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This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we…
We consider a stochastic functional differential equation with an arbitrary Lipschitz diffusion coefficient depending on the past. The drift part contains a term with superlinear growth and satisfying a dissipativity condition. We prove…
Based on the notion of a construction process consisting of the stepwise addition of particles to the pure fluid, a discrete model for the apparent viscosity as well as for the maximum packing fraction of polydisperse suspensions of…
We define a Gaussian invariant measure for the two-dimensional averaged-Euler equation and show the existence of its solution with initial conditions on the support of the measure. An invariant surface measure on the level sets of the…
We study a prominent two-parametric family of diffusion processes $X_{z,z'}$ on an infinite-dimensional Thoma simplex. The family was constructed by Borodin and Olshanski in 2007 and it closely resembles Ethier-Kurtz's…
We consider a one-dimensional diffusion which solves a stochastic differential equation with Borel-measurable coefficients in an open interval. We allow for the endpoints to be inaccessible or absorbing. Given a Borel-measurable function…
Denoising Diffusion Probabilistic Models (DDPM) are powerful state-of-the-art methods used to generate synthetic data from high-dimensional data distributions and are widely used for image, audio, and video generation as well as many more…
We construct explicit jointly invariant measures for the periodic KPZ equation (and therefore also the stochastic Burgers' and stochastic heat equations) for general slope parameters and prove their uniqueness via a one force--one solution…
The present article proposes a partitioned Dirichlet-Neumann algorithm, that allows to address unique challenges arising from a novel mixed-dimensional coupling of very slender fibers embedded in fluid flow using a regularized mortar-type…
We consider a family S=S(a) of 2-valued transformations of special form on the segment [0,1] with measure $\mu=\int p(x) d\lambda$, which is absolutely continuous with respect to the Lebesgue measure $\lambda$. We endow S with a set of…
Nonlinear deformations of a two-dimensional gas bubble are investigated in the framework of a Hamiltonian formulation involving surface variables alone. The Dirichlet--Neumann operator is introduced to accomplish this dimensional reduction…
We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller…
Sampling viable 3D structures (e.g., molecules and point clouds) with SE(3)-invariance using diffusion-based models proved promising in a variety of real-world applications, wherein SE(3)-invariant properties can be naturally characterized…
We generalize the Beckner's type Poincar\'e inequality \cite{Beckner} to a large class of probability measures on an abstract Wiener space of the form $\mu\star\nu$, where $\mu$ is the reference Gaussian measure and $\nu$ is a probability…
In this article, using kernel convolution of order based dependent Dirichlet process (Griffin and Steel (2006)) we construct a nonstationary, nonseparable, nonparametric space-time process, which, as we show, satisfies desirable properties,…
We prove the invariance principle for a \emph{random Lorentz-gas} particle in 3 dimensions under the Boltzmann-Grad limit and simultaneous diffusive scaling. That is, for the trajectory of a point-like particle moving among infinite-mass,…
We discuss the appropriate techniques for modelling the geometry of open ended elastic polymer molecules. The molecule is assumed to have fixed endpoints on a boundary surface. In particular we discuss the concept of the winding number, a…
We study the diffusion of a linear polymer in the presence of permeable membranes without excluded volume interactions, using scaling theory and Monte Carlo simulations. We find that the average time it takes for a chain with polymerization…
In this paper, the development of a mathematical method is presented to explore spatially non-uniform phases with no long-range order in mathematical models of first order phase transitions. We use essential results regarding the…
The diffraction of stochastic point sets, both Bernoulli and Markov, and of random tilings with crystallographic symmetries is investigated in rigorous terms. In particular, we derive the diffraction spectrum of 1D random tilings, of…