Optimal prediction of the last r-excursion time of Brownian motion models

Morphological effects on bacterial Brownian motion: Validation of a chiral two-body model

The backward problem of a stochastic space-fractional diffusion equation driven by fractional Brownian motion

Principal Component Analysis (PCA) is one of the most widely used approaches for multivariate datasets. Biologists use PCA to visualize data, identify patterns in large datasets, determine independent axes of variation, and reduce dimensionality for further statistical analyses. Phylogenetic PCA is an extension of regular PCA that seeks to identify the major axes of variation independent of the phylogeny. We extend these methods by estimating PCA parameters using an explicit probability modeling framework. We implement multiple models of trait evolution (Brownian motion, Ornstein-Uhlenbeck, Early Burst, and Pagel's λ) and use the Akaike Information Criterion (AIC) for model selection. We also introduce a probabilistic approach to select the number of principal components to retain from a PCA. We demonstrate the advantages of probabilistic PCA, such as incorporating the error, or noise, arising from dimensionality reduction, which is ignored in regular PCA. We use extensive simulations and an empirical dataset with 35 traits to show the method's performance. We implemented the new approach in the R package "do3PCA" available from the RCran repository.

The middle Permian represents a critical interval in therapsid evolution, when gorgonopsians emerged as some of the first specialized apex predators within terrestrial ecosystems. Despite their significance, the early diversification of Gorgonopsia in Gondwana remains poorly understood due to scarcity and fragmentary material. Here, we describe a nearly complete skull (BP/1/8260) with an occluded lower jaw from the lower Abrahamskraal Formation (Tapinocephalus Assemblage Zone). The specimen exhibits a distinctive combination of cranial features, including a transversely narrow snout, small orbital and temporal openings, V-shaped palatine bosses, and a vertically oriented occiput that distinguish it from all known gorgonopsians. Based on its unique morphology, a new taxon, Jirahgorgon ceto sp. nov., is established for this specimen. In our phylogenetic analysis, the new taxon forms a clade with Phorcys dubei, which is here named Phorcyidae fam. nov. Members of Phorcyidae are unique among basal African gorgonopsians in combining a vertical occiput and rubidgeine-like cranial proportions, indicating that large-bodied gorgonopsians were present in the Wordian and overturning notions that they were exclusively small carnivores until the Wuchiapingian. Basal skull length analyses indicate that body size evolution in Gorgonopsia, while largely random through time (Brownian motion evolutionary model), is nonetheless structured by shared ancestry. The discovery of Jirahgorgon illustrates the complexity of gorgonopsian evolution, showing an early appearance of large-bodied, robust morphotypes and highlighting the lower Abrahamskraal Formation as a key resource for understanding the initial radiation of theriodonts.

Abstract Stochastically gated interfaces play an important role in a variety of cellular transport processes, including diffusion through membrane ion channels and intercellular gap junctions. Most studies of stochastically-gated interfaces are based on macroscopic models that track the particle concentration averaged with respect to different realisations of the gate dynamics. In this paper we develop a novel probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for one-dimensional BM with an interface at the origin, which effectively sews together a sequence of BMs on the half-line with a totally absorbing boundary at $x=0$. Each time the particle is absorbed, the stochastic process is immediately restarted according to the following rule: if the gate is closed then BM restarts on the same side of the interface, whereas if the gate is open then BM restarts on either side of the interface with equal probability. In order to ensure that diffusion restarts in a state that avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is instantaneously shifted a distance $\epsilon$ from the origin. We explicitly solve the renewal equation for $\epsilon>0$ and show how the solution of a corresponding forward Kolmogorov equation is recovered in the limit $\epsilon\rightarrow 0$. However, the renewal equation provides a more general mathematical framework for modelling a stochastically gated interface by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We illustrate this by calculating the non-equilibrium stationary state across an interface in the presence of stochastic resetting. We conclude by discussing some of the mathematical challenges in extending the theory to higher-dimensional interfaces.

