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Generative models such as Variational Autoencoders (VAEs) and Generative Adversarial Networks (GANs) have shown promise in sequential recommendation tasks. However, they face challenges, including posterior collapse and limited…

The separating time for two probability measures on a filtered space is an extended stopping time which captures the phase transition between equivalence and singularity. More specifically, two probability measures are equivalent before…

Speech super-resolution (SR) is the task that restores high-resolution speech from low-resolution input. Existing models employ simulated data and constrained experimental settings, which limit generalization to real-world SR. Predictive…

Diffusion models have achieved remarkable success in high-fidelity image generation but remain computationally demanding due to their multi-step denoising process and large model sizes. Although prior work improves efficiency either by…

Conditional diffusion models provide a natural framework for probabilistic prediction of dynamical systems and have been successfully applied to fluid dynamics and weather prediction. However, in many settings, the available information at…

In the first paper of this series, I investigated whether a wavefunction model of a heavy particle and a collection of light particles might generate "Brownian-Motion-Like" trajectories of the heavy particle. I concluded that it was…

Inverse problems aim to determine parameters from observations, a crucial task in engineering and science. Lately, generative models, especially diffusion models, have gained popularity in this area for their ability to produce realistic…

In this paper, we study the stationary states of diffusive dynamics driven out of equilibrium by reservoirs. For a small forcing, the system remains close to equilibrium and the large deviation functional of the density can be computed…

This paper investigates the probabilistic properties that determine the existence of space-time transformations between diffusion processes. We prove that two diffusions are related by a monotone space-time transformation if and only if…

We establish a large deviation principle for time dependent trajectories (paths) of the empirical density of $N$ particles with long range interactions, for homogeneous systems. This result extends the classical kinetic theory that leads to…

Diffusion models, a powerful and universal generative AI technology, have achieved tremendous success in computer vision, audio, reinforcement learning, and computational biology. In these applications, diffusion models provide flexible…

Network science investigates the architecture of complex systems to understand their functional and dynamical properties. Structural patterns such as communities shape diffusive processes on networks. However, these results hold under the…

Intracellular protein patterns are described by (nearly) mass-conserving reaction-diffusion systems. While these patterns initially form out of a homogeneous steady state due to the well-understood Turing instability, no general theory…

By means of rather general arguments, based on an approach due to Derrida that makes use of samples of finite size, we analyse the effective diffusivity and drift tensors in certain types of random medium in which the motion of the…

We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium…

We consider the long time behavior of solutions to a nonlocal reaction diffusion equation that arises in the study of directed polymers. The model is characterized by convolution with a kernel $R$ and an $L^2$ inner product. In one spatial…

The dynamics of dispersal-structured populations, consisting of competing individuals that are characterized by different diffusion coefficients but are otherwise identical, is investigated. Competition is taken into account through…

The Kob-Andersen model is a fundamental example of a kinetically constrained lattice gas, that is, an interacting particle system with Kawasaki type dynamics and kinetic constraints. In this model, a particle is allowed to jump when…

Diffusion Models represent a significant advancement in generative modeling, employing a dual-phase process that first degrades domain-specific information via Gaussian noise and restores it through a trainable model. This framework enables…

Perturbations of fluid media can give rise to non-equilibrium dynamics, which may in turn cause motion of immersed inclusions. We consider perturbations ("activations") that are local in space and time, of a fluid density which is…