arXiv subjects

Enrico Priola

Publications and source records attributed to Enrico Priola.

At least 19 recordsLinked to original sources

arXiv2026

Exponential integrability of the solution to the stochastic Burgers equation driven by white noise

We study stochastic Burgers equation driven by a rough noise $(-\Delta)^{\gamma} dW_t$, where $\Delta$ is the Laplacian in one dimension with Dirichlet boundary conditions, and $\gamma \in [0,1/4)$. We prove exponential estimates for the solution $X_t^x$, starting from $x \in L^2(0,1)$, by showing that there exists some constant $\lambda >0$ for which \begin{equation} \label{ds} \mathbb{E} \left[\exp\left(\lambda \sup_{t\in[0,T]}\|X_t^x\|_{L^2(0,1)}^2 \right) \right]< \infty. \end{equation} This estimate was known only in the case of trace class noise when $-1/2 <\gamma < -1/4 $ since in that case one can use the It\^o formula. To prove the exponential estimate we combine the Bou\'e-Dupuis method with an argument used in [Da Prato-Debussche, Potential Anal. 2007]. The exponential estimate have important applications in large deviation theory, among others. We also deduce a new Lipschitz regularizing effect for the corresponding Markov semigroup.

math.PR↗

arXiv2025

Stochastic transport equation with Lévy noise

We study the stochastic transport equation with globally $β$-Hölder continuous and bounded vector field driven by a non-degenerate pure-jump Lévy noise of $α$-stable type. Whereas the deterministic transport equation may lack uniqueness, we prove the existence and pathwise uniqueness of a weak solution in the presence of a multiplicative pure jump noise, assuming $\fracα{2}+β>1$. Notably, our results cover the entire range $α\in (0,2)$, including the supercritical regime $α\in(0,1)$ where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} $C^{1+δ}$-diffeomorphism and new regularity results for the Jacobian determinant of the stochastic flow associated to its stochastic characteristic equation. These novel probabilistic results are of independent interest and constitute a substantial component of our work. Our results are the first full generalization of the celebrated paper by Flandoli, Gubinelli, and Priola [Invent. Math. 2010] from the Brownian motion to the pure jump Lévy noise. To the best of our knowledge, this appears to be the first example of a partial differential equation of fluid dynamics where well-posedness is restored by the influence of a non-degenerate pure-jump noise.

math.PR↗

arXiv2025

Parameter estimation from local measurements for a class of stochastic Burgers equations

We deal with a class of semilinear SPDEs driven by space-time white noise that includes the one dimensional stochastic Burgers equation. Such equations can have nonlocal and quadratic nonlinearities. We consider the problem of estimation of the diffusivity parameter in front of the second-order spatial derivative. Based on local observations in space, we study the estimator derived in [Altmeyer, Reiß, Ann. Appl. Probab.(2021)] for linear stochastic heat equation that has also been used in [Altmeyer, Cialenco, Pasemann, Bernoulli (2023)] to cover certain class of semilinear SPDEs including stochastic Burgers equations driven by trace class noise. The space-time white noise case we consider has also relevant physical motivations. After we establish new regularity results for the solution, we are able to show that our proposed estimator is strongly consistent and asymptotically normal.

math.ST↗

arXiv2025

Stochastic dissipative systems in Banach spaces driven by L\'evy noise

In this paper, we are interested in the well-posedness of stochastic reaction diffusion equations like \begin{equation} \begin{cases} dX(t)(\xi)=\big(\Delta_\xi X(t)(\xi)-p(X(t)(\xi))\big)dt+RdW(t)+dL(t) , \quad t\in [0,T];\\ X(0)=x\in L^2(\mathcal{O}) \end{cases} \end{equation} where $\mathcal{O}$ is a bounded open domain of $\mathbb{R}^d$ with regular boundary, $d\in\mathbb{N}$, $p:\mathbb{R}\rightarrow\mathbb{R}$ is a polynomial of odd degree with positive leading coefficient, $R$ is a linear bounded operator on $L^2(\mathcal{O})$, $\{W(t)\}_{t\geq 0}$ is a $L^2(\mathcal{O})$-cylindrical Wiener process, $\{L(t)\}_{t\geq 0}$ is a pure-jump L\'evy process on $L^2(\mathcal{O})$. We complement the equation with suitable boundary conditions on $\partial \mathcal{O}.$ Some papers in literature analize existence and uniqueness of mild solutions for every $x\in L^p(\mathcal{O})$, for some suitable $p\geq 2$. The results of this paper allow to study reaction diffusion equations also on the space of continuous function $C(\overline{O})$. This seems to be new in the L\'evy case (it is already done in the Wiener case).\\ We also discuss and review the previous cited works with the aim of unifying the different frameworks. We underline that when $R=0$ for every $x\in C(\overline{O})$ (or $x\in L^p(\mathcal{O})$) the mild solution to the equation has a c\`adl\`ag modifications in $C(\overline{O})$ (or $\in L^p(\mathcal{O})$), even if $\{L(t)\}_{t \geq 0}$ is not a L\'evy process taking values in $C(\overline{O})$ (or $\in L^p(\mathcal{O})$). This phenomenon for the linear problem (i.e., $F\equiv 0$ in the SPDE) has been investigated in other papers.