The height gap of planar Brownian motion is $$\frac{5}{\pi }$$

This study scrutinizes the effect of thermal radiation and Stefan blowing on the chemical reactive flow of Boger nanofluid across a stretched sheet with Darcy Forchheimer medium and heat generation using an intelligent computational framework based on Artifice neural network-Bayesian regularization. Furthermore, Brownian motion and thermophoresis properties have been examined. The suggested model of how Stefan blowing affects the chemical reactive flow of a Boger nanofluid with thermophoresis effects and Brownian motion has useful applications in a number of industrial and engineering operations. In chemical reactors, nano-coating technologies, and polymer processing, this model is essential for improving heat and mass transport processes. While the Boger nanofluid model accurately depicts non-Newtonian behaviour pertinent to biofluids and complex lubricants, Stefan blowing consideration offers insights on evaporation or suction effects. For the purpose of maximizing nanoparticle dispersion in cooling systems, fuel cells, and medicinal devices like targeted drug delivery systems where exact control over particle motion and chemical reactivity is crucial, Brownian motion and thermophoresis are also critical. The velocity profile improves as the Stefan blowing parameter values rise, but the thermal and concentration profiles decrease.

This paper investigates the finite-horizon survival probability for a system of correlated arithmetic Brownian motions with heterogeneous drifts and volatilities, focusing on the event in which one component remains strictly below all others. Using a whitening transformation of the covariance structure, we reduce the problem to the survival of a standard Brownian motion in a simplicial cone, characterized by its spherical cross-section. While explicit solutions are available in low dimensions, we address the computationally challenging tetrahedral angular case. We derive a semi-analytic formula for the survival probability via an eigenfunction expansion of the Dirichlet Laplace–Beltrami operator on this curved domain. For efficient implementation, we construct a diffeomorphism from the spherical tetrahedron to a fixed Euclidean tetrahedron, enabling the computation of angular eigenpairs through a stable finite-element scheme. For higher-dimensional regimes, we also introduce a covariance-based difficulty index and geometric bounds based on an inscribed spherical cap to assess spectral convergence and estimate long-time decay rates. Numerical experiments show that this offline–online approach achieves high accuracy and substantial speedups relative to Monte Carlo benchmarks.

Strong-damping limit of quantum Brownian motion in a disordered environment

Abstract There are a wide range of first passage time (FPT) problems in the physical and life sciences that can be modelled in terms of a Brownian particle binding to a reactive target surface and initiating a downstream event (absorption). However, prior to absorption, the particle may undergo several rounds of surface attachment (adsorption), detachment (desorption) and diffusion. That is, the surface is effectively ``sticky''. Alternatively, the surface may be stochastically gated so that absorption can only occur when the gate is open. In both cases one can view each attachment to the surface as a renewal event. In this paper we develop a renewal theory for stochastically gated target problems along analogous lines to previous work on sticky targets. We proceed by constructing a first renewal equation that relates the joint probability density for particle position and the state of a gate to the probability density and FPT density for a totally absorbing (non-gated) boundary. This essentially decomposes sample paths into an alternating sequence of bulk diffusion and instantaneous adsorption/desorption events, which is terminated when adsorption coincides with an open gate. In order to ensure that diffusion restarts in a state that avoids immediate re-adsorption, we assume that whenever the particle reaches a closed boundary it is instantaneously shifted a distance $\epsilon$ from the boundary (desorption-induced resetting). We explicitly solve the renewal equation in the one-dimensional case and show how the solution to the original gated FPT problem is recovered in the limit $\epsilon\rightarrow 0$. We then calculate the MFPT for absorption (assuming it exists) and determine its dependence on $\epsilon$ and the switching rate of the gate. We also show how spectral methods can be used to solve the renewal equation in higher spatial dimensions. We thus establish renewal theory as a general mathematical framework for modelling both sticky and stochastically gated targets.

