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In the first part of this paper, we consider a family of continuous-time dynamical systems coupled with diffusion-transmutation processes. Under certain conditions, such randomly perturbed dynamical systems can be interpreted as an averaged…
In this work we investigate an inverse problem of identifying a spatially variable order in the one-dimensional subdiffusion model from the boundary flux measurement. The model involves a generalized Caputo derivative in time, and arises in…
We present a new method of deriving a boundary condition at a thin membrane for diffusion from experimental data. Based on experimental results obtained for normal diffusion of ethanol in water, we show that the derived boundary condition…
Diffusive representations of fractional differential and integral operators can provide a convenient means to construct efficient numerical algorithms for their approximate evaluation. In the current literature, many different variants of…
Diffusion probability models have shown significant promise in offline reinforcement learning by directly modeling trajectory sequences. However, existing approaches primarily focus on time-domain features while overlooking frequency-domain…
The aim of this tutorial survey is to revisit the basic theory of relaxation processes governed by linear differential equations of fractional order. The fractional derivatives are intended both in the Rieamann-Liouville sense and in the…
Super-diffusion, characterized by a spreading rate $t^{1/\alpha}$ of the probability density function $p(x,t) = t^{-1/\alpha} p \left( t^{-1/\alpha} x , 1 \right)$, where $t$ is time, may be modeled by space-fractional diffusion equations…
Ambiguity in medical image segmentation calls for models that capture full conditional distributions rather than a single point estimate. We present Prior-Guided Residual Diffusion (PGRD), a diffusion-based framework that learns voxel-wise…
We study a simple nonlocal-in-time dynamic system proposed for the effective modeling of complex diffusive regimes in heterogeneous media. We present its solutions and their commonly studied statistics such as the mean square distance. This…
In a fractional Cauchy problem, the usual first order time derivative is replaced by a fractional derivative. This problem was first considered by \citet{nigmatullin}, and \citet{zaslavsky} in $\mathbb R^d$ for modeling some physical…
We introduce and study the first model of an experimentally realizable three-dimensional time-dependent nonturbulent fluid flow to display the phenomenon of global diffusion of passive-scalar particles at arbitrarily small values of the…
It has been suggested that generative image models such as diffusion models can improve performance on clinically relevant tasks by offering deep learning models supplementary training data. However, most conditional diffusion models treat…
We introduce an operator-theoretic framework for analyzing directional dependence in multivariate time series based on order-constrained spectral non-invariance. Directional influence is defined as the sensitivity of second-order dependence…
In the present article, we study the diffusion equations with fractional time derivatives. The aim of this paper is to investigate the best possible regularity for the initial value/boundary value problems with non-homogeneous Dirichlet…
This paper is concerned with analysis of coupled fractional reaction-diffusion equations. It provides analytical comparison for the fractional and regular reaction-diffusion systems. As an example, the reaction-diffusion model with cubic…
We consider the problem of learning two families of time-evolving random measures from indirect observations. In the first model, the signal is a Fleming--Viot diffusion, which is reversible with respect to the law of a Dirichlet process,…
In this work, we study an inverse problem of recovering a space-time dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian-Caputo fractional derivative…
This paper addresses the problem of distributed state estimation for discrete-time linear time-invariant systems. Building on the framework proposed in Gao & Yang (2025), we exploit the Jordan canonical form of the system matrix to develop…
Local diffusion coefficients in disordered systems such as spin glass systems and living cells are highly heterogeneous and may change over time. Such a time-dependent and spatially heterogeneous environment results in irreproducibility of…
Spatiotemporal data imputation plays a crucial role in various fields such as traffic flow monitoring, air quality assessment, and climate prediction. However, spatiotemporal data collected by sensors often suffer from temporal…