Stochastic Process Research Articles

Research on the Efficacy of Markov Chains in Analyzing and Predicting Chemical Reaction Rates

Abstract. The Markov chain is a powerful mathematical model used to describe stochastic processes in which the probability of each event depends only on the state attained in the previous event. The chemical reaction rate is a measure of how quickly or slowly a chemical reaction proceeds in multiple stages. This research paper uses the case study of enzymes to investigate the efficacy of the Markov chain in analyzing and predicting chemical reaction rates. By modeling chemical reaction pathways using Markov chains, this study tries to explore how Markov chains help capture and predict a chemical systems dynamic behavior. Scientists have been using multiple methods to predict chemical reaction rates, which involve methods like the Arrhenius equation, transition state theory, molecular dynamics, quantum mechanics, and machine learning. In conclusion, after comparing the advantages and disadvantages of the Markov chain predicting method with those methods above, Markov chains did stand out and provide valuable insights into the understanding of reaction dynamics.

Impact of organic carbon composition on bacterial taxa assembly upon phosphorus addition in organic and mineral soil layers of a Robinia pseudoaccacia plantation

Increasing phosphorus (P) inputs significantly affect the nutrient cycle of terrestrial ecosystems. The community structure and assembly processes of soil bacteria, which are the primary participants in nutrient cycling within terrestrial ecosystems, are affected by P inputs. However, the assembly processes and driving mechanisms of soil bacterial communities in artificial forest ecosystems in response to P inputs remain unclear. Therefore, in this study, we conducted in situ experimental P additions (0, 0.5, 1, 2, and 4 g P m-2 year-1) to artificial Robinia pseudoacacia forests on the Loess Plateau. We investigated the effects of P addition on the soil organic carbon (SOC) composition, bacterial community characteristics (total, abundant, and rare taxa), and assembly processes in the organic horizon (OH) and mineral horizon (MH) layers of the soil. The results showed that P addition significantly changed soil physicochemical factors, such as available P (AP), SOC content, and SOC composition in terms of the labile (LC) and recalcitrant carbon (RC) fractions. The proportion of the LC fraction increased in OH and decreased in MH, whereas the RC fraction showed the opposite trends. P addition increased the proportion of eutrophic bacteria in OH, whereas it decreased the proportion of oligotrophic bacteria. In contrast to those in OH, the changes in eutrophic and oligotrophic bacteria in MH exhibited the opposite trend. With increasing P levels, the diversity of the bacterial communities in the OH did not change significantly. In the MH, total and rare taxa diversity increased, while the diversity of abundant taxa decreased. During P addition, the total and rare taxa bacterial communities showed stochastic assembly, whereas the assembly of abundant taxa shifted from stochastic to deterministic processes in the OH and from deterministic to stochastic processes in the MH. The structural equation model revealed that during P addition, the assembly of total and rare bacterial communities in the OH and MH was regulated by AP and the LC fraction, respectively. Meanwhile, the assembly of abundant taxa in OH and MH was influenced by the LC fraction and RC fraction, respectively. Our findings highlight the critical role of SOC composition across different bacterial taxa, with abundant bacterial taxa exhibiting heightened sensitivity to environmental filtration and the availability of SOC resources. Our study provides new insights into the community assembly and driving mechanisms of soil bacteria in artificial forest soils during P addition.

Strict monotonicity of stochastic process extreme distributions

Strict monotonicity is proved for the distributions of extremes of processes consisting of series of bounded function with independent random coefficients, in particular for zero mean continuous Gaussian processes over compact metric space. These results have wide applications to global inference problems on unknown functions.

How the ETAS models were created, used, and evolved -- Personal views and perspectives

Statistical seismology originated in Japan 130 years ago. Through the marriage with stochastic point processes in about half a century, we can now provide on-line estimation, real-time forecasting and direct diagnosis of seismic activity. This manuscript describes the origin and recent development of the ETAS models, their relationship to forecasting problems, and their diagnostic studies of the physical phenomena of seismic activity. I will take this opportunity to focus on my research experiences and views.