math.PR↗

arXiv2025

Mild solutions of HJB equations associated with cylindrical stable Lévy noise in infinite dimensions

We study the optimal control of an infinite-dimensional stochastic system governed by an SDE in a separable Hilbert space driven by cylindrical stable noise. We establish the existence and uniqueness of a mild solution to the associated HJB equation. This result forms the basis for the proof of the Verification Theorem, which is the subject of ongoing research and will provide a sufficient condition for optimality.

math.PR↗

arXiv2024

Regular stochastic flow and Dynamic Programming Principle for jump diffusions

Given a Brownian motion $W$ and a stationary Poisson point process $p$ with values in ${\mathbb R}^d$, we prove a Dynamic Programming Principle (DPP) in a strong formulation for a stochastic control problem involving controlled SDEs of the form \begin{align} \label{ci1} \nonumber dX_{t}=&\,b(t, X_{t}, a_t) dt + α\left(t, X_{t}, a_t \right) dW_t+ \! \! \int_{ |z| \le 1} g\left(X_{t-},t,z, a_t \right)\widetilde{N}_p\left(dt,dz\right) \\ & + \int_{ |z| >1 } f\left(X_{t-},t,z, a_t \right){N}_p\left(dt,dz\right), \quad \; X_s=x\in\mathbb{R}^d,\,0\le s \le t \le T. \;\;\;\;\;\;\;\;\;\; (1) \end{align} Here $N_p$ [resp., $\widetilde{N}_p$] is the Poisson [resp., compensated Poisson] random measure associated with $p$. We consider arbitrary predictable controls $a \in {\mathcal P}_T$ with values in a closed convex set $C \subset {\mathbb R}^{l}$. The coefficients $b$, $α$, and $g$ satisfy linear growth and Lipschitz--type conditions in the $x-$variable, and are continuous in the control variable. To prove the DPP for the value function $ v(s,x)=\sup_{a \in {\mathcal P}_T} \, \mathbb{E}\big[\int_{s}^{T}h\left(r,X_r^{s,x,a}, a_r\right)dr + j\left(X_T^{s,x,a}\right)\big] $, assuming that $h$ and $j$ are bounded and continuous, we establish the existence of a regular stochastic flow for (1) when the coefficients are independent of the control $a$. Notably, this regularity result is new even when there is no large--jumps component, i.e., $f\equiv0$ (cf. Kunita's recent book on stochastic flows). The proof of the DPP is completed by introducing an approach that relies on a suitable subclass of finitely generated step controls in $\mathcal{P}_T$. These controls allow us to apply a basic measurable selection theorem by L. D. Brown and R. Purves. We believe that this novel method is of independent interest and could be adapted to prove DPPs arising in other stochastic control problems.

math.PR↗

arXiv2024

One-side Liouville theorems under an exponential growth condition for Kolmogorov operators

It is known that for a possibly degenerate hypoelliptic Ornstein-Uhlenbeck operator $$ L= \frac{1}{2}\text{ tr} (QD^2 ) + \langle Ax, D \rangle = \frac{1}{2}\text{ div} (Q D ) + \langle Ax, D \rangle,\;\; x \in R^N, $$ all (globally) bounded solutions of $Lu=0$ on $R^N$ are constant if and only if all the eigenvalues of $A$ have non-positive real parts (i.e., $s(A) \le 0)$. We show that if $Q$ is positive definite and $s(A) \le 0$, then any non-negative solution $v$ of $Lv=0$ on $R^N$ which has at most an exponential growth is indeed constant. Thus under a non-degeneracy condition we relax the boundedness assumption on the harmonic functions and maintain the sharp condition on the eigenvalues of $A$. We also prove a related one-side Liouville theorem in the case of hypoelliptic Ornstein-Uhlenbeck operators.

