Enrico Priola

Well-posedness of the transport equation by stochastic perturbation

We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, 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 a PDE of fluid dynamics that becomes well-posed under the influence of a (multiplicative) noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

Strong uniqueness for stochastic evolution equations in Hilbert spaces perturbed by a bounded measurable drift

We prove pathwise (hence strong) uniqueness of solutions to stochastic evolution equations in Hilbert spaces with merely measurable bounded drift and cylindrical Wiener noise, thus generalizing Veretennikov’s fundamental result on Rd to infinite dimensions. Because Sobolev regularity results implying continuity or smoothness of functions do not hold on infinite-dimensional spaces, we employ methods and results developed in the study of Malliavin–Sobolev spaces in infinite dimensions. The price we pay is that we can prove uniqueness for a large class, but not for every initial distribution. Such restriction, however, is common in infinite dimensions.

Densities for Ornstein–Uhlenbeck processes with jumps

We consider an Ornstein-Uhlenbeck process with values in ℝ n driven by a Levy process (Z t ) taking values in ℝ d with d possibly smaller than n. The Levy noise can have a degenerate or even vanishing Gaussian component. Under a controllability rank condition and a mild assumption on the Levy measure of (Z t ), we prove that the law of the Ornstein-Uhlenbeck process at any time t > 0 has a density on ℝ n . Moreover, when the Levy process is of α-stable type, α ∈ (0, 2), we show that such density is a C ∞ -function.

Null controllability with vanishing energy

Linear, null controllable systems, for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy, are characterized. Theorems are stated in terms of an associated algebraic Riccati equation and in terms of the spectrum of the linear part of the system. The results so obtained allow us to determine Ornstein--Uhlenbeck operators for which the Liouville theorem about bounded harmonic functions holds.

Elliptic and Parabolic Second-Order PDEs with Growing Coefficients

We consider a second-order parabolic equation in ℝ d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this case. The proof introduces a new localization procedure. Our results show that the constant appearing in the classical Schauder estimates is in fact independent of the L ∞-norms of the lower order coefficients. We also give a proof of uniqueness which is of independent interest even in the case of bounded coefficients.

Regularity of stochastic kinetic equations

We consider regularity properties of stochastic kinetic equations with multiplicative noise and drift term which belongs to a space of mixed regularity (

On Weak Uniqueness for Some Degenerate SDEs by Global Lp Estimates

L^p

On a Dirichlet Problem Involving an Ornstein–Uhlenbeck Operator

-regularity in the velocity-variable and Sobolev regularity in the space-variable). We prove that, in contrast with the deterministic case, the SPDE admits a unique weakly differentiable solution which preserves a certain degree of Sobolev regularity of the initial condition without developing discontinuities. To prove the result we also study the related degenerate Kolmogorov equation in Bessel-Sobolev spaces and construct a suitable stochastic flow.

Linear Operator Inequality and Null Controllability with Vanishing Energy for Unbounded Control Systems

We prove uniqueness in law for possibly degenerate SDEs having a linear part in the drift term. Diffusion coefficients corresponding to non-degenerate directions of the noise are assumed to be continuous. When the diffusion part is constant we recover the classical degenerate Ornstein-Uhlenbeck process which only has to satisfy the Hörmander hypoellipticity condition. In the proof we also use global Lp-estimates for hypoelliptic Ornstein-Uhlenbeck operators recently proved in Bramanti et al. (Math. Z. 266, 789–816 2010) and adapt the localization procedure introduced by Stroock and Varadhan. Appendix contains a quite general localization principle for martingale problems.

A Sharp Liouville Theorem for Elliptic Operators

We consider an elliptic Dirichlet problem which involves Ornstein–Uhlenbeck operators of special form in a half space of Rn. We obtain necessary and sufficient conditions under which global Schauder estimates in spaces of Hölder continuous and bounded functions hold. For this purpose we use analytical tools, in particular semigroups and interpolation theory. Moreover we extend a theorem on the analiticity of subordinated semigroups (see Carasso and Kato; Trans. Amer. Math. Soc.327 (1990, 867–877)) to a class of Markov type semigroups. We also provide explicit formulas for the Poisson kernels.