Random-Coupled Neural Network

Improving the efficiency of current neural networks and modeling them on biological neural systems have become prominent research directions in recent years. The pulse-coupled neural network (PCNN) is widely used to mimic the computational characteristics of the human brain in computer vision and neural network fields. However, PCNN faces limitations such as limited neural connections, high computational costs, and a lack of stochastic properties. This study proposes a random-coupled neural network (RCNN) to address these limitations. RCNN employs a stochastic inactivation process, selectively inactivating neural connections using a random inactivation weight matrix. This method reduces the computational burden and allows for extensive neural connections. RCNN encodes constant stimuli as periodic spike trains and periodic stimuli as chaotic spike trains, reflecting the information encoding characteristics of biological neural systems. Our experiments applied RCNN to image segmentation and fusion tasks, demonstrating its robustness, efficiency, and high noise resistance. Results indicate that RCNN surpasses traditional methods in performance across these applications.

Stochastic Processes Dominate the Assembly of Soil Bacterial Communities of Land Use Patterns in Lesser Khingan Mountains, Northeast China

To meet the demands of a growing population, natural wetlands are being converted to arable land, significantly impacting soil biodiversity. This study investigated the effects of land use changes on bacterial communities in wetland, arable land, and forest soils in the Lesser Khingan Mountains using Illumina MiSeq 16S rRNA sequencing. Soil physicochemical properties and enzyme activities were measured using standard methods, while microbial diversity was assessed through sequencing analysis. Our findings revealed that forest soils had significantly higher levels of total potassium (2.62 g·kg−1), electrical conductivity (8.22 mS·cm−1), urease (0.18 mg·g−1·d−1), and nitrate reductase (0.13 mg·g−1·d−1), attributed to rich organic matter and active microbial communities. Conversely, arable soils showed lower total potassium (1.94 g·kg−1), reduced electrical conductivity, and suppressed enzyme activities due to frequent tilling and fertilization. Wetland soils exhibited the lowest values primarily due to water saturation, which limits organic matter decomposition and microbial activity. Land use changes notably reduced microbial diversity, with conversion from forest to arable land leading to habitat loss. Forest soils supported higher abundances of Proteobacteria (37.59%) and Actinobacteriota (34.73%), while arable soils favored nitrogen-fixing bacteria. Wetlands were characterized by chemoheterotrophic and anaerobic bacteria. Overall, these findings underscore the profound influence of land use on soil microbial communities and their functional roles, highlighting the need for sustainable management practices.

Dynamic programming principle for optimal control of uncertain random differential equations and its application to optimal portfolio selection

This study aimed to examine an uncertain stochastic optimal control problem premised on an uncertain stochastic process. The proposed approach is used to solve an optimal portfolio selection problem. This paper’s research is relevant because it outlines the procedure for solving optimal control problems in uncertain random environments. We implement Bellman’s principle of optimality method in dynamic programming to derive the principle of optimality. Then the resulting Hamilton-Jacobi-Bellman equation (the equation of optimality in uncertain stochastic optimal control) is used to solve a proposed portfolio selection problem. The results of this study show that the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in optimal portfolio selection. Also, the study results indicate that the optimal fraction of investment is independent of wealth. The main conclusion of this study is that, in Itô-Liu financial markets, the dynamic programming principle for optimal control of uncertain stochastic differential equations can be applied in solving the optimal portfolio selection problem.

On strongly m-convex stochastic processes

AbstractIn this paper, we introduce the concept of strongly m-convex stochastic processes and present some basic properties of these stochastic processes. We derive Hermite-Hadamard type inequalities for stochastic processes whose first derivatives in absolute values are strongly m-convex. The results presented in this paper are a generalization and extension of previously known results.