math.AP↗

arXiv2023

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 $A^{-1+\beta}$ is of trace class for some $\beta>0$, $b\colon H\to H$ 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 $$(P\circ Z_x^{-1})(\Phi) =\int_X\Phi(h+e^{\cdot A}x)\, \exp\left\{ -\tfrac12|\gamma_x(h)|^2_{ H_{Q_T}} + I(\gamma_x)(h)\right\} N_{Q_T}(dh), $$ where $ N_{Q_T}$ is the law of $W_A$ in $C([0,T],H)$, $ H_{Q_T}$ its Cameron--Martin space, $$ [\gamma_x(k)](t)=\int_0^t e^{(t-s)A}b(k(s)+e^{sA}x) ds,\quad t\in[0,T], \; k \in C([0,T],H) $$ and $I(\gamma_x) $ is the It\^o integral of $\gamma_x$. 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 $\nu$ of the Markov semigroup $(P_t)$. Some concluding remarks are devoted to a similar problem with colored noise.

math.PR↗

arXiv2023

About the regularity of degenerate non-local Kolmogorov operators under diffusive perturbations

We study here the effects of a time-dependent second order perturbation to a degenerate Ornstein-Uhlenbeck type operator whose diffusive part can be either local or non-local. More precisely, we establish that some estimates, such as the Schauder and Sobolev ones, already known for the non-perturbed operator still hold, and with the same constants, when we perturb the Ornstein-Uhlenbeck operator with second order diffusions with coefficients only depending on time in a measurable way. The aim of the current work is twofold: we weaken the assumptions required on the perturbation in the local case which has been considered already in [KP17] and we extend the approach presented therein to a wider class of degenerate Kolmogorov operators with non-local diffusive part of symmetric stable type.

math.AP↗

arXiv2023

A BSDEs approach to pathwise uniqueness for stochastic evolution equations

We prove strong well-posedness for a class of stochastic evolution equations in Hilbert spaces H when the drift term is Holder continuous. This class includes examples of semilinear stochastic damped wave equations which describe elastic systems with structural damping (for such equations even existence of solutions in the linear case is a delicate issue) and semilinear stochastic 3D heat equations. In the deterministic case, there are examples of non-uniqueness in our framework. Strong (or pathwise) uniqueness is restored by means of a suitable additive Wiener noise. The proof of uniqueness relies on the study of related systems of infinite dimensional forward-backward SDEs (FBSDEs). This is a different approach with respect to the well-known method based on the Ito formula and the associated Kolmogorov equation (the so-called Zvonkin transformation or Ito-Tanaka trick). We deal with approximating FBSDEs in which the linear part generates a group of bounded linear operators in H; such approximations depend on the type of SPDEs we are considering. We also prove Lipschitz dependence of solutions from their initial conditions.

math.PR↗

arXiv2023

Correction to "Well-posedness of semilinear stochastic wave equations with Hölder continuous coefficients''

We prove that semilinear stochastic abstract wave equations, including wave and plate equations, are well-posed in the strong sense with an $α$-Hölder continuous drift coefficient, if $α\in (2/3,1)$. The uniqueness may fail for the corresponding deterministic PDE and well-posedness is restored by adding an external random forcing of white noise type. This shows a kind of regularization by noise for the semilinear wave equation. To prove the result we introduce an approach based on backward stochastic differential equations, differentiability along subspaces and control theoretic results. We stress that the well-posedness holds despite the Markov semigroup associated to the linear stochastic wave equation is not strong Feller.

math.PR↗

arXiv2022

Correction to "An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs"

We show uniqueness in law for the critical SPDE \begin{eqnarray} \label{qq1} dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, \end{eqnarray} where $A$ $ : \text{dom}(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be locally Hölder continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers equations and for three-dimensional stochastic Cahn-Hilliard type equations which have interesting applications. We do not know if uniqueness holds under the sole assumption of continuity of $F$ plus growth condition as stated in [Priola, Ann. of Prob. 49 (2021)]. To get weak uniqueness we use an infinite dimensional localization principle and an optimal regularity result for the Kolmogorov equation $ λu - L u = f$ associated to the SPDE when $F = z \in H$ is constant and $λ>0$. This optimal result is similar to a theorem of [Da Prato, J. Evol. Eq. 3 (2003)].

