Searcharxiv⌕ Search

arXiv subjects

Giuseppe Da Prato

Publications and source records attributed to Giuseppe Da Prato.

At least 37 records · Page 2Linked to original sources
arXiv2012

A basic identity for Kolmogorov operators in the space of continuous functions related to RDEs with multiplicative noise

We consider the Kolmogorov operator associated with a reaction-diffusion equation having polynomially growing reaction coefficient and perturbed by a noise of multiplicative type, in the Banach space $E$ of continuous functions. By analyzing the smoothing properties of the associated transition semigroup, we prove a modification of the classical identité du carré di champs that applies to the present non-Hilbertian setting. As an application of this identity, we construct the Sobolev space $W^{1,2}(E;μ)$, where $μ$ is an invariant measure for the system, and we prove the validity of the Poincaré inequality and of the spectral gap.

math.AP↗
arXiv2012

Pathwise uniqueness for stochastic reaction-diffusion equations in Banach spaces with an Hölder drift component

We prove pathwise uniqueness for an abstract stochastic reaction-diffusion equation in Banach spaces. The drift contains a bounded Hölder term; in spite of this, due to the space-time white noise it is possible to prove pathwise uniqueness. The proof is based on a detailed analysis of the associated Kolmogorov equation. The model includes examples not covered by the previous works based on Hilbert spaces or concrete SPDEs.

math.AP↗
arXiv2012

A mild Ito formula for SPDEs

This article introduces a certain class of stochastic processes, which we suggest to call mild Ito processes, and a new - somehow mild - Ito type formula for such processes. Examples of mild Ito processes are mild solutions of SPDEs and their numerical approximation processes.

math.PR↗
arXiv2012

Maximal $L^2$ regularity for Dirichlet problems in Hilbert spaces

We consider the Dirichlet problem $λU - {\mathcal{L}}U= F$ in \mathcal{O}, U=0 on $\partial \mathcal{O}$. Here $F\in L^2(\mathcal{O}, μ)$ where $μ$ is a nondegenerate centered Gaussian measure in a Hilbert space $X$, $\mathcal{L}$ is an Ornstein-Uhlenbeck operator, and $\mathcal{O}$ is an open set in $X$ with good boundary. We address the problem whether the weak solution $U$ belongs to the Sobolev space $W^{2,2}(\mathcal{O}, μ)$. It is well known that the question has positive answer if $\mathcal{O} = X$; if $\mathcal{O} \neq X$ we give a sufficient condition in terms of geometric properties of the boundary $\partial \mathcal{O}$. The results are quite different with respect to the finite dimensional case, for instance if \mathcal{O} is the ball centered at the origin with radius $r$ we prove that $U\in W^{2,2}(\mathcal{O}, μ)$ only for small $r$.

math.AP↗
arXiv2010

On the Dirichlet semigroup for Ornstein -- Uhlenbeck operators in subsets of Hilbert spaces

We consider a family of self-adjoint Ornstein--Uhlenbeck operators $L_α $ in an infinite dimensional Hilbert space H having the same gaussian invariant measure $μ$ for all $α\in [0,1]$. We study the Dirichlet problem for the equation $λϕ- L_αϕ= f$ in a closed set K, with $f\in L^2(K, μ)$. We first prove that the variational solution, trivially provided by the Lax---Milgram theorem, can be represented, as expected, by means of the transition semigroup stopped to K. Then we address two problems: 1) the regularity of the solution $φ$ (which is by definition in a Sobolev space $W^{1,2}_α(K,μ)$) of the Dirichlet problem; 2) the meaning of the Dirichlet boundary condition. Concerning regularity, we are able to prove interior $W^{2,2}_α$ regularity results; concerning the boundary condition we consider both irregular and regular boundaries. In the first case we content to have a solution whose null extension outside K belongs to $W^{1,2}_α(H,μ)$. In the second case we exploit the Malliavin's theory of surface integrals which is recalled in the Appendix of the paper, then we are able to give a meaning to the trace of $ϕ$ at the boundary of K and to show that it vanishes, as it is natural.

math.AP↗
arXiv2010

Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces

We develop a general technique to prove uniqueness of solutions for Fokker--Planck equations on infinite dimensional spaces. We illustrate this method by implementing it for Fokker--Planck equations in Hilbert spaces with Kolmogorov operators with irregular coefficients and both non-degenerate or degenerate second order part.