The purpose of this work is to investigate the controllability of Langevin-type stochastic neutral impulsive integro-differential equations governed by the Caputo fractional derivative and driven by fractional Brownian motion, which arise naturally in systems exhibiting memory, impulsive effects, and stochastic disturbances. Using resolvent operators and fixed-point techniques, necessary and sufficient controllability conditions are established for the associated linear system, while the controllability of the nonlinear system is demonstrated via the Banach contraction principle. The theoretical results confirm that appropriate control functions can steer the system to a desired state within a finite time interval. Finally, illustrative numerical examples are provided to demonstrate the applicability and effectiveness of the obtained results, highlighting their relevance to practical stochastic control problems.

Athletes have better dynamic visual acuity (DVA) while seated compared to the general population. However, it is unclear whether athletes maintain superior DVA during standing and walking.PurposeTo examine DVA performance while standing and walking relative to seated for athletes compared to students from the general population.MethodsInteruniversity athletes (Athlete = 16; age = 20.7 ± 1.4) and recreationally-active students (Student = 17; age = 21.3 ± 1.4) performed a custom DVA task. A Tumbling 'E' was presented in four possible orientations moving in random (R) or horizontal (H) motion at 30°/s. DVA was performed during four conditions: seated, standing, low-intensity and moderate-intensity treadmill walking. Change in DVA from seated was calculated as the difference in log of the minimum angle of resolution (logMAR) from each condition and response time (RT, ms) was recorded using a keypad. Repeated measures mixed ANOVAs were conducted to compare DVA change scores and RT between groups for each condition.ResultsDuring R-motion, the Athlete group improved DVA change scores from seated to all conditions, whereas the Student group had worse DVA change scores (p = 0.015, f = 0.51). Both groups responded significantly faster for R-motion during moderate-intensity walking (p = 0.001). For H-motion, no differences were observed in DVA change score or RT.ConclusionAthletes improved performance on an random and unpredictable DVA task from seated to standing and walking compared to recreationally-active students who exhibited worse changes in DVA accuracy despite faster responses. Spatiotemporal properties of the DVA task appeared to modulate performance based on level of complexity.

Abstract In this paper we propose a refracted skew Brownian motion as a risk model with endogenous regime switching, which generalizes the refracted diffusion risk process introduced by Gerber and Shiu. We consider an optimal dividend problem for the refracted skew Brownian risk model and identify sufficient conditions, respectively, for barrier strategy, band strategy, and their variants to be optimal.

Abstract We consider Brownian motion with partial resetting, which has recently attracted a lot of attention in physics as well as the mathematics literature. We analyze the speed of convergence of this process towards stationarity as well as its quasistationary behavior. In particular, we prove the existence of a Yaglom limit and hence of a minimal quasistationary distribution. We use these results to study our main topic, namely the process conditioned on staying positive using methods which are well adapted to this specific process. It turns out that this process can be described explicitly as a three‐dimensional Bessel process with partial resetting with the same parameter but a modified resetting rate. This can be interpreted as an effect due to entropic repulsion.

Optimizing epidemic control: Nash game approach to stochastic modeling with Brownian motion

Thermally induced thermophoresis and Brownian motion in bio-convection with motile organisms over a progressively curved surface

Clarifying space use concepts in ecology: Range vs. occurrence distributions.

Averaging principle for stochastic fractional differential equations driven by Tempered Fractional Brownian Motion with two-time-scale Markov switching

ABSTRACT In this paper, we focus on estimating some unknown parameters of a geometric bifractional Brownian motion. A geometric bifractional Brownian motion satisfies a stochastic differential equation driven by a bifractional Brownian motion. Firstly, using the method of quadratic variation for a Gaussian process and the maximum likelihood method, we give the estimators for the unknown parameters. Then, we prove the asymptotic properties of the estimators. Secondly, the Monte Carlo method is used for simulation. Compared with the single maximum likelihood estimation method, the results show that the method in this paper is effective, reliable, and superior. Finally, we conduct an empirical study of financial markets with real financial data from Danimer Scientific Inc‐A (DNMR.N). By using path simulation, Euclidean distance and out‐of‐sample forecasting compared to other classical models, we effectively validate the superiority of the model in this paper in describing financial time series.