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Massimiliano Gubinelli

Publications and source records attributed to Massimiliano Gubinelli.

At least 55 records · Page 3Linked to original sources
arXiv2017

Probabilistic approach to the stochastic Burgers equation

We review the formulation of the stochastic Burgers equation as a martingale problem. One way of understanding the difficulty in making sense of the equation is to note that it is a stochastic PDE with distributional drift, so we first review how to construct finite-dimensional diffusions with distributional drift. We then present the uniqueness result for the stationary martingale problem of [Gubinelli, Perkowski 2016], but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of [Gubinelli, Perkowski 2016] to a non-stationary regime.

math.PR↗
arXiv2016

One-dimensional reflected rough differential equations

We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some ideas we introduced in a previous work dealing with rough kinetic PDEs [arXiv:1604.00437].

math.PR↗
arXiv2016

The Kardar-Parisi-Zhang equation as scaling limit of weakly asymmetric interacting Brownian motions

We consider a system of infinitely many interacting Brownian motions that models the height of a one-dimensional interface between two bulk phases. We prove that the large scale fluctuations of the system are well approximated by the solution to the KPZ equation provided the microscopic interaction is weakly asymmetric. The proof is based on the martingale solutions of Goncalves and Jara and the corresponding uniqueness result of Gubinelli and Perkowski.

math.PR↗
arXiv2015

KPZ reloaded

We analyze the one-dimensional periodic Kardar-Parisi-Zhang equation in the language of paracontrolled distributions, giving an alternative viewpoint on the seminal results of Hairer. Apart from deriving a basic existence and uniqueness result for paracontrolled solutions to the KPZ equation we perform a thorough study of some related problems. We rigorously prove the links between KPZ equation, stochastic Burgers equation, and (linear) stochastic heat equation and also the existence of solutions starting from quite irregular initial conditions. We also build a partial link between energy solutions as introduced by Goncalves and Jara and paracontrolled solutions. Interpreting the KPZ equation as the value function of an optimal control problem, we give a pathwise proof for the global existence of solutions and thus for the strict positivity of solutions to the stochastic heat equation. Moreover, we study Sasamoto-Spohn type discretizations of the stochastic Burgers equation and show that their limit solves the continuous Burgers equation possibly with an additional linear transport term. As an application, we give a proof of the invariance of the white noise for the stochastic Burgers equation which does not rely on the Cole-Hopf transform.

math.PR↗
arXiv2014

A Fourier approach to pathwise stochastic integration

We develop a Fourier approach to rough path integration, based on the series decomposition of continuous functions in terms of Schauder functions. Our approach is rather elementary, the main ingredient being a simple commutator estimate, and it leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Itô and of Stratonovich type. We apply it to solve stochastic differential equations in a pathwise manner.

math.PR↗
arXiv2014

Controlled viscosity solutions of fully nonlinear rough PDEs

We propose a definition of viscosity solutions to fully nonlinear PDEs driven by a rough path via appropriate notions of test functions and rough jets. These objects will be defined as controlled processes with respect to the driving rough path. We show that this notion is compatible with the seminal results of Lions and Souganidis and with the recent results of Friz and coauthors on fully non-linear SPDEs with rough drivers.

math.PR↗
arXiv2013

Remarks on the stochastic transport equation with Hölder drift

We consider a stochastic linear transport equation with a globally Hölder continuous and bounded vector field. Opposite to what happens in the deterministic case where shocks may appear, we show that the unique solution starting with a C^{1}-initial condition remains of class $C^{1}$ in space. We also improve some results of Flandoli-Gubinelli-Priola (2009) about well-posedness. Moreover, we prove a stability property for the solution with respect to the initial datum.

math.AP↗
arXiv2010

Rough evolution equations

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 noise. As an illustration, we discuss a class of linear and nonlinear 1d SPDEs driven by a space--time Gaussian noise with singular space covariance and Brownian time dependence.

math.PR↗
arXiv2009

Global existence for rough differential equations under linear growth conditions

We prove existence of global solutions for differential equations driven by a geometric rough path under the condition that the vector fields have linear growth. We show by an explicit counter-example that the linear growth condition is not sufficient if the driving rough path is not geometric. This settle a long-standing open question in the theory of rough paths. So in the geometric setting we recover the usual sufficient condition for differential equation. The proof rely on a simple mapping of the differential equation from the Euclidean space to a manifold to obtain a rough differential equation with bounded coefficients.

