The increasing volume and sampling frequency of telemetry data require a critical evaluation of algorithms for estimating animal home range (HR), especially for highly mobile species with complex territorial behavior. In this study, we conducted a comprehensive comparative analysis of the performance of 6 HR estimation algorithms using tracking data from Golden Eagles (Aquila chrysaetos). Model evaluation was performed using aggregated multi-criteria decision analysis (MCDA), spatial indices (SQI, M1/M2), predictive power metrics (AUC, LL), and the two-sample Kolmogorov-Smirnov test to assess behavioral realism. The analysis was conducted on both original high-frequency data and artificially rarefied (downsampled) tracks (up to 1 location per 6 hours). The results revealed fundamental differences in the spatial topology of the models. Classical approaches (KDE h ref, BBMM) demonstrated excessive oversmoothing of the HR core area (50%). In contrast, dynamic Brownian Bridge Movement Models (dBBMM) were highly sensitive to sampling density. On high-frequency data, the algorithm falls into spatial overfitting, classifying over 90% of actual transit locations as statistical noise. Extreme downsampling of tracks revealed a paradox of algorithm convergence and an inversion of their scale nesting. Also, it led to a critical loss of behavioral microstructure across all interpolative models. Based on the aggregation of all metrics, the AKDE method was recognized as the absolute leader. By integrating the temporal autocorrelation structure (OU/OUF models), AKDE proved to be the only algorithm that preserved the reliable spatial extent of the HR and predictive accuracy under sparse data conditions. This study demonstrates that for highly mobile avian predators, route interpolation becomes ecologically inadequate, and it is precisely AKDE that allows the transformation of discrete transit locations into a biologically robust HR model.

Multimodal Brownian bridge diffusion model for controllable synthetic medical image generation

On the distribution of the integral of a function with respect to a Brownian bridge

Augmented bridge matching, as an extension of bridge matching, not only retains the flexibility to connect arbitrary pairs of distributions but also preserves the coupling information. It has garnered significant attention in generative learning. In this work, we present a comprehensive end-to-end convergence guarantee for the augmented Brownian bridge diffusion model. Our results show that for general joint distributions with bounded support, under the mild Lipschitz conditions of the drift term, the generated distribution of augmented Brownian bridge diffusion model is guaranteed to converge to the target distribution in terms of the second-order Wassersteindistance.

Abstract As a prevalent non‐invasive screening technique, Wireless Capsule Endoscopy is often hindered by poor image quality, including under‐/overexposure and low light condition. While illumination correction based on diffusion modeling or frequency‐domain decomposition has shown effectiveness, existing methods often (1) underexploit structural information, and (2) lack adaptive strategies for varying illumination degradations, leading to suboptimal restoration and unnecessary computation. To this end, we propose Brownian Bridge Diffusion Transformer‐Mixture‐of‐Experts (BiT‐MoFE), a unified adaptive framework that integrates the merits of the two paradigms for endoscopic illumination correction. We adopt a Brownian Bridge Diffusion framework, in which an efficient Transformer serves as the backbone network, and design a frequency‐decomposed MoFEs module to explicitly handle illumination and image structure simultaneously. By dynamically selecting the most suitable experts conditioned on exposure cues and diffusion timesteps, our framework achieves a strong balance between restoration fidelity and computational efficiency. Extensive experiments on multiple public datasets demonstrate that BiT‐MoFE achieves state‐of‐the‐art performance on both exposure correction and low‐light enhancement tasks.

Urbanization has contributed to the decline of many wildlife species through habitat loss. To examine how Bald Eagles (Haliaeetus leucocephalus) respond to urbanization we used dynamic Brownian bridge movement models to estimate home range size and core-use areas of 24 territorial Bald Eagles affixed with GPS/GSM transmitters during five different stages of the annual cycle. We then used a mixed-effects linear regression model to identify the land cover characteristics associated with home range size and core-use areas. We found that Bald Eagle home ranges and core-use areas varied in size and were often discontinuous. Home ranges and core-use areas tended to be larger during the pre-nesting and non-nesting stages and smaller during the nestling and post-fledge stages, with these differences being more pronounced in females. Home ranges and core-use areas were smaller in areas containing more water and lower canopy cover. Home ranges, but not core-use areas, were also smaller in areas containing higher amounts of herbaceous wetlands. Home range size was not strongly associated with impervious cover, a metric of human development intensity, but core-use area size increased with impervious cover. Eagle core-use areas did not contain more than an average of 39 % impervious cover, defined as low development intensity by the U.S. Geological Survey. In our study, home ranges covered areas with higher levels of impervious surface compared to core-use areas, demonstrating that while eagles will inhabit areas with moderate levels of development, they concentrate their use in areas with lower levels of development. These findings suggest that Bald Eagles can use urbanizing landscapes, but care should be taken to protect key nesting and foraging sites from high density development.

