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Stéphane Menozzi

Publications and source records attributed to Stéphane Menozzi.

18 recordsLinked to original sources
arXiv2026

Weak Error on the densities for the Euler scheme of stable additive SDEs with Besov drift

We are interested in the Euler-Maruyama dicretization of the formal SDE, $dX_t=b(t,X_t)dt+dZ_t$, where $Z$ is a symmetric isotropic d dimensional stable process of index $α\in (1,2)$, and $b$ is distributional. It belongs to a mix Lebesgue-Besov space. The associated parameters satisfy some constraints which guarantee weak-well posedness. Defining an appropriate Euler scheme, we obtain a convergence rate for the weak error on the densities. The rate depends on the parameters.

math.AP↗
arXiv2026

On Heat kernel Estimtes for Brownian SDEs with Distributional Drift

We establish heat-kernel bounds and regularity estimates for the transition densities of the diffusion associated with the martingale problem corresponding to the generator of a formal multidimensional Brownian SDE with singular drift. As a by-product, we also derive Schauder estimates for the associated Kolmogorov (kinetic) Cauchy problem. We consider both the cases of non-degenerate and degenerate noise (e.g. kinetic-type models), in the so-called Young regime. Namely, we consider a time inhomogeneous drift in L $\infty$ [0,T ] C $β$ o for some fixed time horizon T , where ), with o standing for an underlying distance, namely the usual Euclidean one in the non degenerate setting, and the scale-homogeneous one in the kinetic case. Importantly, the estimates are obtained by employing as parametrix the transition density of the SDE (with variable coefficients) without singular perturbation, as opposed to the standard Levi parametrix obtained by freezing the noise. Finally, since the noise is multiplicative, the weak well-posedness of the singular SDE is a novel result in itself, and the density estimates directly imply irreducibility and strong Feller property of its solutions.

math.AP↗
arXiv2026

Weak error for SDEs with additive stable noise and singular drift: choose the test function in the same space as the drift!

We emphasize that for a stochastic differential equation with isotropic stable additive noise and non Lipschitz drift, when considering an appropriate discretization scheme and the associated weak error, it is somehow natural to consider a test function having the same spatial regularity as the drift involved. We will in particular focus on drifts belonging to Lebsegue, H{ö}lder or Besov spaces with negative regularity index in their spatial variable. Choosing such a test function allows to improve the convergence rate previously obtained on the densities (for Lebesgue or H{ö}lder drifts) or preserve the rate for possibly singular generalized test functions (for Besov spaces with negative regularity).

math.PR↗
arXiv2025

Strong regularization by noise for a class of kinetic SDEs driven by symmetric α-stable processes

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $α$-stable process under Hölder regularity conditions for the drift term. We partially recover the thresholds for the Hölder regularity that are optimal for weak uniqueness. In general dimension, we only consider $α= 2$ and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe-Yamada technique.

math.PR↗
arXiv2024

Weak well-posedness and weak discretization error for stable-driven SDEs with Lebesgue drift

We are interested in the discretization of stable driven SDEs with additive noise for $α$ $\in$ (1, 2) and Lq -- Lp drift under the Serrin type condition $α$/q + d/p < $α$ -- 1. We show weak existence and uniqueness as well as heat kernel estimates for the SDE and obtain a convergence rate of order (1/$α$)*($α$ -- 1 -- $α$/q - d/p) for the difference of the densities for the Euler scheme approximation involving suitably cutoffed and time randomized drifts.

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↗
arXiv2022

Non Linear Singular Drifts and Fractional Operators

We consider parabolic PDEs associated with fractional type operators drifted by non-linear singular first order terms. When the drift enjoys some boundedness properties in appropriate Lebesgue and Besov spaces, we establish by exploiting a priori Besov-type estimates, the H{ö}lder continuity of the solutions. In particular, we handle the almost critical case in whole generality.

math.AP↗
arXiv2022

On Multidimensional stable-driven Stochastic Differential Equations with Besov drift

We establish well-posedness results for multidimensional non degenerate $α$-stable driven SDEs with time inhomogeneous singular drifts in $\mathbb{L}^r-{\mathbb B}_{p,q}^{-1+γ}$ with $γ<1$ and $α$ in $(1,2]$, where $\mathbb{L}^r$ and ${\mathbb B}_{p,q}^{-1+γ} $ stand for Lebesgue and Besov spaces respectively. Precisely, we first prove the well-posedness of the corresponding martingale problem and then give a precise meaning to the dynamics of the SDE. This allows us in turn to define an ad hoc notion of weak solution, for which well-posedness holds as well. Our results rely on the smoothing properties of the underlying PDE, which is investigated by combining a perturbative approach with duality results between Besov spaces.

math.PR↗
arXiv2022

Heat kernel of supercritical SDEs with unbounded drifts

Let $α\in(0,2)$ and $d\in{\mathbb N}$. Consider the following SDE in ${\mathbb R}^d$:$${\rm d}X_t=b(t,X_t){\rm d} t+a(t,X_{t-}){\rm d} L^{(α)}_t,\ \ X_0=x,$$where $L^{(α)}$ is a $d$-dimensional rotationally invariant $α$-stable process, $b:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ and $a:{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ are H{ö}lder continuous functions in space, with respective order $β,γ\in (0,1)$ such that $(β\wedge γ)+α>1$, uniformly in $t$. Here $b$ may be unbounded.When $a$ is bounded and uniformly elliptic, we show that the unique solution $X_t(x)$ of the above SDE admits a continuous density, which enjoys sharp two-sided estimates. We also establish sharp upper-bound for the logarithmic derivative. In particular, we cover the whole supercritical range $α\in (0,1) $.Our proof is based on ad hoc parametrix expansions and probabilistic techniques.

