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We introduce a Langevin equation characterized by a time dependent drift. By assuming a temporal power-law dependence of the drift we show that a great variety of behavior is observed in the dynamics of the variance of the process. In…

The purpose of this paper is to investigate the long time behaviour for a self-interacting diffusion and a self-interacting velocity jump process. While the diffusion case has already been studied for some particular potential function, the…

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…

A diffusion process for charge distributions in a phase space is examined. The corresponding charge moves in a force field and under an action of a random field. There are the diffusion motions for coordinates and for momenta. In our model,…

The combination of protrusions and retractions in the movement of polarized cells leads to understand the effect of possible synchronisation between the two ends of the cells. This synchronisation, in turn, could lead to different dynamics…

Using statistical physics methods, we study generative diffusion models in the regime where the dimension of space and the number of data are large, and the score function has been trained optimally. Our analysis reveals three distinct…

Generative diffusion models have achieved spectacular performance in many areas of machine learning and generative modeling. While the fundamental ideas behind these models come from non-equilibrium physics, variational inference and…

The effect of diffusional relaxation on the random sequential deposition process is studied in the limit of fast deposition. Expression for the coverage as a function of time are analytically derived for both the short-time and long-time…

We explore the fractional advection-diffusion equation and rare events associated with the ACTRW model. When waiting times have a finite mean but infinite variance, and the displacements follow a narrow distribution, the fractional operator…

A discrete time branching process where the offspring distribution is generation-dependent, and the number of reproductive individuals is controlled by a random mechanism is considered. This model is a Markov chain but, in general, the…

This work presents Causal Drift Generator (CaDrift), a time-dependent synthetic data generator framework based on Structural Causal Models (SCMs). The framework produces a virtually infinite combination of data streams with controlled shift…

Depending on their mechanism of self-propulsion, active particles can exhibit a time-dependent, often periodic, propulsion velocity. The precise propulsion velocity profile determines their mean square displacement and their effective…

While diffusion models can successfully generate data and make predictions, they are predominantly designed for static images. We propose an approach for efficiently training diffusion models for probabilistic spatiotemporal forecasting,…

The drift diffusion model (DDM) is a model of sequential sampling with diffusion (Brownian) signals, where the decision maker accumulates evidence until the process hits a stopping boundary, and then stops and chooses the alternative that…

This paper proposes an adaptive time-stepping mothods for stochastic diffusion systems whose drift and diffusion coefficients are locally Lipschitz continuous and may exhibit polynomial growth. By controlling the growth of both the drift…

Animals learn the timing between consecutive events very easily. Their precision is usually proportional to the interval to time (Weber's law for timing). Most current timing models either require a central clock and unbounded accumulator…

We study diffusion processes driven by a Brownian motion with regular drift in a finite dimension setting. The drift has two components on different time scales, a fast conservative component and a slow dissipative component. Using the…

Stochastic resetting models diverse phenomena across numerous scientific disciplines. Current understanding stems from the renewal framework, which relates systems subject to global resetting to their non-resetting counterparts. Yet, in…

Non-linear renewal theory is extended to include random walks perturbed by both a slowly changing sequence and a stationary one. Main results include a version of the Key Renewal Theorem, a derivation of the limiting distribution of the…

In a reaction-diffusion system, fluctuations in both diffusion and reaction events, have important effects on the steady-state statistics of the system. Here, we argue through extensive lattice simulations, mean-field type arguments, and…