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We consider the stochastically forced Burgers equation with an emphasis on spatially rough driving noise. We show that the law of the process at a fixed time $t$, conditioned on no explosions, is absolutely continuous with respect to the stochastic heat equation obtained by removing the nonlinearity from the equation. This establishes a form of ellipticity in this infinite dimensional setting. The results follow from a recasting of the Girsanov Theorem to handle less spatially regular solutions while only proving absolute continuity at a fixed time and not on path-space. The results are proven by decomposing the solution into the sum of auxiliary processes which are then shown to be absolutely continuous in law to a stochastic heat equation. The number of levels in this decomposition diverges to infinite as we move to the stochastically forced Burgers equation associated to the KPZ equation, which we conjecture is just beyond the validity of our results (and certainly the current proof). The analysis provides insights into the structure of the solution as we approach the regularity of KPZ. A number of techniques from singular SPDEs are employed as we are beyond the regime of classical solutions for much of the paper., Comment: 49 Pages

We consider the Mean Field limit of Gibbsian ensembles of 2-dimensional (2D) point vortices on the torus. It is a classical result that in such limit correlations functions converge to 1, that is, point vortices decorrelate: We compute the rate at which this convergence takes place by means of Gaussian integration techniques, inspired by the correspondence between the 2D Coulomb gas and the Sine-Gordon Euclidean field theory.

We consider Canonical Gibbsian ensembles of Euler point vortices on the 2-dimensional torus or in a bounded domain of R 2 . We prove that under the Central Limit scaling of vortices intensities, and provided that the system has zero global space average in the bounded domain case (neutrality condition), the ensemble converges to the so-called Energy-Enstrophy Gaussian random distributions. This can be interpreted as describing Gaussian fluctuations around the mean field limit of vortices ensembles. The main argument consists in proving convergence of partition functions of vortices and Gaussian distributions., 27 pages, to appear on Communications in Mathematical Physics

We extend the validity of a simple method for the existence of a density for stochastic differential equations, first introduced in [DebRom2014], by proving local estimate for the density, existence for the density with summable drift, and by improving the regularity of the density., Comment: Added a section with examples and applications

We give a rigorous proof of the validity of the point vortex description for a class of inviscid generalized surface quasi-geostrophic models on the whole plane.

We analyse the effect of random initial conditions on the local well-posedness of semi-linear PDEs, to investigate to what extent recent ideas on singular stochastic PDEs can prove useful in this framework.In particular, in some cases, stochastic initial conditions extend the validity of the fixed-point argument to larger spaces than deterministic initial conditions would allow, but in general, it is never possible to go beyond the threshold that is predicted by critical scaling, as in our general class of equations we are not exploiting any special structure present in the equation.We also give a specific example where the level of regularity for the fixed-point argument reached by random initial conditions is not yet critical, but it is already sharp in the sense that we find infinitely many random initial conditions of slightly lower regularity, where there is no solution at all. Thus criticality cannot be reached even by random initial conditions.The existence and uniqueness in a critical space is always delicate, but we can consider the Burgers equation in logarithmically sub-critical spaces, where existence and uniqueness hold, and again random initial conditions allow to extend the validity to spaces of lower regularity which are still logarithmically sub-critical.

The existence of suitable weak solutions of 3D Navier-Stokes equations, driven by a random body force, is proved. These solutions satisfy a local balance of energy. Moreover it is proved also the existence of a statistically stationary solution., 28 pages. Submitted on January 2001; still waiting for the the referee's answer

We explain the construction of some solutions of the Stokes system with a given set of singular points, in the sense of Caffarelli, Kohn and Nirenberg. By means of a partial regularity theorem (proved elsewhere), it turns out that we are able to show the existence of a suitable weak solution to the Navier-Stokes equations with a singular set of positive one dimensional Hausdorff measure., 14 pages, 1 figure

We prove that the density of the law of any finite-dimensional projection of solutions of the Navier–Stokes equations with noise in dimension three is Holder continuous in time with values in the natural space L 1. When considered with values in Besov spaces, Holder continuity still holds. The Holder exponents correspond, up to arbitrarily small corrections, to the expected, at least with the known regularity, diffusive scaling.