math.AP↗
arXiv2021

Convergence Rate of the Euler-Maruyama Scheme Applied to Diffusion Processes with L Q -- L $ρ$ Drift Coefficient and Additive Noise

We are interested in the time discretization of stochastic differential equations with additive d-dimensional Brownian noise and L q -- L $ρ$ drift coefficient when the condition d $ρ$ + 2 q < 1, under which Krylov and R{ö}ckner [26] proved existence of a unique strong solution, is met. We show weak convergence with order 1 2 (1 -- (d $ρ$ + 2 q)) which corresponds to half the distance to the threshold for the Euler scheme with randomized time variable and cutoffed drift coefficient so that its contribution on each time-step does not dominate the Brownian contribution. More precisely, we prove that both the diffusion and this Euler scheme admit transition densities and that the difference between these densities is bounded from above by the time-step to this order multiplied by some centered Gaussian density.

math.PR↗
arXiv2020

Sharp Schauder Estimates for some Degenerate Kolmogorov Equations

We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.

math.AP↗
arXiv2019

Well-posedness of some non-linear stable driven SDEs

We prove the well-posedness of some non-linear stochastic differential equations in the sense of McKean-Vlasov driven by non-degenerate symmetric $α$-stable Lévy processes with values in $R^d$ under some mild H{ö}lder regularity assumptions on the drift and diffusion coefficients with respect to both space and measure variables. The methodology developed here allows to consider unbounded drift terms even in the so-called super-critical case, i.e. when the stability index $α\in (0, 1)$. New strong well-posedness results are also derived from the previous analysis.

math.AP↗
arXiv2019

Schauder estimates for drifted fractional operators in the supercritical case

We consider a non-local operator $L_{ α}$ which is the sum of a fractional Laplacian $\triangle^{α/2} $, $α\in (0,1)$, plus a first order term which is measurable in the time variable and locally $β$-Hölder continuous in the space variables. Importantly, the fractional Laplacian $Δ^{ α/2} $ does not dominate the first order term. We show that global parabolic Schauder estimates hold even in this case under the natural condition $α+ β>1$. Thus, the constant appearing in the Schauder estimates is in fact independent of the $L^{\infty}$-norm of the first order term. In our approach we do not use the so-called extension property and we can replace $\triangle^{α/2} $ with other operators of $α$-stable type which are somehow close, including the relativistic $α$-stable operator. Moreover, when $α\in (1/2,1)$, we can prove Schauder estimates for more general $α$-stable type operators like the singular cylindrical one, i.e., when $\triangle^{α/2} $ is replaced by a sum of one dimensional fractional Laplacians $\sum_{k=1}^d (\partial_{x_k x_k}^2 )^{α/2}$.

math.AP↗
arXiv2017

Non Linear Singular Drifts and Fractional Operators: when Besov meets Morrey and Campanato

Within the global setting of singular drifts in Morrey-Campanato spaces presented in [6], we study now the H{ö}lder regularity properties of the solutions of a transport-diffusion equation with nonlinear singular drifts that satisfy a Besov stability property. We will see how this Besov information is relevant and how it allows to improve previous results. Moreover, in some particular cases we show that as the nonlinear drift becomes more regular, in the sense of Morrey-Campanato spaces, the additional Besov stability property will be less useful.

math.AP↗
arXiv2016

Fractional operators with singular drift: Smoothing properties and Morrey-Campanato spaces

We investigate some smoothness properties for a transport-diffusion equation involving a class of non-degerate L{é}vy type operators with singular drift. Our main argument is based on a duality method using the molecular decomposition of Hardy spaces through which we derive some H{ö}lder continuity for the associated parabolic PDE. This property will be fulfilled as far as the singular drift belongs to a suitable Morrey-Campanato space for which the regularizing properties of the L{é}vy operator suffice to obtain global H{ö}lder continuity.

math.AP↗
arXiv2009

Stopped diffusion processes: boundary corrections and overshoot

For a stopped diffusion process in a multidimensional time-dependent domain $\D$, we propose and analyse a new procedure consisting in simulating the process with an Euler scheme with step size $Δ$ and stopping it at discrete times $(iΔ)_{i\in\N^*}$ in a modified domain, whose boundary has been appropriately shifted. The shift is locally in the direction of the inward normal $n(t,x)$ at any point $(t,x)$ on the parabolic boundary of $\D$, and its amplitude is equal to $0.5826 (...) |n^*σ|(t,x)\sqrt Δ$ where $σ$ stands for the diffusion coefficient of the process. The procedure is thus extremely easy to use. In addition, we prove that the rate of convergence w.r.t. $Δ$ for the associated weak error is higher than without shifting, generalizin g previous results by \cite{broa:glas:kou:97} obtained for the one dimensional Brownian motion. For this, we establish in full generality the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. Numerical experiments support these results.

math.PR↗
arXiv2006

A forward--backward stochastic algorithm for quasi-linear PDEs

We propose a time-space discretization scheme for quasi-linear parabolic PDEs. The algorithm relies on the theory of fully coupled forward--backward SDEs, which provides an efficient probabilistic representation of this type of equation. The derivated algorithm holds for strong solutions defined on any interval of arbitrary length. As a bypass product, we obtain a discretization procedure for the underlying FBSDE. In particular, our work provides an alternative to the method described in [Douglas, Ma and Protter (1996) Ann. Appl. Probab. 6 940--968] and weakens the regularity assumptions required in this reference.

math.PR↗