Soil total phosphorus mediate the assembly processes of rhizosphere microbial communities of ficus species in a tropical rainforest

Revealing the assembly processes of plant rhizosphere microbial communities and the underlying influencing factors is essential for understanding the biodiversity and function of forest ecosystem. However, it remains unclear how deterministic and stochastic processes shape community structure and their relative importance in phosphorus-limited tropical environments. Here, we investigated the diversity, composition, and assembly processes of rhizosphere microbial communities of Ficus species in the Xishuangbanna region of southwest China, using methods such as high-throughput sequencing, variance partitioning analysis and null model analysis. We found that the community assembly processes of bacteria and fungi were primarily dominated by deterministic processes, with the fungal group being more deterministic than the bacteria group. Soil total phosphorus (TP) was the primary determinant of the composition and assembly of the rhizosphere microbial community, explaining 12.58% and 21.35% of the compositional variation in bacterial and fungal communities, respectively, and accounting for 14% of the microbial community assembly, but has a minor impact on their alpha diversity. This study highlights the distinct environmental driving factors of community composition and community assembly. The exposed positive relationship between soil TP and microbial deterministic process has inspiration for link of microbial community functions to soil function and sustainable forest management.

Characterization of a wiener process taking values in a Hilbert space

This paper characterize wiener process by taking values in a Hilbert space. A standard wiener process is stochastic process indesed by nonnegative real numbers t with the following properties: W0=0 With probability 1, the function t àWt is continuous in t. The process has stationary, independent increments. The increments - Ws has the NORMAL (0,t) distribution

A Method for Predicting the Remaining Service Life of Finite Data Products Based on Gaussian Stochastic Processes

There is a problem of poor predictive performance when predicting the remaining service life of products. Therefore, this paper proposes a finite data product remaining service life prediction method based on Gaussian stochastic processes. Analyze the types of product failure and degradation under limited data, and determine the factors affecting the degradation performance of product life through constant stress accelerated degradation tests, step stress accelerated degradation tests, and sequential stress accelerated aging tests; using the Wiener process to analyze the performance degradation characteristics of products, setting product failure thresholds, defining the remaining life of the product based on the degradation process, and using equal interval partitioning methods to determine the product health index, and clarifying the product failure process under limited data; by calculating the mathematical expected value, the numerical characteristics of the remaining life of the product are defined. Variance and moments are used to determine the predicted second-order central moment and origin moment. Based on the monotonic increasing characteristics of the inverse Gaussian process, the probability distribution of the random variable of product failure is determined after the Gaussian process. Under limited data, the actual working time of the product and the degradation amount at the time scale are clarified, and a preset threshold is set for the degradation amount. Build a residual life prediction model for product degradation and output the prediction results. The experimental results show that this method can accurately predict the remaining service life of the product and has good predictive performance.

Simulation of topological data using a stochastic model applied to protein engineering

Topological data analysis (TDA) is a powerful tool used to extract meaningful properties from the topological structure of data, enabling the discovery of complex relationships and patterns within datasets. It provides a unique perspective by focusing on the shape and structure of data rather than traditional statistical methods. In our research, we aim to explore the impact of stochastic processes on TDA. Specifically, we investigate how introducing stochastic calculations can enhance the analysis of topological features, offering a more nuanced understanding of the data. Stochastic processes account for randomness and variability, which are inherent in many real-world datasets. By integrating these processes into TDA, we can uncover more precise relationships between data points and obtain deeper insights into the underlying topological structures. Our work highlights the advantages of this approach, particularly in improving the robustness and accuracy of TDA in the presence of noise and variability. The results of this research have the potential to broaden the applicability of TDA in fields such as biology, neuroscience, and machine learning, where complex data structures are common, and stochastic behavior often plays a critical role in shaping the data’s characteristics.

Fokker-Planck Central Moment Lattice Boltzmann Method for Effective Simulations of Fluid Dynamics