math.PR↗

arXiv2022

A counterexample to $L^{\infty}$-gradient type estimates for Ornstein-Uhlenbeck operators

Let $(λ_k)$ be a strictly increasing sequence of positive numbers such that $\sum_{k=1}^{\infty} \frac{1}{λ_k} < \infty.$ Let $f $ be a bounded smooth function and denote by $u= u^f$ the bounded classical solution to $u(x) - \frac{1}{2}\sum_{k=1}^m D^2_{kk} u(x) + \sum_{k =1}^m λ_k x_k D_k u(x) = f(x), $ $ x \in \R^m$. It is known that the following dimension-free estimate holds: $$ \displaystyle \int_{\R^m} \Big (\sum_{k=1}^m λ_k \, (D_k u (y))^2 \Big)^{p/2} μ_m (dy) \le (c_p)^p \, \int_{\R^m} |f( y)|^p μ_m (dy),\;\;\; 1 < p < \infty; $$ here $μ_m$ is the "diagonal" Gaussian measure determined by $λ_1, \ldots, λ_m$ and $c_p > 0$ is independent of $f$ and $m$. This is a consequence of generalized Meyer's inequalities [Chojnowska-Michalik, Goldys, J. Funct. Anal. 182 (2001)]. We show that, if $λ_k \sim k^2$, then such estimate does not hold when $p= \infty$. Indeed we prove $$ \sup_{\substack{f \in C^{ 2}_b(\R^m),\;\; \|f\|_{\infty} \leq 1}} \Big \{ \sum_{k=1}^m λ_k \, (D_k u^f (0))^2 \Big \} \to \infty \;\; \text {as} \; m \to \infty. $$ This is in contrast to the case of $λ_k = λ>0$, $k \ge 1$, where a dimension-free bound holds for $p =\infty$.

math.PR↗

arXiv2022

HJB equations and stochastic control on half-spaces of Hilbert spaces

In this paper we study a first extension of the theory of mild solutions for HJB equations in Hilbert spaces to the case when the domain is not the whole space. More precisely, we consider a half-space as domain, and a semilinear Hamilton-Jacobi-Bellman (HJB) equation. Our main goal is to establish the existence and the uniqueness of solutions to such HJB equations, that are continuously differentiable in the space variable. We also provide an application of our results to an exit time optimal control problem and we show that the corresponding value function is the unique solution to a semilinear HJB equation, possessing sufficient regularity to express the optimal control in feedback form. Finally, we give an illustrative example.

math.OC↗

arXiv2022

Gradient formula for transition semigroup corresponding to stochastic equation driven by a system of independent Lévy processes

Let $(P_t)$ be the transition semigroup of the Markov family $(X^x(t))$ defined by SDE $$ d X= b(X) dt + d Z, \qquad X(0)=x, $$ where $Z=\left(Z_1, \ldots, Z_d\right)^*$ is a system of independent real-valued Lévy processes. Using the Malliavin calculus we establish the following gradient formula $$ \nabla P_tf(x)= \mathbb{E}\, f\left(X^x(t)\right) Y(t,x), \qquad f\in B_b(\mathbb{R}^d), $$ where the random field $Y$ does not depend on $f$. Sharp estimates on $\nabla P_tf(x)$ when $Z_1, \ldots , Z_d$ are $α$-stable processes, $α\in (0,2)$, are also given.

math.PR↗

arXiv2021

An optimal regularity result for Kolmogorov equations and weak uniqueness for some critical SPDEs

We show uniqueness in law for the critical SPDE $$ dX_t = AX_t dt + (-A)^{1/2}F(X(t))dt + dW_t,\;\; X_0 =x \in H, $$ where $A$ $ : dom(A) \subset H \to H$ is a negative definite self-adjoint operator on a separable Hilbert space $H$ having $A^{-1}$ of trace class and $W$ is a cylindrical Wiener process on $H$. Here $F: H \to H $ can be continuous with at most linear growth (some functions $F$ which grow more than linearly can also be considered). This leads to new uniqueness results for generalized stochastic Burgers' equations and for three-dimensional stochastic Cahn-Hilliard type equations which have interesting applications. To get weak uniqueness we also establish a new optimal regularity result for the Kolmogorov equation $ λu - Lu = f$ on $H$, where $λ>0$, $ f: H \to {\mathbb R}$ is Borel and bounded and $L$ is the Ornstein-Uhlenbeck operator related to the SPDE when $F=0$. In particular we show that the first derivative $Du : H \to H$ verifies $Du(x) \in \text{dom}((-A)^{1/2})$, for any $x \in H,$ and moreover $$ \sup_{x \in H} |(-A)^{1/2}Du (x)|_H = \| (-A)^{1/2}Du \|_{0} \le C \, \| f\|_{0}. $$

