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Giuseppe Da Prato

Publications and source records attributed to Giuseppe Da Prato.

At least 19 recordsLinked to original sources
arXiv2026

A mild Girsanov formula

We consider a well posed SPDE$\colon dZ=(AZ+b(Z)) dt+dW(t),\,Z_0=x, $ on a separable Hilbert space $H$, where $A\colon H\to H$ is self-adjoint, negative and such that $A^{-1+β}$ is of trace class for some $β>0$, $b\colon H\to H$ is Lipschitz continuous and $W$ is a cylindrical Wiener process on $H$. We denote by $W_A(t)=\int_0^te^{(t-s)A}\,dW(s),\,t\in[0,T],$ the stochastic convolution. We prove, with the help of a formula for nonlinear transformations of Gaussian integrals due to R. Ramer, the following identity $$(P\circ Z_x^{-1})(Φ) =\int_XΦ(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|γ_x(h)|^2_{ H_{Q_T}} + I(γ_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [γ_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(γ_x) $ is the Itô integral of $γ_x$. Some applications are discussed; in particular, when $b$ is dissipative we provide an explicit formula for the law of the stationary process and the invariant measure $ν$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.

math.PR↗
arXiv2019

Absolutely continuous solutions for continuity equations in Hilbert spaces

We prove existence of solutions to continuity equations in a separable Hilbert space. We look for solutions which are absolutely continuous with respect to a reference measure γwhich is Fomin-differentiable with exponentially integrable partial logarithmic derivatives. We describe a class of examples to which our result applies and for which we can prove also uniqueness. Finally, we consider the case where γis the invariant measure of a reaction-diffusion equation and prove uniqueness of solutions in this case. We exploit that the gradient operator D_x is closable with respect to L^p(H,γ) and a recent formula for the commutator D_xP_t - P_tD_x where P_t is the transition semigroup corresponding to the reaction-diffusion equation, [DaDe14]. We stress that P_t is not necessarily symmetric in this case. This uniqueness result is an extension to such γof that in [DaFlRo14] where γwas the Gaussian invariant measure of a suitable Ornstein-Uhlenbeck process.

math.PR↗
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

Gradient estimates for SDEs without monotonicity type conditions

We prove gradient estimates for transition Markov semigroups $(P_t)$ associated to SDEs driven by multiplicative Brownian noise having possibly unbounded $C^1$-coefficients, without requiring any monotonicity type condition. In particular, first derivatives of coefficients can grow polynomially and even exponentially. We establish pointwise estimates with weights for $D_x P_tφ$ of the form \[ {\sqrt{t}} \, |D_x P_t φ(x) | \le c \, (1+ |x|^k) \, \| φ\|_{\infty} \] $t \in (0,1]$, $φ\in C_b ({\mathbb R}^d)$, $x \in {\mathbb R}^d.$ To prove the result we use two main tools. First, we consider a Feynman--Kac semigroup with potential $V$ related to the growth of the coefficients and of their derivatives for which we can use a Bismut-Elworthy-Li type formula. Second, we introduce a new regular approximation for the coefficients of the SDE. At the end of the paper we provide an example of SDE with additive noise and drift $b$ having sublinear growth together with its derivative such that uniform estimates for $D_x P_t φ$ without weights do not hold.

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

Construction of a surface integral under local Malliavin assumption and integration by parts formulae

In this paper, we consider convex sets $K_r = \{g \ge r\}$ in an infinite dimensional Hilbert space, where $g$ is suitably related to a reference Gaussian measure $μ$ in $H$. We first show how to define a surface measure on the level sets $\{g = r\}$ that is related to $μ$. This allows to introduce an integration-by-parts formula in $H$. This formula can be applied in several important constructions, as for instance the case where $μ$ is the law of a (Gaussian) stochastic process and $H$ is the space of its trajectories

math.PR↗
arXiv2015

An integral inequality for the invariant measure of some finite dimensional stochastic differential equation

We prove an integral inequality for the invariant measure $ν$ of a stochastic differential equation with additive noise in a finite dimensional space $H=\R^d$. As a consequence, we show that there exists the Fomin derivative of $ν$ in any direction $z\in H$ and that it is given by $v_z=\langle D\logρ,z\rangle$, where $ρ$ is the density of $ν$ with respect to the Lebesgue measure. Moreover, we prove that $v_z\in L^p(H,ν)$ for any $p\in[1,\infty)$. Also we study some properties of the gradient operator in $L^p(H,ν)$ and of his adjoint.

math.PR↗
arXiv2015

An integral inequality for the invariant measure of a stochastic reaction--diffusion equation

We consider a reaction--diffusion equation perturbed by noise (not necessarily white). We prove an integral inequality for the invariant measure $ν$ of a stochastic reaction--diffusion equation. Then we discuss some consequences as an integration by parts formula which extends to $ν$ a basic identity of the Malliavin Calculus. Finally, we prove the existence of a surface measure for a ball and a half-space of $H$.

math.PR↗
arXiv2014

Estimate for $P_tD$ for the stochastic Burgers equation

We consider the Burgers equation on $H=L^2(0,1)$ perturbed by white noise and the corresponding transition semigroup $P_t$. We prove a new formula for $P_tDφ$ (where $φ:H\to\R$ is bounded and Borel) which depends on $φ$ but not on its derivative. Then we deduce some new consequences for the invariant measure $ν$ of $P_t$ as its Fomin differentiability and an integration by parts formula which generalises the classical one for gaussian measures.

math.PR↗
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↗
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

An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations

We prove a new uniqueness result for solutions to Fokker-Planck-Kolmogorov (FPK) equations for probability measures on infinite-dimensional spaces. We consider infinite-dimensional drifts that admit certain finite-dimensional approximations. In contrast to most of the previous work on FPK-equations in infinite dimensions, we include cases with non-constant coefficients in the second order part and also include degenerate cases where these coefficients can even be zero. Also a new existence result is proved. Some applications to Fokker-Planck-Kolmogorov equations associated with SPDEs are presented.

math.PR↗
arXiv2013

Uniqueness for continuity equations in Hilbert spaces with weakly differentiable drift

We prove uniqueness for continuity equations in Hilbert spaces $H$. The corresponding drift $F$ is assumed to be in a first order Sobolev space with respect to some Gaussian measure. As in previous work on the subject, the proof is based on commutator estimates which are infinite dimensional analogues to the classical ones due to DiPerna-Lions. Our general approach is, however, quite different since, instead of considering renormalized solutions, we prove a dense range condition implying uniqueness. In addition, compared to known results by Ambrosio-Figalli and Fang-Luo, we use a different approximation procedure, based on a more regularizing Ornstein-Uhlenbeck semigroup and consider Sobolev spaces of vector fields taking values in $H$ rather than the Cameron-Martin space of the Gaussian measure. This leads to different conditions on the derivative of $F$, which are incompatible with previous work on the subject. Furthermore, we can drop the usual exponential integrability conditions on the Gaussian divergence of $F$, thus improving known uniqueness results in this respect.

math.AP↗