Prob-BBDM: A Probabilistic Brownian Bridge Diffusion Model for MRI sequence image-to-image translation.

We tracked two rescued spot-billed pelicans, Pelecanus philippensis, a hand-raised juvenile and a rehabilitated adult, using GPS telemetry for approximately two years after release from Kokkare-Bellur Community Reserve, southern India. Dynamic Brownian Bridge Movement Models revealed distinct age-dependent utilization distribution strategies. The juvenile exhibited progressive spatial expansion, from localized movements in the first year to distant wetlands in the second year. The adult dispersed immediately and remained largely within the Bengaluru–Mysuru corridor, with daily distance traveled increasing during the breeding season. Results demonstrate successful post-release dispersal and landscape use, highlighting the importance of wetland network connectivity for the conservation of the pelican.

Abstract Understanding a population's distribution depends on observing the presence and movement of individuals throughout their range. For highly mobile marine species, these observations typically rely on high effort monitoring programs. Tracking enough individuals to understand trends in movement behavior is not always logistically feasible, and animals are less likely to be observed in migratory or transitional habitats. To optimize observation data we built a Brownian bridge movement model to generate spatially and temporally explicit space use estimations of individuals along tracks created from intermittent sightings of North Atlantic right whales from 1980 to 2022. Right whales can be identified by unique callosities and markings, providing a noninvasive opportunity to link sightings to individual movement. This model generated location probability estimates of medium‐ and large‐scale movements in biologically plausible habitats. A total of 351,214 d of occurrence distributions were calculated from 67,840 sightings attributed to 806 individuals, representing more than a five‐fold increase in individual spatial information. From 1980 to 2022, the model generated space use estimates for at least one whale on 75.7% of days, compared to the underlying visual sightings data, which only provided observations on 38.7% of days. Model outputs compared to tracks of tagged whales demonstrated proficiency estimating space use across regions. The occurrence distributions produced depict known changes in seasonal and decadal space use and estimate transitional space use where observations are sparse. These methods improve spatial distribution predictions in intermittently observed animals and expand the utility of stationary sightings without constraining predictions to historical relationships with the environment.

Quantifying animal movements is necessary for answering a wide array of research questions in ecology and conservation biology. Consequently, ecologists have made considerable efforts to identify the best way to estimate an animal's home range, and many methods of estimating home ranges have arisen over the past half a century. Most of these methods fall into two distinct categories of estimators that have only recently been described in statistical detail: those that measure range distributions (methods such as kernel density estimation that quantify the long-run behavior of a movement process that features restricted space use) and those that measure occurrence distributions (methods such as Brownian bridge movement models and the Correlated Random Walk Library that quantify uncertainty in an animal movement path during a specific period of observation). In this paper, we use theory, simulations, and empirical analysis to demonstrate the importance of appropriately using these two categories of distributions and their estimators. Conflating range and occurrence distributions can have serious consequences for ecological inference and conservation practice. For example, in most situations, home ranges estimated using estimators of occurrence distributions are too small, and this problem is exacerbated by ongoing improvements in tracking technology that enable more frequent and more accurate data on animal movements. We encourage researchers to use estimators of range distributions to quantify home ranges and estimators of occurrence distributions to answer other questions in movement ecology, such as when and where an animal crossed a linear feature, visited a location of interest, or interacted with other animals.

Abstract The eastern box turtle ( Terrapene carolina carolina ) is a long-lived terrestrial turtle species distributed throughout the eastern United States that has experienced widespread population decline. Many eastern box turtle populations are persisting as remanent populations in small, fragmented urban green spaces. We investigated the movement and resource selection of eastern box turtles within a mid-Atlantic region urban forest in the eastern United States. We used a combination of turtle occurrence data (via visual encounter surveys) and radio telemetry to create resource selection functions. Additionally, we applied a simulation modeling approach and modeled activity areas via dynamic Brownian Bridge Movement Models to quantify interactions between turtles and roads or trails. We also used these models to determine the propensity for turtles to move outside of the managed urban forest boundary and into surrounding development. We observed that turtles selected for deciduous forest patches and avoided roads and trails despite the urban forest having very little available areas where anthropogenic features could be avoided. We also demonstrated observed (and probable) movements outside of the urban forest boundary. Although eastern box turtles are persisting within the urban green space we examined, our work determined that interactions with roads and trails, and movements outside of protected boundaries into developed areas present challenges to individuals navigating the urban forest.

