• Corpus ID: 119287396

Martingale-driven approximations of singular stochastic PDEs

@article{Matetski2018MartingaledrivenAO,
  title={Martingale-driven approximations of singular stochastic PDEs},
  author={Konstantin Matetski},
  journal={arXiv: Probability},
  year={2018}
}
  • K. Matetski
  • Published 28 August 2018
  • Mathematics, Computer Science
  • arXiv: Probability
We define multiple stochastic integrals with respect to c\`{a}dl\`{a}g martingales and prove moment bounds and chaos expansions, which allow to work with them in a way similar to Wiener stochastic integrals. In combination with the discretization framework of Erhard and Hairer (2017), our results give a tool for proving convergence of interacting particle systems to stochastic PDEs using regularity structures. As examples, we prove convergence of martingale-driven discretizations of the $3… 
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References

SHOWING 1-10 OF 47 REFERENCES
PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES
We introduce an approach to study certain singular partial differential equations (PDEs) which is based on techniques from paradifferential calculus and on ideas from the theory of controlled rough
Renormalization Group and Stochastic PDEs
We develop a renormalization group (RG) approach to the study of existence and uniqueness of solutions to stochastic partial differential equations driven by space-time white noise. As an example, we
An analytic BPHZ theorem for regularity structures
We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated
Discretisation of regularity structures
  • D. Erhard, Martin Hairer
  • Computer Science, Mathematics
    Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
  • 2019
TLDR
A general framework allowing to apply the theory of regularity structures to discretisations of stochastic PDEs and a "black box" describing the behaviour of the authors' discretised objects at scales below $\varepsilon $ is introduced.
Stopping Times and Tightness. II
To establish weak convergence of a sequence of martingales to a continuous martingale limit, it is sufficient (under the natural uniform integrability condition) to establish convergence of
A theory of regularity structures
We introduce a new notion of “regularity structure” that provides an algebraic framework allowing to describe functions and/or distributions via a kind of “jet” or local Taylor expansion around each
Renormalising SPDEs in regularity structures
The formalism recently introduced in arXiv:1610.08468 allows one to assign a regularity structure, as well as a corresponding "renormalisation group", to any subcritical system of semilinear
Stochastic Equations in Infinite Dimensions
Preface Introduction Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. Stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with
Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems
We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear
A central limit theorem for the KPZ equation
We consider the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in
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