arXiv subjects

Vlad Bally

Publications and source records attributed to Vlad Bally.

At least 19 recordsLinked to original sources

arXiv2026

Ergodic approximation for the invariant distribution: An abstract framework for law-dependent dynamics

This paper studies the approximation of invariant distributions for a broad class of law-dependent dynamics, including McKean-Vlasov stochastic differential equations and Boltzmann-type equations. We consider discrete-time approximation schemes with decreasing time steps and analyse the convergence of their associated ergodic (or occupation) measure towards the invariant distribution of the underlying continuous-time process. Under a general coupling assumption, we prove convergence in the expected $p$-Wasserstein distance ($p \ge 1$) and derive explicit convergence rates. Our approach combines estimates on ergodic averages, regularization techniques for discrete measures, and a generalized discrete Gronwall lemma to control the error between the self-interacting scheme and the target invariant measure. We show that our framework applies to a wide range of models: McKean-Vlasov SDEs, a Boltzmann type equation, and a neuronal model.

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arXiv2026

Emergence of Common Noise: Quantitative Conditional Propagation of Chaos

We study a dynamical invariance principle for interacting particle systems with mean-field interactions and common noise emerging through a collective stochastic perturbation. The particle dynamics combine autonomous evolution with a weakly scaled random bombardment whose cumulative effect generates a Brownian common noise in the large population limit. Working in a general abstract framework based on stochastic flows and the stochastic sewing lemma, we establish quantitative conditional propagation of chaos estimates for both discrete Euler schemes and their continuous-time flow limits. Our approach yields explicit Wasserstein convergence rates and applies in particular to jump-diffusion models motivated by interacting spiking neuron systems. The analysis relies on quantitative central limit theorems of Rio and Bonis together with a stochastic sewing argument adapted to $L^2$-valued flows.

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arXiv2025

Euler-type approximation for the invariant measure: An abstract framework

We establish a general framework to study the rate of convergence of a Euler type approximation scheme with decreasing time steps to the invariant measure, for a general class of stochastic systems. The error is measured in general Wasserstein distances, which enables to encompass cases with non global contractivity conditions. Our main assumption is a coupling property which is expressed in terms of the one-step approximation. We show that the proposed set-up can be applied to a wide range of equations that may be law dependent, such as Langevin equations, reflected equations, Boltzmann type equations and for a recent McKean Vlasov type model for neuronal activity.

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arXiv2024

Stochastic sewing lemma on Wasserstein space

The stochastic sewing lemma recently introduced by Le~(2020) allows to construct a unique limit process from a doubly indexed stochastic process that satisfies some regularity. This lemma is stated in a given probability space on which these processes are defined. The present paper develops a version of this lemma for probability measures: from a doubly indexed family of maps on the set of probability measures that have a suitable probabilistic representation, we are able to construct a limit flow of maps on the probability measures. This result complements and improves the existing result coming from the classical sewing lemma. It is applied to the case of law-dependent jump SDEs for which we obtain weak existence result as well as the uniqueness of the marginal laws.

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arXiv2023

Total variation distance between a jump-equation and its Gaussian approximation

We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the approximate random variable $X^\varepsilon\_t$ converges to the original random variable $X\_t$ in total variation distance and we estimate the error. We also give an estimate of the distance between the densities of the laws of the two random variables. These are done by using some integration by parts techniques in Malliavin calculus.

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arXiv2023

Approximation for the invariant measure with applications for jump processes (convergence in total variation distance)

In this paper, we establish an abstract framework for the approximation of the invariant probability measure for a Markov semigroup. Following Pag{è}s and Panloup [40] we use an Euler scheme with decreasing step (unadjusted Langevin algorithm). Under some contraction property with exponential rate and some regularization properties, we give an estimate of the error in total variation distance. This abstract framework covers the main results in [40] and [14]. As a specific application we study the convergence in total variation distance to the invariant measure for jump type equations. The main technical difficulty consists in proving the regularzation properties-this is done under an ellipticity condition, using Malliavin calculus for jump processes.