We present a new formulation of the central moment lattice Boltzmann (LB) method based on a minimal continuous Fokker-Planck (FP) kinetic model, originally proposed for stochastic diffusive-drift processes (e.g., Brownian dynamics), by adapting it as a collision model for the continuous Boltzmann equation (CBE) for fluid dynamics. The FP collision model has several desirable properties, including its ability to preserve the quadratic nonlinearity of the CBE, unlike that based on the common Bhatnagar-Gross-Krook model. Rather than using an equivalent Langevin equation as a proxy, we construct our approach by directly matching the changes in different discrete central moments independently supported by the lattice under collision to those given by the CBE under the FP-guided collision model. This can be interpreted as a new path for the collision process in terms of the relaxation of the various central moments to “equilibria”, which we term as the Markovian central moment attractors that depend on the products of the adjacent lower order moments and a diffusion coefficient tensor, thereby involving of a chain of attractors; effectively, the latter are nonlinear functions of not only the hydrodynamic variables, but also the non-conserved moments; the relaxation rates are based on scaling the drift coefficient by the order of the moment involved. The construction of the method in terms of the relevant central moments rather than via the drift and diffusion of the distribution functions directly in the velocity space facilitates its numerical implementation and analysis. We show its consistency to the Navier-Stokes equations via a Chapman-Enskog analysis and elucidate the choice of the diffusion coefficient based on the second order moments in accurately representing flows at relatively low viscosities or high Reynolds numbers. We will demonstrate the accuracy and robustness of our new central moment FP-LB formulation, termed as the FPC-LBM, using the D3Q27 lattice for simulations of a variety of flows, including wall-bounded turbulent flows. We show that the FPC-LBM is more stable than other existing LB schemes based on central moments, while avoiding numerical hyperviscosity effects in flow simulations at relatively very low physical fluid viscosities through a refinement to a model founded on kinetic theory.

Construction and sample path properties of Brownian house-moving between two curves

AbstractThis study aims to construct a stochastic process called “Brownian house-moving,” which is a Brownian bridge conditioned to stay between two curves. To construct this process, statements are prepared on the weak convergence of conditioned Brownian motions, conditioned Brownian bridges, and conditioned three-dimensional Bessel bridges. Moreover, the sample path properties of Brownian house-moving are studied as well.

The impact of hydroclimate-driven periodic runoff on hydropower production and management

This study evaluates the impact of hydroclimate-driven periodic runoff on hydropower operations and production, with a focus on how the forecasted biennial periodicity of runoff time series could affect the efficiency of hydropower generation. Hydrologic stochastic processes are utilized to forecast long-term runoff, and seven hydroclimate scenarios are developed to be input into a production management model, allowing for an analysis of how periodic hydroclimate variations influence hydropower management and output. The results reveal that the biennial alternation between wet and dry years is a key factor affecting hydropower operations in the Dalälven River Basin. Notable differences between wet- and dry-year scenarios were observed in terms of power efficiency, production output, and forecasting accuracy. Operating hydropower systems based on dry-year runoff forecasts in wet years results in a 1.63% decrease in production efficiency and a reduction of 9,104 MWh in power generation. Conversely, applying wet-year forecasts in dry years slightly boosts production efficiency by 0.31% and increases power generation by 7,832 MWh. Scenarios that adhere to biennial periodicity offer the highest forecasting accuracy, particularly when applying dry-year forecasts in dry years in winter and spring, which produce the most precise predictions. In contrast, using dry-year forecasts in wet years results in the lowest forecasting accuracy.

Modeling and analysis of uncertain Bass diffusion model driven by uncertain Liu process

The Bass diffusion model is one of the most influential models for predicting the diffusion of new products and technologies, but the diffusion process will be affected by some uncertain factors. Usually, such kinds of uncertain factors are characterized as stochastic processes. However, in many cases, this approach can lead to uncertain issues. Therefore, in this article, a Bass diffusion model that incorporates the uncertainty theory is studied. First, we present an uncertain Bass diffusion model where the uncertain disturbance is modeled as an uncertain Liu process. Then, the existence and uniqueness of the uncertain Bass diffusion model are analyzed. Moreover, the uncertain Bass diffusion model is solved by using the 99-α-Runge-Kutta method. Finally, we carry out a numerical simulation on the output of the integrated circuit industry in China by using uncertain differential equations and stochastic differential equations. The results show that modeling with uncertain differential equations is superior to using stochastic differential equations.

A Denseness Property of Stochastic Processes

AbstractIn this work, we show the following general property of stochastic processes: Denseness of probability distributions extends to denseness of stochastic processes. In particular, for every $$d\in \mathbb {N}$$ d ∈ N , we consider any class of probability distributions which is dense in the space of probability distributions on $$\mathbb {R}^{d}$$ R d with respect to convergence in distribution. Using these classes, we construct various explicit families of times series, continuous processes and càdlàg processes that are dense in their respective spaces with respect to convergence in distribution. This is particularly interesting because, for processes, finite-dimensional distributions convergence does not automatically imply convergence in distribution.