We study local existence and uniqueness for a surface growth model with space-time white noise in 2D. Unfortunately, the direct fixed-point argument for mild solutions fails here, as we do not have sufficient regularity for the stochastic forcing. Nevertheless, one can give a rigorous meaning to the stochastic PDE and show uniqueness of solutions in that setting. Using spectral Galerkin method and any other types of regularization of the noise, we obtain always the same solution.

We prove three results on the existence of densities for the laws of finite dimensional functionals of the solutions of the stochastic Navier-Stokes equations in dimension 3. In particular, under very mild assumptions on the noise, we prove that finite dimensional projections of the solutions have densities with respect to the Lebesgue measure which have some smoothness when measured in a Besov space. This is proved thanks to a new argument inspired by an idea introduced in Fournier and Printems (2010)., 20 pages

We consider the dyadic model, which is a toy model to test issues of well-posedness and blow-up for the Navier-Stokes and Euler equations. We prove well-posedness of positive solutions of the viscous problem in the relevant scaling range which corresponds to Navier-Stokes. Likewise we prove well-posedness for the inviscid problem (in a suitable regularity class) when the parameter corresponds to the strongest transport effect of the non-linearity.

We show the existence and uniqueness of solutions (either local or global for small data) for an equation arising in different aspects of surface growth. Following the work of Koch and Tataru we consider spaces critical with respect to scaling and we prove our results in the largest possible critical space such that weak solutions are defined. The uniqueness of global weak solutions remains unfortunately open, unless the initial conditions are sufficiently small., Comment: 17 pages

The paper analyses a model in surface growth, where uniqueness of weak solutions seems to be out of reach. We provide the existence of a weak martingale solution satisfying energy inequalities and having the Markov property. Furthermore, under non-degeneracy conditions on the noise, we establish that any such solution is strong Feller and has a unique invariant measure., Comment: 33 pages, 1 figure

The paper contains several regularity results and blow-up criterions for a surface growth model, which seems to have similar properties to the 3D Navier-Stokes, although it is a scalar equation. As a starting point we focus on energy methods and Lyapunov-functionals.

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier-Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier-Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness., 59 pages; corrected several errors and typos, added references

We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially fast. Moreover, we give a well-posedness criterion for the equations in terms of invariant measures. We also analyse the energy balance and identify the term which ensures equality in the balance., 32 pages

The solutions to a large class of non-linear parabolic PDE s are given in terms of expectations of suitable functionals of a tree of branching particles. A sufficient, and in some cases necessary, condition is given for the integrability of the stochastic representation, using a comparison scalar PDE . In cases where the representation fails to be integrable, a sequence of pruned trees is constructed, producing approximate stochastic representations that in some cases converge, globally in time, to the solution of the original PDE .

We prove ergodicity of the finite dimensional approximations of the three dimen- sional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential. The uniqueness of statistical steady states for the Navier-Stokes equations is a less known but nevertheless important problem in the mathematical theory of turbulence. The question is completely open in dimension three, mainly because, due to the lack of uniqueness of the equations, there is no way yet to give meaning to the mathematical objects involved in the subject. In the present paper the property of ergodicity is proved for the finite dimensional approxi- mations of the three-dimensional Navier-Stokes equations, driven by a random force. The same problem has been solved in two dimensions by E and Mattingly (2). Such result can have a qualitative interest for the statistical behaviour of an incompressible fluid. Indeed, if the Kolmogorov theory of turbulence is taken into account, one can believe that the cascade of energy, responsible of the transport of the energy through the scales, is effective in the inertial range so that at smaller scales only the dissipation ends up to be relevant. Hence the long-time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field. In some sense, if the ultraviolet cut-off is sufficiently large, in order to capture all the important modes, the corresponding invariant measure gives the real behaviour of the fluid. In view of these considerations, the conclusions of the paper can give also both a hint and a possible starting point for the analysis of the infinite dimensional case. We consider a finite dimensional truncation of the three dimensional Navier-Stokes equations, driven by a random force, with periodic boundary conditions. The proof of ergodicity is classical and it is developed in two steps. Firstly we prove that the transition probability densities are regular, by checking that the diffusion operator is hypoelliptic (the Hormander condition). Then we show that the Markov process is irreducible, in the sense that each open set is visited with positive probability at each time. For this aim we study the associated control problem (see Section 6). Irreducibility for the infinite dimensional equations was firstly proved by Flandoli (4), under the assumption that the noise acts on all modes.