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The inherent complexity of biological agents often leads to motility behavior that appears to have random components. Robust stochastic inference methods are therefore required to understand and predict the motion patterns from time…

The migration of active particles in slowly moving, crowded, and heterogeneous media is fundamental to various biological processes and technological applications, such as cargo transport. In this study, we numerically investigate the…

It is our intention to provide via fractional calculus a generalization of the pure and compound Poisson processes, which are known to play a fundamental role in renewal theory, without and with reward, respectively. We first recall the…

We discuss velocity-jump models for chemotaxis of bacteria with an internal state that allows the velocity jump rate to depend on the memory of the chemoattractant concentration along their path of motion. Using probabilistic techniques, we…

Discrete diffusion models have recently emerged as a powerful class of generative models for chemistry and biology data. In these fields, the goal is to generate various samples with high rewards (e.g., drug-likeness in molecules), making…

The interactions between diffusing molecules and membrane-bound receptors drive numerous cellular processes. In this work, we develop a spatial model of molecular interactions with membrane receptors by homogenizing the cell membrane and…

We present the detailed analysis of the diffusive transport of spatially inhomogeneous fluid mixtures and the interplay between structural and dynamical properties varying on the atomic scale. The present treatment is based on different…

By formulating data samples' formation as a Markov denoising process, diffusion models achieve state-of-the-art performances in a collection of tasks. Recently, many variants of diffusion models have been proposed to enable controlled…

Diffusion models excel at capturing the natural design spaces of images, molecules, DNA, RNA, and protein sequences. However, rather than merely generating designs that are natural, we often aim to optimize downstream reward functions while…

In this paper diffusion processes with changing modes are studied involving the variable order partial differential equations. We prove the existence and uniqueness theorem of a solution of the Cauchy problem for fractional variable order…

This paper introduces the use of statistical distributions based on transport differential equations for clear distinction of transport modes within transient kinetic experiments. More specifically,novel techniques are developed for the…

We study the contribution of advection by thermal velocity fluctuations to the effective diffusion coefficient in a mixture of two indistinguishable fluids. The enhancement of the diffusive transport depends on the system size L and grows…

In this paper we use an individual-based model and its associated kinetic equation to study the generation of long jumps in the motion of E. coli. These models relate the run-and-tumble process to the intracellular reaction where the…

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…

Active particle systems of interacting self-propelled particles offer a versatile framework for modeling complex systems. When employed to describe aspects of animal behavior, the complexity of animal movement and decision-making often…

In this paper, we argue that iterative computation with diffusion models offers a powerful paradigm for not only generation but also visual perception tasks. We unify tasks such as depth estimation, optical flow, and amodal segmentation…

To explain the differentiation of stem cells in terms of dynamical systems theory, models of interacting cells with intracellular protein expression dynamics are analyzed and simulated. Simulations were carried out for all possible protein…

Consider a particle diffusing in a confined volume which is divided into two equal regions. In one region the diffusion coefficient is twice the value of the diffusion coefficient in the other region. Will the particle spend equal…

Adapting pretrained diffusion models to downstream objectives such as inverse problems often requires expensive test-time guidance or optimization. We propose a principled framework for generating high-quality reward-aligned samples at…

The main motivation of this work is to assess the validity of a LWR traffic flow model to model measurements obtained from trajectory data, and propose extensions of this model to improve it. A formulation for a discrete dynamical system is…