Estimating the age of a given clade is not trivial. Most dating studies rely on a node dating approach, which is highly sensitive to fossil placement uncertainties. A perfect example of that can be found in bark beetles (Curculionidae: Scolytinae). Molecular dating of the subfamily has often been anchored to a single fossil, †Cylindrobrotus pectinatus (~125 Ma), systematically used as a calibration point on the stem node of the subfamily. However, its phylogenetic position has never been tested through formal analyses, casting uncertainty over previous estimates. Here, we rely on total-evidence dating, integrating morphological and molecular data along with 19 scolytine fossils, to refine the divergence time estimates of the subfamily. We additionally used a traditional node-dating approach, and a fossil-only dating method, the Bayesian Brownian Bridge model, to estimate the age of the clade and compare the different results. Our findings suggest that †C. pectinatus is more closely related to Dryocoetini s.l. or Ipini-Dryocoetini s.l. clades rather than being a stem lineage of Scolytinae. In our case, when †C. pectinatus is inferred inside the subfamily, total-evidence dating with different pattern of branch rates across the tree (unlinking clocks for each gene and the morphological data), appeared more appropriate than node dating for estimating the age of Scolytinae, with ages ranging from ~149.9 to ~134.9 Ma. Under the same priors, node dating produced earlier ages, from ~195.4 to ~131.7 Ma, while the Bayesian Brownian Bridge produced ages ranging from ~168.4 to ~141.2 Ma. Across all methods, Scolytinae are constantly inferred to have diversified before the Angiosperm Terrestrial Revolution. These results highlight the importance of integrating multiple dating approaches to mitigate biases inherent to any single method, ultimately leading to more reliable divergence estimates.

A novel Brownian bridge diffusion-based generative inpainting algorithm for ancient murals

Tokenizing continuous time-series signals for Large Language Models (LLMs) remains a fundamental challenge due to the mismatch between numerical temporal data and discrete token-based architectures. Existing approaches (such as patching, symbolic aggregation, or quantization) address this problem in isolation, often sacrificing numerical fidelity, temporal coherence, or computational efficiency. We propose Hierarchical Symbolic–Quantized Patching (HSQP), a unified tokenization framework that hierarchically integrates patch segmentation, adaptive Brownian Bridge–based symbolic aggregation (ABBA), and affine quantization into a jointly embedded, non-separable token representation. Unlike sequential preprocessing pipelines, HSQP fuses symbolic cluster identities and quantized numerical descriptors into compact dual-semantic tokens that simultaneously encode structural temporal patterns and bounded numerical precision. The hierarchical design reduces sequence length, constrains reconstruction drift through patch-level anchoring, and introduces bounded quantization noise that stabilizes downstream Transformer attention. We provide theoretical analysis of quantization error propagation and demonstrate that reconstruction error remains locally bounded under Lipschitz continuity assumptions. Extensive experiments on six benchmark datasets show that HSQP consistently improves reconstruction fidelity and downstream forecasting accuracy while achieving stable compression ratios and balanced token entropy. Integrated as a plug-and-play module with frozen LLM backbones, HSQP enhances predictive performance without architectural modification. These results establish HSQP as an efficient, interpretable, and scalable tokenization paradigm for adapting continuous time-series data to LLM-based forecasting frameworks.

We introduce an ensemble of spatial networks built from the junctions of hindered-rotation chains, incorporating directional correlations between bonds, an aspect ignored in the standard network modeling paradigm. The emergent random networks support geodesics with a wandering exponent ξ=1/2, and a travel-time fluctuation exponent χ=0, consistent with the KPZ relation, yet violating the bound χ≥1/8 predicted in the Poissonian framework. Transverse deviations follow the Kolmogorov distribution, indicating similarities between Brownian bridge excursions and geodesics in a random medium with correlated edges orientations. These results reveal a new universality class of Euclidean first-passage percolation, where local orientational memory reshapes transport properties and challenges existing bounds for random spatial networks.

