相关论文: Estimates for distribution of suprema of solutions…

We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They…

Analysis of extremal behavior of stochastic processes is a key ingredient in a wide variety of applications, including probability, statistical physics, theoretical computer science, and learning theory. In this paper, we consider centered…

In this work we study the solutions to some fractional higher-order equations. Special cases in which time-fractional derivatives take integer values are also examined and the explicit solutions are presented. Such solutions can be…

We discuss the solvability of an infinite system of first order ordinary differential equations on the half line, subject to nonlocal initial conditions. The main result states that if the nonlinearities possess a suitable "sub-linear"…

We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation whose nonlinearity is the determinant of the Hessian matrix of the solution. We…

We consider constrained partial differential equations of hyperbolic type with a small parameter $\varepsilon>0$, which turn parabolic in the limit case, i.e., for $\varepsilon=0$. The well-posedness of the resulting systems is discussed…

A semilinear initial-boundary value problem with a Caputo time derivative of fractional order $\alpha\in(0,1)$ is considered, solutions of which typically exhibit a singular behaviour at an initial time. For L1-type discretizations of this…

We consider an implicit finite difference scheme on uniform grids in time and space for the Cauchy problem for a second order parabolic stochastic partial differential equation where the parabolicity condition is allowed to degenerate. Such…

Motivated by the study of the propagation of convexity by semi-groups of stochastic differential equations and convex comparison between the distributions of solutions of two such equations, we study the comparison for the convex order…

In this paper Gaussian models of retarded and accelerated anomalous diffusion are considered. Stochastic differential equations of fractional order driven by single or multiple fractional Gaussian noise terms are introduced to describe…

In this paper we study the randomized heat equation with homogeneous boundary conditions. The diffusion coeffcient is assumed to be a random variable and the initial condition is treated as a stochastic process. The solution of this…

This paper is devoted to the prediction problem in extreme value theory. Our main result is an explicit expression of the regular conditional distribution of a max-stable (or max-infinitely divisible) process $\{\eta(t)\}_{t\in T}$ given…

In arXiv:2305.03945 [math.NA], a first-order optimization algorithm has been introduced to solve time-implicit schemes of reaction-diffusion equations. In this research, we conduct theoretical studies on this first-order algorithm equipped…

This work leverages recent advances in probabilistic machine learning to discover conservation laws expressed by parametric linear equations. Such equations involve, but are not limited to, ordinary and partial differential,…

We obtain error approximation bounds between expected suprema of canonical processes that are generated by random vectors with independent coordinates and expected suprema of Gaussian processes. In particular, we obtain a sharper proximity…

We analyze here different forms of fractional relaxation equations of order {\nu}\in(0,1) and we derive their solutions both in analytical and in probabilistic forms. In particular we show that these solutions can be expressed as crossing…

This note is concerned with an important for modelling question of existence of solutions of stochastic partial differential equations as proper stochastic processes, rather than processes in the generalized sense. We consider a first order…

We establish a priori upper bounds for solutions to the spatially inhomogeneous Landau equation in the case of moderately soft potentials, with arbitrary initial data, under the assumption that mass, energy and entropy densities stay under…

One goal in Bayesian machine learning is to encode prior knowledge into prior distributions, to model data efficiently. We consider prior knowledge from systems of linear partial differential equations together with their boundary…

We investigate nonlinear, higher-order dispersive equations with measure (or even less regular) potentials and initial data with low regularity. Our approach is of distributional nature and relies on the phase space analysis (via Gabor wave…