Related papers: What is the most optimal diffusion?
Diffusion probabilistic models (DPMs) represent a class of powerful generative models. Despite their success, the inference of DPMs is expensive since it generally needs to iterate over thousands of timesteps. A key problem in the inference…
This is the second part of study on the optimal convergence rate of the explicit Euler discretization in time for the convection-diffusion equations [Appl. Math. Lett. \textbf{131} (2022) 108048] which focuses on high-dimensional…
Since the famous 1926 paper by Richardson, the relative diffusion of two particles in a turbulent liquid has attracted a lot of interest. The motion of a single particle on the other hand is usually considered not to be especially…
We consider a one-dimensional stationary stochastic process $x(\tau)$ of duration $T$. We study the probability density function (PDF) $P(t_{\rm m}|T)$ of the time $t_{\rm m}$ at which $x(\tau)$ reaches its global maximum. By using a path…
Mathematical models based on probability density functions (PDF) have been extensively used in hydrology and subsurface flow problems, to describe the uncertainty in porous media properties (e.g., permeability modelled as random field).…
Continuous time random walks are non-Markovian stochastic processes, which are only partly characterized by single-time probability distributions. We derive a closed evolution equation for joint two-point probability density functions of a…
We study the dynamics of an athermal inertial run-and-tumble particle moving in a shear-thickening medium in $d=1$. The viscosity of the medium is represented by a nonlinear function $f(v)\sim\tan(v)$, while a symmetric dichotomous noise of…
We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in…
The problem of anomalous diffusion in the momentum space is considered on the basis of the appropriate probability transition function (PTF). New general equation for description of the diffusion of heavy particles in the gas of the light…
Fractional Fokker-Planck equation plays an important role in describing anomalous dynamics. To the best of our knowledge, the existing discussions mainly focus on this kind of equation involving one diffusion operator. In this paper, we…
The Weibull distribution can be obtained using a power transformation from the standard exponential distribution. In this article, we will consider a symmetrized power transformation of a random variable with the standard normal…
We consider a process given as the solution of a stochastic differential equation with irregular, path dependent and time-inhomogeneous drift coefficient and additive noise. Explicit and optimal bounds for the Lebesgue density of that…
According to modern developments in turbulence theory, the "dissipation" scales (u.v. cut-offs) $\eta$ form a random field related to velocity increments $\delta_{\eta}u$. In this work we, using Mellin's transform combined with the Gaussain…
We aim at estimating the invariant density associated to a stochastic differential equation with jumps in low dimension, which is for $d=1$ and $d=2$. We consider a class of jump diffusion processes whose invariant density belongs to some…
In this paper we study the randomized non-autonomous complete linear differential equation. The diffusion coefficient and the source term in the differential equation are assumed to be stochastic processes and the initial condition is…
The problem of anomalous diffusion in momentum (velocity) space is considered based on the master equation and the appropriate probability transition function (PTF). The approach recently developed by the author for coordinate space, is…
Sub-diffusion equations are used in a large range of applications including fluids, plasma physics and biology. Their mathematical analysis is advanced even if a much larger literature addresses super-diffusions. The goal of this paper is…
The problem of anomalous diffusion in momentum space is considered for plasma-like systems on the basis of a new collision integral, which is appropriate for consideration of the probability transition function (PTF) with long tails in…
We consider a particle living in $\mathbb{R}_+$, whose velocity is a positive recurrent diffusion with heavy-tailed invariant distribution when the particle lives in $(0,\infty)$. When it hits the boundary $x=0$, the particle restarts with…
We obtain a fast diffusion equation (FDE) as scaling limit of a sequence of zero-range process with symmetric unit rate. Fast diffusion effect comes from the fact that the diffusion coefficient goes to infinity as the density goes to zero.…