math.PR↗

arXiv2020

Harnack inequality and Liouville-type theorems for Ornstein-Uhlenbeck and Kolmogorov operators

We prove, with a purely analytic technique, a one-side Liouville theorem for a class of Ornstein--Uhlenbeck operators ${\mathcal L_0}$ in $\mathbb{R}^N$, as a consequence of a Liouville theorem at "$t=- \infty$" for the corresponding Kolmogorov operators ${\mathcal L_0} - \partial_t$ in $\mathbb{R}^{N+1}$. In turn, this last result is proved as a corollary of a global Harnack inequality for non-negative solutions to $({\mathcal L_0} - \partial_t) u = 0$ which seems to have an independent interest in its own right. We stress that our Liouville theorem for ${\mathcal L_0}$ cannot be obtained by a probabilistic approach based on recurrence if $N>2$. We provide a self-contained proof of a Liouville theorem involving recurrent Ornstein--Uhlenbeck stochastic processes in the Appendix.

math.AP↗

arXiv2020

Weak Well-Posedness of Multidimensional Stable Driven SDEs in the Critical Case

We establish weak well-posedness for critical symmetric stable driven SDEs in R d with additive noise Z, d $\ge$ 1. Namely, we study the case where the stable index of the driving process Z is $α$ = 1 which exactly corresponds to the order of the drift term having the coefficient b which is continuous and bounded. In particular, we cover the cylindrical case when Zt = (Z 1 t ,. .. , Z d t) and Z 1 ,. .. , Z d are independent one dimensional Cauchy processes. Our approach relies on L p-estimates for stable operators and uses perturbative arguments. 1. Statement of the problem and main results We are interested in proving well-posedness for the martingale problem associated with the following SDE: (1.1) X t = x + t 0 b(X s)ds + Z t , where (Z s) s$\ge$0 stands for a symmetric d-dimensional stable process of order $α$ = 1 defined on some filtered probability space ($Ω$, F, (F t) t$\ge$0 , P) (cf. [2] and the references therein) under the sole assumptions of continuity and boundedness on the vector valued coefficient b: (C) The drift b : R d $\rightarrow$ R d is continuous and bounded. 1 Above, the generator L of Z writes: L$Φ$(x) = p.v. R d \{0} [$Φ$(x + z) -- $Φ$(x)]$ν$(dz), x $\in$ R d , $Φ$ $\in$ C 2 b (R d), $ν$(dz) = d$ρ$ $ρ$ 2$μ$ (d$θ$), z = $ρ$$θ$, ($ρ$, $θ$) $\in$ R * + x S d--1. (1.2) (here $\times$, $\times$ (or $\times$) and | $\times$ | denote respectively the inner product and the norm in R d). In the above equation, $ν$ is the L{é}vy intensity measure of Z, S d--1 is the unit sphere of R d and$μ$ is a spherical measure on S d--1. It is well know, see e.g. [20] that the L{é}vy exponent $Φ$ of Z writes as: (1.3) $Φ$($λ$) = E[exp(i $λ$, Z 1)] = exp -- S d--1 | $λ$, $θ$ |$μ$(d$θ$) , $λ$ $\in$ R d , where $μ$ = c 1$μ$ , for a positive constant c 1 , is the so-called spectral measure of Z. We will assume some non-degeneracy conditions on $μ$. Namely we introduce assumption (ND) There exists $κ$ $\ge$ 1 s.t. (1.4) $\forall$$λ$ $\in$ R d , $κ$ --1 |$λ$| $\le$ S d--1 | $λ$, $θ$ |$μ$(d$θ$) $\le$ $κ$|$λ$|. 1 The boundedness of b is here assumed for technical simplicity. Our methodology could apply, up to suitable localization arguments, to a drift b having linear growth.

math.PR↗