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Alessandra Lunardi

Publications and source records attributed to Alessandra Lunardi.

At least 19 recordsLinked to original sources
arXiv2024

Schrödinger semigroups and the Hörmander hypoellipticity condition

We introduce a class of (possibly) degenerate dispersive equations with a drift. We prove that, under the Hörmander hypoellipticity condition, the relevant Cauchy problem can be uniquely solved in the Schwartz class, and the solution operator can be uniquely extended to a strongly continuous semigroup $\{\mathcal T(t)\}_{t\ge 0}$ in $L^2(\Rm)$. Finally, we prove that for $t>0$ the operator $\mathcal T(t)$ satisfies a sharp form of dispersive estimate in $L^p$, for any $1\le p\le 2$, and an uncertainty principle.

math.AP↗
arXiv2021

Schauder theorems for a class of (pseudo-)differential operators on finite and infinite dimensional state spaces

We prove maximal regularity results in Hölder and Zygmund spaces for linear stationary and evolution equations driven by a large class of differential and pseudo-differential operators L, both in finite and in infinite dimension. The assumptions are given in terms of the semigroup generated by L. We cover the cases of fractional Laplacians and Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and several types of local and nonlocal Ornstein-Uhlenbeck operators, as well as the Gross Laplacian and its negative powers, in infinite dimension.

math.AP↗
arXiv2020

Time regularity for generalized Mehler semigroups

We study continuity and Hölder continuity of $t\mapsto P_tf$, where $P_t$ is a generalized Mehler semigroup in $C_b(X)$, the space of the continuous and bounded functions from a Banach space $X$ to $R$, and $f\in C_b(X)$. The generators $L$ of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators $L$ to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups, in infinite dimension.

math.FA↗
arXiv2020

Ornstein-Uhlenbeck semigroups in infinite dimension

This is a survey paper about Ornstein-Uhlenbeck semigroups in infinite dimension, and their generators. We start from the classical Ornstein-Uhlenbeck semigroup in Wiener spaces and then discuss the general case in Hilbert spaces. Finally, we present some results for O-U semigroups in Banach spaces.

math.AP↗
arXiv2018

On the law of the minimum of the solutions to a class of unidimensional SDEs

We prove that the law of the minimum $m:=\min_{t\in[0,1]} ξ(t)$ of the solution $ξ$ to a one-dimensional ODE with good nonlinearity has continuous density with respect to the Lebesgue measure. As a byproduct of the procedure, we show that the sets $ \{ x\in C([0,1]):\; \min x > r\}$ have finite perimeter with respect to the law $ν$ of the solution $ξ(\cdot)$ in $L^2(0,1)$.

math.PR↗
arXiv2018

BV functions in Hilbert spaces

We study $BV$ functions in a Hilbert space $X$ endowed with a probability measure $ν$, assuming that $ν$ is Fomin differentiable along suitable directions. We establish basic characterizations, and we apply the general theory to relevant examples, including invariant measures of some stochastic PDEs.

math.FA↗
arXiv2016

Malliavin Calculus for non Gaussian differentiable measures and surface measures in Hilbert spaces

We construct surface measures in a Hilbert space endowed with a probability measure $ν$. The theory fits for invariant measures of some stochastic partial differential equations such as Burgers and reaction--diffusion equations. Other examples are weighted Gaussian measures and special product measures $ν$ of non Gaussian measures; in this case we exhibit a Markov process having $ν$ as invariant measure. In any case we prove integration by parts formulae on sublevel sets of good functions (including spheres and hyperplanes) that involve surface integrals.

math.PR↗
arXiv2016

Averaging principle for non autonomous slow-fast systems of stochastic RDEs: the almost periodic case

We study the validity of an averaging principle for a slow-fast system of stochastic reaction diffusion equations. We assume here that the coefficients of the fast equation depend on time, so that the classical formulation of the averaging principle in terms of the invariant measure of the fast equation is not anymore available. As an alternative, we introduce the time depending evolution family of measures associated with the fast equation. Under the assumption that the coefficients in the fast equation are almost periodic, the evolution family of measures is almost periodic. This allows to identify the appropriate averaged equation and prove the validity of the averaging limit.