It is well-known that the detection of change-points in heavy-tailed time series is an open problem since the traditional tests may not have a power. This article introduces a winsorized cumulative sum (CUSUM) approach to solve this problem. We begin by investigating the winsorized CUSUM process and then use it to construct the Kolmogorov-Smirnov (KS) test and the Self-normalized (SN) test. Under the null hypothesis, it is shown that each weakly converges to the maximum of a function related to the standard Brownian bridge. Under the alternative, we first study the behavior of tests after applying the winsorization technique, and then show that our tests have a power approaching to 1 as the sample size n → ∞ . Furthermore, we extend the winsorizing technique to test for multiple change-points without prior knowledge of the number of change points. Our framework is general and its assumptions are mild, so that our tests can be applied to a wide range of linear and nonlinear time series. The empirical results illustrate the effectiveness of our proposed procedures for change-point detection.

ABSTRACT We develop a procedure for testing and estimating the change point in the autoregressive moving average (ARMA) model with heavy‐tailed general generalized autoregressive conditional heteroskedasticity (G‐GARCH) noises. Based on the self‐weighted least absolute deviation estimator (SLADE), we propose two score‐type test statistics for change point detection. Under the null hypothesis, we show that one of them converges weakly to the maxima of a Brownian bridge, and the other one converges weakly to an extreme distribution. Furthermore, we prove that the SLADE of the change point converges weakly to the location of the maxima of a double‐sided random walk, and the SLADE of other parameters is asymptotically normal. Our two score‐type tests and SLADE are applicable for the data with an infinite variance, while not specifying the form of G‐GARCH noises. Finally, we use simulations and two real examples to demonstrate the usefulness of the proposed tests and estimator in handling the heavy‐tailed data with an infinite variance.

Abstract We establish large deviations for dynamical Schrödinger problems driven by perturbed Brownian motions when the noise parameter tends to zero. Our results show that Schrödinger bridges charge exponentially small masses outside the support of the limiting law that agrees with the optimal solution to the dynamical Monge–Kantorovich optimal transport problem. Our proofs build on mixture representations of Schrödinger bridges and establishing exponential continuity of Brownian bridges with respect to the initial and terminal points.

Understanding how wild species utilize their resources is crucial for designing protected natural areas. Using the dynamic Brownian bridge motion model (DBBMM), this study employed satellite telemetry to calculate and analyze the home ranges and core areas of black bear individuals in Mexico. Various environmental factors (such as altitude, vegetation cover, the modified soil adjusted vegetation index, and distance to water bodies) and anthropogenic variables (including distance to primary and secondary roads and the human modification index) were examined to describe the bears' home ranges. The average home range estimated using MCP was greater in females (245.3 ± 355.0 km2) than in males (130.7 ± 149.3 km2), although there were no significant differences, with the 95% DBBMM (females, 68.5 ± 61.3 km2; males, 69.4 ± 117 km2; W = 20, p = 0.65). In contrast, the 50% DBBMM showed a tendency toward higher values in females (4.73 ± 1.12 km2) than in males (3.05 ± 4.8 km2), but without statistical significance. The generalized linear models revealed that for females and males, proximity to bodies of water influences the selection of core areas, as does proximity to primary roads. Females also use sites with low human modification, and vegetation was not a determining factor for either sex. This study provides insights into the environmental and anthropogenic variables that influence the establishment and size of core areas and home ranges of black bears in northern Mexico.

Infrared images have garnered significant interest due to their superior performance, driving extensive research on visible-to-infrared image translation. However, existing cross-domain generation methods lack specialization for infrared image generation under aerial perspective, leading to distribution inconsistencies between synthetic and real infrared images and failing to mitigate challenges like small-target blurring and background interference under aerial views. To address these issues, we propose an RGB-to-infrared image generation method based on the Brownian bridge diffusion model for aerial perspective. Technically, we optimize the diffusion coefficient and variance scheduling of the Brownian bridge by introducing a parabolic function, design a Laplacian of Gaussian (LOG) loss that fuses high-, medium-, and low-frequency features, and construct two core modules: a modality enhancement module that integrates spectral involution and cross-modal fusion, and an information guidance module based on wavelet decomposition. Experimental results demonstrate state-of-the-art performance: the method achieves a PSNR of 15.06 and an SSIM of 49.47, which are 1.5% and 1.2% higher than the suboptimal baseline BBDM-VQ4, respectively; its FID is reduced to 36.83, representing a 25.6% decrease compared to BBDM-VQ4, and its LPIPS is 2.0% lower than that of BBDM-VQ4. This approach effectively eliminates distribution biases induced by small-target blurring and background interference under aerial perspective while ensuring the semantic consistency of generated infrared images.