math.PR↗
arXiv2009

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Holder continuous and bounded vector field. While this deterministic PDE may not be well-posed, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influece of noise. The key tool is a differentiable stochastic flow constructed and analysed by means of a special transformation of the drift of Ito-Tanaka type.

math.AP↗
arXiv2008

Large scale behavior of semiflexible heteropolymers

We consider a general discrete model for heterogeneous semiflexible polymer chains. Both the thermal noise and the inhomogeneous character of the chain (the disorder) are modeled in terms of random rotations. We focus on the quenched regime, i.e., the analysis is performed for a given realization of the disorder. Semiflexible models differ substantially from random walks on short scales, but on large scales a Brownian behavior emerges. By exploiting techniques from tensor analysis and non-commutative Fourier analysis, we establish the Brownian character of the model on large scale and we obtain an expression for the diffusion constant. We moreover give conditions yielding quantitative mixing properties.

math.PR↗
arXiv2007

Gibbs measures on Brownian currents

Motivated by applications to quantum field theory we consider Gibbs measures for which the reference measure is Wiener measure and the interaction is given by a double stochastic integral and a pinning external potential. In order properly to characterize these measures through DLR equations, we are led to lift Wiener measure and other objects to a space of configurations where the basic observables are not only the position of the particle at all times but also the work done by test vector fields. We prove existence and basic properties of such Gibbs measures in the small coupling regime by means of cluster expansion.

math-ph↗
arXiv2007

On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model

We study the pathwise regularity of the map $$ ϕ\mapsto I(ϕ) = \int_0^T < ϕ(X_t), dX_t>$$ where $ϕ$ is a vector function on $\R^d$ belonging to some Banach space $V$, $X$ is a stochastic process and the integral is some version of a stochastic integral defined via regularization. A \emph{stochastic current} is a continuous version of this map, seen as a random element of the topological dual of $V$. We give sufficient conditions for the current to live in some Sobolev space of distributions and we provide elements to conjecture that those are also necessary. Next we verify the sufficient conditions when the process $X$ is a $d$-dimensional fractional Brownian motion (fBm); we identify regularity in Sobolev spaces for fBm with Hurst index $H \in (1/4,1)$. Next we provide some results about general Sobolev regularity of Brownian currents. Finally we discuss applications to a model of random vortex filaments in turbulent fluids.

math.PR↗
arXiv2005

Critical Behavior of the Two-Dimensional Randomly Driven Lattice Gas

We investigate the critical behavior of the two-dimensional randomly driven lattice gas, in which particles are driven along one of the lattice axes by an infinite external field with randomly changing sign. A finite-size scaling (FSS) analysis provides novel evidences that this model is not in the same universality class as the driven lattice gas with a constant drive (DLG), contrarily to what has been recently reported in the literature. Indeed, the FSS functions of transverse observables (i.e., related to order-parameter fluctuations with wave vector perpendicular to the direction of the field) differ from the mean-field behavior predicted and observed within the DLG universality class. At variance with the DLG case, FSS is attained on lattices with fixed aspect ratio and anisotropy exponent equal to 1 and the transverse Binder cumulant does not vanish at the critical point.

cond-mat.stat-mech↗
arXiv2004

The evolution of a random vortex filament

We study an evolution problem in the space of continuous loops in three-dimensional Euclidean space modelled upon the dynamics of vortex lines in 3d incompressible and inviscid fluids. We establish existence of a local solution starting from Hölder regular loops with index greater than 1/3. When the Hölder regularity of the initial condition X is smaller or equal 1/2 we require X to be a rough path in the sense of Lyons. The solution will then live in an appropriate space of rough paths. In particular we can construct (local) solution starting from almost every Brownian loop.

math.PR↗
arXiv2004

Young integrals and SPDEs

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t$, where $X$ is a $γ$-Hölder continuous function of the time parameter, with values in a distribution space, and $-A$ the generator of an analytical semigroup. Then, we will give some sharp conditions on $X$ in order to solve the above equation in a function space, first in the linear case (for any value of $γ$ in $(0,1)$), and then when $B$ satisfies some Lipschitz type conditions (for $γ>1/2$). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.

math.PR↗
arXiv2004

Statistics of a vortex filament model

A random field composed by Poisson distributed Brownian vortex filaments is constructed. The filament have a random thickness, length and intensity, governed by a measure $γ$. Under appropriate assumptions on $γ$ we compute the scaling law of the structure function and get the multifractal scaling as a particular case.

math.PR↗