Complex stochastic optimal control foundation of quantum mechanics

Abstract Recent studies have extended the use of the stochastic Hamilton-Jacobi-Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies often assume that it is valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers when finding optimal controls. This paper explores the application of the HJB equation in the context of complex variables. It provides an in-depth investigation of the stochastic movement of quantum particles within the framework of stochastic optimal control theory. We obtain the complex diffusion coefficient in the stochastic equation of motion using the Cauchy-Riemann theorem, considering that the particle’s stochastic movement is described by two perfectly correlated real and imaginary stochastic processes. During the development of the covariant form of the HJB equation, we demonstrate that if the temporal stochastic increments of the two processes are perfectly correlated, then the spatial stochastic increments must be perfectly anti-correlated, and vice versa. The diffusion coefficient we derive has a form that enables the linearization of the HJB equation. The method for linearizing the HJB equation, along with the subsequent derivation of the Dirac equation, was developed in our previous work [V. Yordanov, Scientific Reports 14, 6507 (2024)]. These insights deepen our understanding of quantum dynamics and enhance the application of stochastic optimal control theory to quantum mechanics.

Interplay between vanadium distribution and microbial community in soil-plant system

Soil-plant system play an essential role in distribution and transformation of vanadium (V). V shapes the diversity of soil communities, while soil microorganisms mediate V transformation. Plants also absorb V from surrounding soil. However, the study of microbial response to V stress in different soil-plant compartments is limited, and the metabolic functions driving V transformation across these systems remain elusive. The study investigates the distribution of V in soil-plant systems nearby a V smelter. 16S rRNA sequencing and metagenomics are utilized to reveal the microbial adaptation and V transformation in bulk soil, rhizosphere, and endosphere. Bothriochloa ischaemum (L.) Keng. (BK) exhibits higher phytoextraction potential (TF = 0.74 ± 0.26). Environmental variables, including pH, V, OM, and AP, show significant (p < 0.05) influence in soil community composition, with homogeneous selection governing the assembly processes in bulk soil and rhizosphere, while stochastic process dominates endospheric assembly. Metagenomic investigation revealed a coordinated metabolic pathway between functional taxa in soil and plants, which lead to root uptake and translocation. V stress is mitigated through Nocardioide, Microvirga, and Solirubrobacter, putatively harboring V(V) reduction genes n arG and mtrC in soil. In rhizosphere, citrate synthase gltA and alkaline phosphatase phoD exhibit functional potential to facilitate formation of V-complexation to increase V mobility. In endoshere, endophytic Enterobacter further detoxifies V(V), and likely promotes V translocation through siderophore biosynthesis gene, iucA. These findings enhance our understanding on interplay between V and microbial community in soil-plant systems, which is instrumental in developing mitigation plan for V contaminated sites.

Flexible Bayesian modeling for longitudinal binary and ordinal responses

Longitudinal studies with binary or ordinal responses are widely encountered in various disciplines, where the primary focus is on the temporal evolution of the probability of each response category. Traditional approaches build from the generalized mixed effects modeling framework. Even amplified with nonparametric priors placed on the fixed or random effects, such models are restrictive due to the implied assumptions on the marginal expectation and covariance structure of the responses. We tackle the problem from a functional data analysis perspective, treating the observations for each subject as realizations from subject-specific stochastic processes at the measured times. We develop the methodology focusing initially on binary responses, for which we assume the stochastic processes have Binomial marginal distributions. Leveraging the logits representation, we model the discrete space processes through continuous space processes. We utilize a hierarchical framework to model the mean and covariance kernel of the continuous space processes nonparametrically and simultaneously through a Gaussian process prior and an Inverse-Wishart process prior, respectively. The prior structure results in flexible inference for the evolution and correlation of binary responses, while allowing for borrowing of strength across all subjects. The modeling approach can be naturally extended to ordinal responses. Here, the continuation-ratio logits factorization of the multinomial distribution is key for efficient modeling and inference, including a practical way of dealing with unbalanced longitudinal data. The methodology is illustrated with synthetic data examples and an analysis of college students’ mental health status data.