An Introduction to Stochastic PDEs

Abstract

These notes are based on a series of lectures given first at the University of Warwick in spring 2008 and then at the Courant Institute, Imperial College London, and EPFL. It is an attempt to give a reasonably self-contained presentation of the basic theory of stochastic partial differential equations, taking for granted basic measure theory, functional analysis and probability theory, but nothing else. The approach taken in these notes is to focus on semilinear parabolic problems driven by additive noise. These can be treated as stochastic evolution equations in some infinite-dimensional Banach or Hilbert space that usually have nice regularising properties and they already form a very rich class of problems with many interesting properties. Furthermore, this class of problems has the advantage of allowing to completely pass under silence many subtle problems arising from stochastic integration in infinite-dimensional spaces.

Many small fixes. Includes a new section on probability measures on Polish spaces

References in corpus (1)

• Spectral gaps in Wasserstein distances and the 2D stochastic Navier--Stokes equations

Cited by in corpus (47)

• On a generalization of the preconditioned Crank-Nicolson Metropolis algorithm
• A mild Ito formula for SPDEs
• Approximations to the Stochastic Burgers Equation
• Triviality of the 2D stochastic Allen-Cahn equation
• Sharp estimates for metastable lifetimes in parabolic SPDEs: Kramers' law and beyond
• Posterior Consistency for Bayesian Inverse Problems through Stability and Regression Results
• A computational framework for infinite-dimensional Bayesian inverse problems. Part I: The linearized case, with application to global seismic inversion
• Travelling Waves for Reaction-Diffusion Equations Forced by Translation Invariant Noise
• Asymptotic laws for the spatial distribution and the number of connected components of zero sets of Gaussian random functions
• Invariant measure of the stochastic Allen-Cahn equation: the regime of small noise and large system size
• Sampling conditioned hypoelliptic diffusions
• Couplings via comparison principle and exponential ergodicity of SPDEs in the hypoelliptic setting
• Optimal rate of convergence for stochastic Burgers-type equations
• Numerical approximation of stochastic evolution equations: Convergence in scale of Hilbert spaces
• Smoothness and monotonicity of the excursion set density of planar Gaussian fields
• An introduction to singular stochastic PDEs: Allen-Cahn equations, metastability and regularity structures
• Adaptive Monte Carlo augmented with normalizing flows
• Existence and space-time regularity for stochastic heat equations on p.c.f. fractals
• Parameter Estimation in an SPDE Model for Cell Repolarisation
• Regularity of solutions of abstract linear evolution equations
• Time inhomogeneous Generalized Mehler Semigroups
• Functional central limit theorem for Brownian particles in domains with Robin boundary condition
• Importance sampling for thermally induced switching and non-switching probabilities in spin-torque magnetic nanodevices
• Singular perturbations to semilinear stochastic heat equations
• Stochastic Discontinuous Galerkin Methods (SDGM) Based on Fluctuation-Dissipation Balance
• Spatio-Temporal Forecasting by Coupled Stochastic Differential Equations: Applications to Solar Power
• A randomized maximum a posterior method for posterior sampling of high dimensional nonlinear Bayesian inverse problems
• Learning Temporal Evolution of Spatial Dependence with Generalized Spatiotemporal Gaussian Process Models
• Malliavin Calculus and Density for Singular Stochastic Partial Differential Equations
• Large deviation principle of occupation measures for Non-linear monotone SPDEs
• Transportation-cost inequalities for diffusions driven by Gaussian processes
• A Feynman-Kac Type Theorem for ODEs: Solutions of Second Order ODEs as Modes of Diffusions
• Renormalization of determinant lines in Quantum Field Theory
• Stochastic Quantization for the Edwards Measure of Fractional Brownian Motion with $Hd=1$
• Gaussian approximations for transition paths in Brownian dynamics
• Almost sure global well posedness for the BBM equation with infinite $L^2$ initial data
• A Modified log-Harnack inequality and asymptotically strong Feller property
• Stochastic Averaging of The Einstein Vacuum Equations on a Toroidal Manifold with Randomly Perturbed Radial Moduli: Stability Criteria and Induced 'Cosmological Constant' Terms
• Stochastic Control of Tolman-Oppenheimer-Snyder Collapse of Zero-Pressure Stars to Black Holes: Rigorous Criteria for Density Bounds and Singularity Smoothing
• Atomic decomposition of finite signed measures on compacts of ${\mathbb{R}}^n$
• Introduction to SPDEs from Probability and PDE
• A New Class of Backward Stochastic Partial Differential Equations with Jumps and Applications
• On local time for the solution to a white noise driven heat equation
• Inverse Random Source Scattering for Elastic Waves
• A continuous spatio-temporal approach to estimate climate change
• An inverse problem for a hyperbolic system on a vector bundle and energy measurements
• Stochastic PDEs with multiscale structure