math.PR↗
arXiv2015

Strong convergence of solutions to nonautonomous Kolmogorov equations

We study a class of nonautonomous, linear, parabolic equations with unbounded coefficients on $\mathbb R^{d}$ which admit an evolution system of measures. It is shown that the solutions of these equations converge to constant functions as $t\to+\infty$. We further establish the uniqueness of the tight evolution system of measures and treat the case of converging coefficients.

math.AP↗
arXiv2015

Semilinear nonautonomous parabolic equations with unbounded coefficients in the linear part

We study the Cauchy problem for the semilinear nonautonomous parabolic equation $u_t=\mathcal{A}(t)u+ψ(t,u)$ in $[s,τ]\times {{\mathbb R}^d}$, $τ> s $, in the spaces $C_b([s, τ]\times{{\mathbb R}^d})$ and in $L^p((s, τ)\times{{\mathbb R}^d}, ν)$. Here $ν$ is a Borel measure defined via a tight evolution system of measures for the evolution operator $G(t,s)$ associated to the family of time depending second order uniformly elliptic operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large and stability of the null solution are also given in both $C_b$ and $L^p$ contexts. The novelty with respect to the literature is that the coefficients of the operators $\mathcal{A}(t)$ are allowed to be unbounded.

math.AP↗
arXiv2014

Sobolev regularity for a class of second order elliptic PDE's in infinite dimension

We consider an elliptic Kolmogorov equation $λu - Ku = f$ in a separable Hilbert space $H$. The Kolmogorov operator $K$ is associated to an infinite dimensional convex gradient system: $dX = (AX - DU(X))dt + dW (t)$, where $A $ is a self--adjoint operator in $H$ and $U$ is a convex lower semicontinuous function. Under mild assumptions we prove that for $λ>0$ and $f\in L^2(H,ν)$ the weak solution $u$ belongs to the Sobolev space $W^{2,2}(H,ν)$, where $ν$ is the log-concave probability measure of the system. Moreover maximal estimates on the gradient of $u$ are proved. The maximal regularity results are used in the study of perturbed non gradient systems, for which we prove that there exists an invariant measure. The general results are applied to Kolmogorov equations associated to reaction--diffusion and Cahn--Hilliard stochastic PDE's.

math.AP↗
arXiv2014

Surface measures in infinite dimension

We construct surface measures associated to Gaussian measures in separable Banach spaces, and we prove several properties including an integration by parts formula.

math.PR↗
arXiv2014

BV functions on convex domains in Wiener spaces

We study functions of bounded variation defined in an abstract Wiener space X, relating the variation of a function u on a convex open set O in X to the behavior near t=0 of T(t)u, T(t) being the Ornstein--Uhlenbeck semigroup in O.

math.FA↗
arXiv2013

Maximal Sobolev regularity in Neumann problems for gradient systems in infinite dimensional domains

We consider an elliptic Kolmogorov equation lambda u - Ku =f in a convex subset C of a separable Hilbert space X. We prove maximal Sobolev regularity of its weak solution, when lambda >0 and f is in L^2(C,nu), where nu is the log-concave measure associated to the system. Moreover we prove maximal estimates on the gradient of u, that allow to show that u satisfies the Neumann boundary condition in the sense of traces at the boundary of C. The general results are applied to Kolmogorov equations of reaction-diffusion stochastic PDEs and Cahn-Hilliard stochastic PDEs in convex sets of suitable Hilbert spaces.

math.AP↗
arXiv2013

Traces of Sobolev functions on regular surfaces in infinite dimensions

In a Banach space $X$ endowed with a nondegenerate Gaussian measure, we consider Sobolev spaces of real functions defined in a sublevel set $O= \{x\in X:\;G(x) <0\}$ of a Sobolev nondegenerate function $G:X\mapsto \R$. We define the traces at $G^{-1}(0)$ of the elements of $W^{1,p}(O, μ)$ for $p>1$, as elements of $L^1(G^{-1}(0), ρ)$ where $ρ$ is the surface measure of Feyel and de La Pradelle. The range of the trace operator is contained in $L^q(G^{-1}(0), ρ)$ for $1\leq q<p$ and even in $L^p(G^{-1}(0), ρ)$ under further assumptions. If $O$ is a suitable halfspace, the range is characterized as a sort of fractional Sobolev space at the boundary. An important consequence of the general theory is an integration by parts formula for Sobolev functions, which involves their traces at $G^{-1}(0)$.

math.AP↗
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↗