Rough evolution equations

Gubinelli, Massimiliano; Tindel, Samy

doi:10.1214/08-aop437

Cited by 122 publications

(205 citation statements)

References 23 publications

(91 reference statements)

Supporting

Mentioning

200

Contrasting

Order By: Relevance

“…So far, mainly SPDE with "simple" diffusion coefficients (additive or linear multiplicative) are well-understood in this regard, with some notable exceptions [24,30,34]. The usual approach to prove the existence of a stochastic flow for such SPDE relies on a transformation of the SPDE into a random PDE which may then be treated in a pathwise manner (cf.…”

Section: Generation Of An Rdsmentioning

confidence: 99%

Finite Speed of Propagation for Stochastic Porous Media Equations

Gess¹

2013

SIAM J. Math. Anal.

View full text Add to dashboard Cite

Abstract. We prove finite speed of propagation for stochastic porous media equations perturbed by linear multiplicative space-time rough signals. Explicit and optimal estimates for the speed of propagation are given. The result applies to any continuous driving signal, thus including fractional Brownian motion for all Hurst parameters. The explicit estimates are then used to prove that the corresponding random attractor has infinite fractal dimension.

show abstract

Section: Generation Of An Rdsmentioning

confidence: 99%

Finite Speed of Propagation for Stochastic Porous Media Equations

Gess¹

2013

SIAM J. Math. Anal.

View full text Add to dashboard Cite

show abstract

“…One can then show, and this is the content of Proposition 7.13 below, that there exists a process Y with values in C γ 2 such that Y ε → Y in probability in C(R, C γ 2 ) for every γ < 1. We refer to Section 3 below for more details, but the gist of the theory of controlled rough paths is that one can use the process Y in order to define a "rough integral" y x A t (z) dY t (z) as a convergent limit of compensated Riemann sums for every smooth test function ϕ and for every function A t such that, for some δ > 0, there exists A t ∈ C δ and R A t ∈ C 1/2+δ 2 [CF09,CFO11,GT10,Tei11]. In all of these cases, the theory of rough paths was used to deal with the lack of temporal regularity of the equations.…”

Section: Treatment Of the Remaindermentioning

confidence: 99%

Solving the KPZ Equation

Hairer

2013

XVIIth International Congress on Mathematical Physics

View full text Add to dashboard Cite

We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a "universal" measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the fluctuations of systems in the KPZ universality class.As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the FokkerPlanck equation associated to a particle diffusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.

show abstract

“…As is natural, afterwards has been considered some of the possible generalizations of the diffusion processes. For instance, in the literature we can find now papers about PDEs [18,3,5,12], Volterra equations [6,7,2] or systems with delay [10,9,16,14].…”

Section: Introductionmentioning

confidence: 99%