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 partial differential equation that become well-posed under the influence of 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.
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 R d 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. This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Probability, 2013, Vol. 41, No. 5, 3306-3344. This reprint differs from the original in pagination and typographic detail. 1 2 DA PRATO, FLANDOLI, PRIOLA AND RÖCKNER nikov [33] in the case H = R d . We refer to [35] and [32] for the case H = R as well as to the generalizations of [33] to unbounded drifts in [23,34] and also to the references therein; see [17,18]. We note that [32] also includes the case of α-stable noise, α ≥ 1, which in turn was extended to R d in [29]. Explicit cases of parabolic stochastic partial differential equations, with space-time white noise in space-dimension one, have been solved on various levels of generality for the drift by Gyöngy and coworkers, in a series of papers; see [1,16,19,20] and the references therein. The difference of the present paper with respect to these works is that we obtain a general abstract result, applicable, for instance, to systems of parabolic equations or equations with differential operators of higher order than two. As we shall see, the price to pay for this generality is a restriction on the initial conditions. Indeed, using that for B = 0 there exists a unique nondegenerate (Gaussian) invariant measure µ, we will prove strong uniqueness for µ-a.e. initial x ∈ H or random H-valued x with distribution absolutely continuous with respect to µ.At the abstract level, this work generalizes [5] devoted to the case where B is bounded and in addition Hölder continuous, but with no restriction on the initial conditions. To prove our result we use some ideas from [5,10,13,14] and [23].The extension of Veretennikov's result [33] and also of [23] to infinite dimensions has resisted various attempts of its realization for many years. The reason is that the finite-dimensional results heavily depend on advanced parabolic Sobolev regularity results for solutions to the corresponding Kolmogorov equations. Such regularity results, leading to continuity or smoothness of the solutions, however, do not hold in infinite dimensions. A technique different from [33] is used in [14]; see also [5, 10] and [29]. This technique allows us to prove uniqueness for stochastic equations with time independent coefficients by merely using elliptic (not parabolic) regularity results. In the present paper we succeed in e...
Mathematics Subject Classification (2000): 60H15, 60J75, 47D07, 35R60.Key words: Stochastic PDEs with jumps, strong Feller property, regularity of trajectories.Abstract: We consider a class of semilinear stochastic evolution equations driven by an additive cylindrical stable noise. We investigate structural properties of the solutions like Markov, irreducibility, stochastic continuity, Feller and strong Feller properties, and study integrability of trajectories. The obtained results can be applied to semilinear stochastic heat equations with Dirichlet boundary conditions and bounded and Lipschitz nonlinearities. * The paper is almost identical with the paper published under the same title in Probab. Theory Related Fields (see [21]) with the exception of some constants in particular in Theorem 4.16 and Hypothesis 5.6. We also thank Lihu Xu for indicating an error in our previous calculations.
We consider a class of elliptic and parabolic differential operators with unbounded coefficients in R n , and we study the properties of the realization of such operators in suitable weighted L 2 spaces.
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