lambda-Fleming–Viot model (Λ V) in population genetics was mainly driven by the pioneering work of Alison Etheridge, in collaboration with Nick Barton and Amandine Véber about ten years ago (Barton et al., 2010; Barton et al., 2013). The Λ V model provides a sound mathematical framework for describing the evolution of a population of related individuals along a spatial continuum. It alleviates the "pain in the torus" issue with Wright and Malécot's isolation by distance model and is sampling consistent, making it a tool of choice for statistical inference. Yet, little is known about the potential connections between the Λ V and other stochastic processes generating trees and the spatial coordinates along the corresponding lineages. This work focuses on a version of the Λ V whereby lineages move rapidly over small distances. Using simulations, we show that the induced Λ V tree-generating process is well approximated by a birth–death model. Our results also indicate that Brownian motions modelling the movements of lines of descent along birth–death trees do not generally provide a good approximation of the Λ V due to habitat boundaries effects that play an increasingly important role in the long run. Accounting for habitat boundaries through reflected Brownian motions considerably increases the similarity to the Λ V model however. Finally, we describe efficient algorithms for fast simulation of the backward and forward in time versions of the Λ V model. [ABSTRACT FROM AUTHOR]

This paper deals with the probabilistic analysis of a class of compartmental models formulated via a system of linear differential equations with time‐dependent non‐homogeneous terms. For the sake of generality, we assume that initial conditions and rates between compartments are random variables with arbitrary distributions while the source terms defining the flows entering the compartments are stochastic processes. We then take extensive advantage of the so‐called random variable transformation (RVT) technique to determine the first probability distribution of the solution of such a randomized model under very general hypotheses. In the simplest but relevant case of time‐independent source terms, we also obtain the probability distribution of the equilibrium, which is a random vector. Furthermore, we first particularize the aforementioned results for different types of integrable source terms that permit obtaining an explicit solution of the compartmental model: random constants, a train of Dirac delta impulses with random intensities, and a Wiener process. Secondly, we show an alternative approach based on the Liouville–Gibbs equation, which is useful when dealing with source terms that do not admit a closed‐form primitive. All the previous theoretical results are first illustrated through several numerical examples and simulations where a wide range of different probability distributions are assumed for the model parameters. The paper concludes by applying a compartmental model that describes the dynamics of oral drug administration through multiple chronologically spaced doses using synthetic data generated according to pharmacological references. [ABSTRACT FROM AUTHOR]

We define a state space and a Markov process associated to the stochastic quantisation equation of Yang–Mills–Higgs (YMH) theories. The state space S is a nonlinear metric space of distributions, elements of which can be used as initial conditions for the (deterministic and stochastic) YMH flow with good continuity properties. Using gauge covariance of the deterministic YMH flow, we extend gauge equivalence ∼ to S and thus define a quotient space of "gauge orbits" O . We use the theory of regularity structures to prove local in time solutions to the renormalised stochastic YMH flow. Moreover, by leveraging symmetry arguments in the small noise limit, we show that there is a unique choice of renormalisation counterterms such that these solutions are gauge covariant in law. This allows us to define a canonical Markov process on O (up to a potential finite time blow-up) associated to the stochastic YMH flow. [ABSTRACT FROM AUTHOR]