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In this chapter we provide an introduction to fractional dissipative partial differential equations (PDEs) with a focus on trying to understand their dynamics. The class of PDEs we focus on are reaction-diffusion equations but we also…
For a class of particle systems in continuous space with local interactions, we show that the asymptotic diffusion matrix is an infinitely differentiable function of the density of particles. Our method allows us to identify relatively…
A delayed term in a differential equation reflects the fact that information takes significant time to travel from one place to another within a process being studied. Despite de apparent similarity with ordinary differential equations,…
The large deviations principles are established for a class of multidimensional degenerate stochastic differential equations with reflecting boundary conditions. The results include two cases where the initial conditions are adapted and…
Elliptic stochastic differential equations (SDE) make sense when the coefficients are only continuous. We study the corresponding linearized SDE whose coefficients are not assumed to be locally bounded. This leads to existence of…
Self-similar stable mixed moving average processes can be related to nonsingular flows through their minimal representations. Self-similar stable mixed moving averages related to dissipative flows have been studied, as well as processes…
We introduce and study a deterministic lattice model describing the motion of an infinite system of oppositely charged particles under the action of a constant electric field. As an application this model represents a traffic flow of cars…
A Toda flow is constructed starting from a certain class of unbounded initial conditions including sequences growing with power order of less than 1. Unbounded ergodic sequences are allowed, and especially \b{eta}-ensembles matrix models in…
Research on asymptotic model selection in the context of stochastic differential equations (SDEs) is almost non-existent in the literature. In particular, when a collection of SDEs is considered, the problem of asymptotic model selection…
The complex dynamics of physical systems can often be modeled with stochastic differential equations. However, computational constraints inhibit the estimation of dynamics from large time-series datasets. I present a method for estimating…
We extend flow matching to ensembles of linear systems in both deterministic and stochastic settings. Averaging over system parameters induces memory leading to a non-Markovian interpolation problem for the stochastic case. In this setting,…
Flow-based models typically define a latent space with dimensionality identical to the observational space. In many problems, however, the data does not populate the full ambient data space that they natively reside in, rather inhabiting a…
Here we show how an anisotropic fluid in the diffusion limit can be equivalent to an isotropic fluid in the streaming out limit, in spherical symmetry. For a particular equation of state this equivalence is total, from one fluid we can…
We propose new limiting dynamics for stochastic gradient descent in the small learning rate regime called stochastic modified flows. These SDEs are driven by a cylindrical Brownian motion and improve the so-called stochastic modified…
The first-return map, or the Poincar\'e map, is a fundamental concept in the theory of flows. However, it can generally be defined only partially, and additional conditions are required to define it globally. Since this partiality reflects…
We prove that a sequence of semi-discrete approximations converges to a multiplicative functional for reflected Brownian motion, which intuitively represents the Lyapunov exponent for the corresponding stochastic flow. The method of proof…
Flow-based latent generative models such as Stable Diffusion 3 are able to generate images with remarkable quality, even enabling photorealistic text-to-image generation. Their impressive performance suggests that these models should also…
Identifying the drift and diffusion of an SDE from its population dynamics is a notoriously challenging task. Researchers in machine learning and single-cell biology have only been able to prove a partial identifiability result: for…
We present a generative model that is defined on finite sets of exchangeable, potentially high dimensional, data. As the architecture is an extension of RealNVPs, it inherits all its favorable properties, such as being invertible and…
We exhibit conditions under which the flow of marginal distributions of a discontinuous semimartingale $\xi$ can be matched by a Markov process, whose infinitesimal generator is expressed in terms of the local characteristics of $\xi$. Our…