Related papers: Long-Time Asymptotics for Subordinated Fractional …

The behaviour of the solutions of the time-fractional diffusion equation, based on the Caputo derivative, is studied and its dependence on the fractional exponent is analysed. The time-fractional convection-diffusion equation is also solved…

Reaction-diffusion equations are widely used as the governing evolution equations for modeling many physical, chemical, and biological processes. Here we derive reaction-diffusion equations to model transport with reactions on a…

Space-time fractional evolution equations are a powerful tool to model diffusion displaying space-time heterogeneity. We prove existence, uniqueness and stochastic representation of classical solutions for an extension of Caputo evolution…

We find for the first time the asymptotic representation of the solution to the space dependent variable order fractional diffusion and Fokker-Planck equations. We identify a new advection term that causes ultra-slow spatial aggregation of…

Diffusion models have recently emerged as powerful stochastic frameworks for high-dimensional inference and generation. However, existing applications to partial differential equations (PDEs) predominantly rely on physics-informed training…

We propose a novel framework, Continuous_Time Attention, which infuses partial differential equations (PDEs) into the Transformer's attention mechanism to address the challenges of extremely long input sequences. Instead of relying solely…

We study the long time behavior of the solution of a stochastic PDEs with random coefficients assuming that randomness arises in a different independent scale. We apply the obtained results to 2D- Navier--Stokes equations.

We consider time-changed diffusions driven by generators with discontinuous coefficients. The PDE's connections are investigated and in particular some results on the asymptotic analysis according to the behaviour of the coefficients are…

We propose a novel framework for analyzing the dynamics of distribution shift in real-world systems that captures the feedback loop between learning algorithms and the distributions on which they are deployed. Prior work largely models…

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the…

We develop asymptotic approximations that can be applied to sequential estimation and inference problems, adaptive randomized controlled trials, and related settings. In batched adaptive settings where the decision at one stage can affect…

We give a complete study of the asymptotic behavior of a simple model of alignment of unit vectors, both at the level of particles , which corresponds to a system of coupled differential equations, and at the continuum level, under the form…

We present the stochastic solution to a generalized fractional partial differential equation involving a regularized operator related to the so-called Prabhakar operator and admitting, amongst others, as specific cases the fractional…

We study an abstract family of advection-diffusion equations within the framework of the fractional Laplacian. The system involves two independent diffusion parameters: one introduced via a damping operator acting on the scalar unknown and…

This paper investigates the asymptotic behaviour of solutions of periodic evolution equations. Starting with a general result concerning the quantified asymptotic behaviour of periodic evolution families we go on to consider a special class…

This work introduces and analyzes a finite element scheme for evolution problems involving fractional-in-time and in-space differentiation operators up to order two. The left-sided fractional-order derivative in time we consider is employed…

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on…

Many real-world systems modeled using partial differential equations (PDEs) involve unknown parameters that must be estimated from limited, noisy system observations. While typically assumed to be constants, some of these unobserved…

We explore the derivation of distributed parameter system evolution laws (and in particular, partial differential operators and associated partial differential equations, PDEs) from spatiotemporal data. This is, of course, a classical…

For characterizing the Brownian motion in a bounded domain: $\Omega$, it is well-known that the boundary conditions of the classical diffusion equation just rely on the given information of the solution along the boundary of a domain; on…