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Accelerated diffusion models hold the potential to significantly enhance the efficiency of standard diffusion processes. Theoretically, these models have been shown to achieve faster convergence rates than the standard $\mathcal…
We consider a continuous-time Bienaym\'e-Galton-Watson process with logistic competition in a regime of weak competition, or equivalently of a large carrying capacity. Individuals reproduce at random times independently of each other but…
We consider the problem of the Bayesian inference of drift and diffusion coefficient functions in a stochastic differential equation given discrete observations of a realisation of its solution. We give conditions for the well-posedness and…
Diffusion Models (DMs) iteratively denoise random samples to produce high-quality data. The iterative sampling process is derived from Stochastic Differential Equations (SDEs), allowing a speed-quality trade-off chosen at inference. Another…
We present analytical results for the biased diffusion of particles moving under a constant force in a randomly layered medium. The influence of this medium on the particle dynamics is modeled by a piecewise constant random force. The…
The filtering of a Markov diffusion process on a manifold from counting process observations leads to `large' changes in the conditional distribution upon an observed event, corresponding to a multiplication of the density by the intensity…
Diffusion of a two component fluid is studied in the framework of differential equations, but where these equations are systematically derived from a well-defined microscopic model. The model has a finite carrying capacity imposed upon it…
Diffusion models offer stable training and state-of-the-art performance for deep generative modeling tasks. Here, we consider their use in the context of multivariate subsurface modeling and probabilistic inversion. We first demonstrate…
The use of non parametric hidden Markov models with finite state space is flourishing in practice while few theoretical guarantees are known in this framework. Here, we study asymptotic guarantees for these models in the Bayesian framework.…
A variety of enhanced statistical and numerical methods are now routinely used to extract comprehensible and relevant thermodynamic information from the vast amount of complex, high-dimensional data obtained from intensive molecular…
We study the problem of learning a mixture model of non-parametric product distributions. The problem of learning a mixture model is that of finding the component distributions along with the mixing weights using observed samples generated…
We study diffusion processes and stochastic flows which are time-changed random perturbations of a deterministic flow on a manifold. Using non-symmetric Dirichlet forms and their convergence in a sense close to the Mosco-convergence, we…
The paper studies a higher-order diffusion model of Maxwell-Stefan kind. The model is based upon higher-order moment equations of kinetic theory of mixtures, which include viscous dissipation in the model. Governing equations are analyzed…
For a diffusion process $X(t)$ of drift $\mu(x)$ and of diffusion coefficient $D=1/2$, we study the joint distribution of the two local times $A(t)= \int_{0}^{t} d\tau \delta(X(\tau)) $ and $B(t)= \int_{0}^{t} d\tau \delta(X(\tau)-L) $ at…
We address the statistics of continuous weak linear measurement on a few-state quantum system that is subject to a conditioned quantum evolution. For a conditioned evolution, both the initial and final states of the system are fixed: the…
We study the inductive biases of diffusion models with a conditioning-variable, which have seen widespread application as both text-conditioned generative image models and observation-conditioned continuous control policies. We observe that…
The method of potential solutions of Fokker-Planck equations is used to develop a transport equation for the joint probability of N coupled stochastic variables with the Dirichlet distribution as its asymptotic solution. To ensure a bounded…
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 previous publications, we showed that the incremental process of the chaotic diffusion of dissipative solitons in a prototypical complex Ginzburg-Landau equation, known, e.g., from nonlinear optics, is governed by a simple Markov process…
We present analytical expressions for the time-dependent and stationary probability distributions corresponding to a stochastically perturbed one-dimensional flow with critical points, in two physically relevant situations: delayed…