# PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES

@article{Gubinelli2012PARACONTROLLEDDA, title={PARACONTROLLED DISTRIBUTIONS AND SINGULAR PDES}, author={Massimiliano Gubinelli and Peter Imkeller and Nicolas Perkowski}, journal={Forum of Mathematics, Pi}, year={2012}, volume={3} }

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 paths. We illustrate its applicability on some model problems such as differential equations driven by fractional Brownian motion, a fractional Burgers-type stochastic PDE (SPDE) driven by space-time white noise, and a nonlinear version of the parabolic Anderson model with a white noise potential.

## 493 Citations

### Singular stochastic PDEs

- Mathematics
- 2014

We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to…

### Paracontrolled distribution approach to stochastic Volterra equations

- MathematicsJournal of Differential Equations
- 2021

### Asymptotics of PDE in random environment by paracontrolled calculus

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2021

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle…

### Discretisations of rough stochastic PDEs December

- Mathematics
- 2015

We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in…

### Discretisations of rough stochastic PDEs

- Mathematics
- 2015

We develop a general framework for spatial discretisations of parabolic stochastic PDEs whose solutions are provided in the framework of the theory of regularity structures and which are functions in…

### Support theorem for a singular semilinear stochastic partial differential equation

- Mathematics
- 2014

We consider the generalized parabolic Anderson equation (gPAM) in 2 dimensions with periodic boundary. This is an example of a singular semilinear stochastic partial differential equations, solutions…

### Some recent progress in singular stochastic PDEs.

- Mathematics
- 2019

Stochastic PDEs are ubiquitous in mathematical modeling. Yet, many such equations are too singular to admit classical treatment. In this article we review some recent progress in defining,…

### A non-linear parabolic PDE with a distributional coefficient and its applications to stochastic analysis

- MathematicsJournal of Differential Equations
- 2019

### Paracontrolled calculus for quasilinear singular PDEs

- MathematicsStochastics and Partial Differential Equations: Analysis and Computations
- 2022

We develop further in this work the high order paracontrolled calculus setting to deal with the analytic part of the study of quasilinear singular PDEs. A number of continuity results for some…

### The strong Feller property for singular stochastic PDEs

- MathematicsAnnales de l'Institut Henri Poincaré, Probabilités et Statistiques
- 2018

We show that the Markov semigroups generated by a large class of singular stochastic PDEs satisfy the strong Feller property. These include for example the KPZ equation and the dynamical $\Phi^4_3$…

## References

SHOWING 1-10 OF 57 REFERENCES

### Approximating Rough Stochastic PDEs

- MathematicsCommunications on Pure and Applied Mathematics
- 2013

We study approximations to a class of vector‐valued equations of Burgers type driven by a multiplicative space‐time white noise. A solution theory for this class of equations has been developed…

### Rough evolution equations

- Mathematics
- 2008

We generalize Lyons' rough paths theory in order to give a pathwise meaning to some nonlinear infinite-dimensional evolution equation associated to an analytic semigroup and driven by an irregular…

### Rough stochastic PDEs

- MathematicsCommunications on Pure and Applied Mathematics
- 2011

In this article, we show how the theory of rough paths can be used to provide a notion of solution to a class of nonlinear stochastic PDEs of Burgers type that exhibit too‐high spatial roughness for…

### Spatial rough path lifts of stochastic convolutions

- Mathematics
- 2012

We present sufficient conditions for finite controlled rho-variation of the covariance of Gaussian processes with stationary increments, based on concavity or convexity of their variance function.…

### Rough solutions for the periodic Korteweg--de~Vries equation

- Mathematics
- 2006

We show how to apply ideas from the theory of rough paths to the analysis of low-regularity solutions to non-linear dispersive equations. Our basic example will be the one dimensional Korteweg– de…

### Lectures on singular stochastic PDEs

- MathematicsEnsaios Matemáticos
- 2015

These are the notes for a course at the 18th Brazilian School of Probability held from August 3rd to 9th, 2014 in Mambucaba. The aim of the course is to introduce the basic problems of non‐linear…

### Di erential equations driven by rough signals

- Mathematics
- 1998

This paper aims to provide a systematic approach to the treatment of differential equations of the type
dyt = Si fi(yt) dxti
where the driving signal xt is a rough path. Such equations are very…