math.PR↗
arXiv2009

Kolmogorov equation associated to the stochastic reflection problem on a smooth convex set of a Hilbert space

We consider the stochastic reflection problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $K$ with nonempty interior and regular boundary $Σ$ in a Hilbert space $H$. We prove the existence and uniqueness of a smooth solution for the corresponding elliptic infinite-dimensional Kolmogorov equation with Neumann boundary condition on $Σ$.

math.PR↗
arXiv2009

Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces

We consider a stochastic differential equation in a Hilbert space with time-dependent coefficients for which no general existence and uniqueness results are known. We prove, under suitable assumptions, existence and uniqueness of a measure valued solution, for the corresponding Fokker--Planck equation. In particular, we verify the Chapman--Kolmogorov equations and get an evolution system of transition probabilities for the stochastic dynamics informally given by the stochastic differential equation.

math.PR↗
arXiv2009

Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups

We consider stochastic equations in Hilbert spaces with singular drift in the framework of [Da Prato, Röckner, PTRF 2002]. We prove a Harnack inequality (in the sense of [Wang, PTRF 1997]) for its transition semigroup and exploit its consequences. In particular, we prove regularizing and ultraboundedness properties of the transition semigroup as well as that the corresponding Kolmogorov operator has at most one infinitesimally invariant measure $μ$ (satisfying some mild integrability conditions). Finally, we prove existence of such a measure $μ$ for non-continuous drifts.

math.PR↗
arXiv2008

Self-organized criticality via stochastic partial differential equations

Models of self-organized criticality, which can be described as singular diffusions with or without (multiplicative) Wiener forcing term (as e.g. the Bak/Tang/Wiesenfeld- and Zhang-models), are analyzed. Existence and uniqueness of nonnegative strong solutions are proved. Previously numerically predicted transition to the critical state in 1-D is confirmed by a rigorous proof that this indeed happens in finite time with high probability.

math.PR↗
arXiv2008

On the martingale probem associated to the 2D and 3D Stochastic Navier-Stokes equations

We consider the martingale problem associated to the Navier-Stokes in dimension 2 or 3. Existence is well known and it has been recently shown that markovian transition semi group associated to these equations can be constructed. We study the Kolmogorov operator associated to these equations. It can be defined formally as a differential operator on an infinite dimensional Hilbert space. It can be also defined in an abstract way as the infinitesimal generator of the transition semi group. We explicit cores for these abstract operators and identify them with the concrete differential operators on these cores. In dimension 2, the core is explicit and we can use a classical argument to prove uniqueness for the martingale problem. In dimension 3, we are only able to exhibit a core which is defined abstractly and does not allow to prove uniqueness for the martingale problem. Instead, we exhibit a core for a modified Kolmogorov operator which enables us to prove uniqueness for the martingale problem up to the time the solutions are regular.

math.PR↗
arXiv2008

Stochastic Porous Media Equation and Self-Organized Criticality

The existence and uniqueness of nonnegative strong solutions for stochastic porous media equations with noncoercive monotone diffusivity function and Wiener forcing term is proven. The finite time extinction of solutions with high probability is also proven in 1-D. The results are relevant for self-organized critical behaviour of stochastic nonlinear diffusion equations with critical states.

math.PR↗
arXiv2005

Gradient Bounds for Solutions of Elliptic and Parabolic Equations

Let $L$ be a second order elliptic operator on $R^d$ with a constant diffusion matrix and a dissipative (in a weak sense) drift $b \in L^p_{loc}$ with some $p>d$. We assume that $L$ possesses a Lyapunov function, but no local boundedness of $b$ is assumed. It is known that then there exists a unique probability measure $μ$ satisfying the equation $L^*μ=0$ and that the closure of $L$ in $L^1(μ)$ generates a Markov semigroup $\{T_t\}_{t\ge 0}$ with the resolvent $\{G_λ\}_{λ> 0}$. We prove that, for any Lipschitzian function $f\in L^1(μ)$ and all $t,λ>0$, the functions $T_tf$ and $G_λf$ are Lipschitzian and |\nabla T_tf(x)| \leq T_t|\nabla f|(x) and |\nabla G_λf(x)| \leq \frac{1}λ G_λ|\nabla f|(x). An analogous result is proved in the parabolic case.

math.PR↗