Related papers: Introduction to SPDEs from Probability and PDE

This mini-course of 20 lectures aims at highlights of spectral theory for self-adjoint partial differential operators, with a heavy emphasis on problems with discrete spectrum. Part I: Discrete Spectrum (ODE preview, Laplacian - computable…

Stochastic Partial Differential Equations (SPDEs) driven by random noise play a central role in modeling physical processes with rough spatio-temporal dynamics, such as turbulence flows, superconductors, and quantum dynamics. Although…

Recently delivered lectures on Self-Referential Mathematics, [2], at the Department of Mathematics and Applied Mathematics, University of Pretoria, are briefly presented. Comments follow on the subject, as well as on Inconsistent…

These lecture notes grew out of notes for courses around Integrable PDEs and the KdV equation given by the authors during the past five years at the University of Antwerp (Belgium). Comments and suggestions are welcome.

This paper proposes a novel low-rank approximation to the multivariate State-Space Model. The Stochastic Partial Differential Equation (SPDE) approach is applied component-wise to the independent-in-time Mat\'ern Gaussian innovation term in…

Notes for a Course on Probability and Statistics: L1: Elements of Probability; L2: Bayesian Inference; L3: Monte Carlo Methods

We propose a novel framework for discovering Stochastic Partial Differential Equations (SPDEs) from data. The proposed approach combines the concepts of stochastic calculus, variational Bayes theory, and sparse learning. We propose the…

This is lecture notes on the course "Stochastic Processes". In this format, the course was taught in the spring semesters 2017 and 2018 for third-year bachelor students of the Department of Control and Applied Mathematics, School of Applied…

These lecture notes provide an introduction to free probability theory, with a focus on tools and techniques useful in the study of large random matrices. Topics include freeness, free cumulants, additive and multiplicative free…

One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and…

Correlation and smoothness are terms used to describe a wide variety of random quantities. In time, space, and many other domains, they both imply the same idea: quantities that occur closer together are more similar than those further…

These lecture notes have been prepared for a series of lectures given at the Summer School "From kinetic equations to statistical mechanics", (see https://www.lebesgue.fr/content/sem2021-equat_cynet ) organised by the Henri Lebesgue Center…

We consider parameter estimation for a linear parabolic second-order stochastic partial differential equation (SPDE) in two space dimensions driven by two types $Q$-Wiener processes based on high frequency data in time and space. We first…

In this paper, we study the option pricing problems for rough volatility models. As the framework is non-Markovian, the value function for a European option is not deterministic; rather, it is random and satisfies a backward stochastic…

Introductory statistical inference texts and courses treat the point estimation, hypothesis testing, and interval estimation problems separately, with primary emphasis on large-sample approximations. Here I present an alternative approach…

The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is…

In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport…

We introduce and test methods for the calibration of the diffusion term in Stochastic Partial Differential Equations (SPDEs) describing fluids. We take two approaches, one uses ideas from the singular value decomposition and the Biot-Savart…

We deal with parametric estimation for a parabolic linear second order stochastic partial differential equation (SPDE) with a small dispersion parameter based on high frequency data which are observed in time and space. By using the thinned…

This note deals with existence and uniqueness of (variational) solutions to the following type of stochastic partial differential equations on a Hilbert space H dX(t) = A(t,X(t))dt + B(t,X(t))dW(t) + h(t) dG(t) where A and B are random…