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arXiv2022

Total variation distance between a jump-equation and its Gaussian approximation

We deal with stochastic differential equations with jumps. In order to obtain an accurate approximation scheme, it is usual to replace the "small jumps" by a Brownian motion. In this paper, we prove that for every fixed time $t$, the approximate random variable $X^\varepsilon_t$ converges to the original random variable $X_t$ in total variation distance and we estimate the error. We also give an estimate of the distance between the densities of the laws of the two random variables. These are done by using some integration by parts techniques in Malliavin calculus.

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arXiv2022

Transfer of regularity for Markov semigroups

We study the regularity of a Markov semigroup $(P_t)_{t>0}$, that is, when $P_t(x,dy)=p_t(x,y)dy$ for a suitable smooth function $p_t(x,y)$. This is done by transferring the regularity from an approximating Markov semigroup sequence $(P^n_t)_{t>0}$, $n\in\mathbb{N}$, whose associated densities $p^n_t(x,y)$ are smooth and can blow up as $n\to\infty$. We use an interpolation type result and we show that if there exists a good equilibrium between the blow up and the speed of convergence, then $P_{t}(x,dy)=p_{t}(x,y)dy$ and $p_{t}$ has some regularity properties.

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arXiv2021

Construction of Boltzmann and McKean Vlasov type flows (the sewing lemma approach)

We are concerned with a mixture of Boltzmann and McKean-Vlasov type equations, this means (in probabilistic terms) equations with coefficients depending on the law of the solution itself,and driven by a Poisson point measure with the intensity depending also on the law of the solution. Both the analytical Boltzmann equation and the probabilistic interpretation initiated by Tanaka (1978) have intensively been discussed in the literature for specific models related to the behavior of gas molecules. In this paper, we consider general abstract coefficients that may include mean field effects and then we discuss the link with specific models as well. In contrast with the usual approach in which integral equations are used in order to state the problem, we employ here a new formulation of the problem in terms of flows of endomorphisms on the space of probability measure endowed with the Wasserstein distance. This point of view already appeared in the framework of rough differential equations. Our results concern existence and uniqueness of the solution, in the formulation of flows, but we also prove that the "flow solution" is a solution of the classical integral weak equation and admits a probabilistic interpretation. Moreover, we obtain stability results and regularity with respect to the time for such solutions. Finally we prove the convergence of empirical measures based on particle systems to the solution of our problem, and we obtain the rate of convergence. We discuss as examples the homogeneous and the inhomogeneous Boltzmann (Enskog) equation with hard potentials.

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arXiv2019

Regularization lemmas and convergence in total variation

We provide a simple abstract formalism of integration by parts under which we obtain some regularization lemmas. These lemmas apply to any sequence of random variables $(F_n)$ which are smooth and non-degenerated in some sense and enable one to upgrade the distance of convergence from smooth Wasserstein distances to total variation in a quantitative way. This is a well studied topic and one can consult for instance Bally and Caramellino [Electron. J. Probab. 2014], Bogachev, Kosov and Zelenov [Trans. Amer. Math. Soc. 2018], Hu, Lu and Nualart [J. Funct. Anal. 2014], Nourdin and Poly [Stoch. Proc. Appl. 2013] and the references therein for an overview of this issue. Each of the aforementioned references share the fact that some non-degeneracy is required along the whole sequence. We provide here the first result removing this costly assumption as we require only non-degeneracy at the limit. The price to pay is to control the smooth Wasserstein distance between the Malliavin matrix of the sequence and the Malliavin matrix of the limit which is particularly easy in the context of Gaussian limit as their Malliavin matrix is deterministic. We then recover, in a slightly weaker form, the main findings of Nourdin, Peccati and Swan [J. Funct. Anal. 2014]. Another application concerns the approximation of the semi-group of a diffusion process by the Euler scheme in a quantitative way and under the Hörmander condition.

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arXiv2019

A generic construction for high order approximation schemes of semigroups using random grids

Our aim is to construct high order approximation schemes for general semigroups of linear operators $P_{t},t\geq 0$. In order to do it, we fix a time horizon $T $ and the discretization steps $h_{l}=\frac{T}{n^{l}},l\in \mathbb{N}$ and we suppose that we have at hand some short time approximation operators $Q_{l}$ such that $P_{h_{l}}=Q_{l}+O(h_{l}^{1+\alpha })$ for some $\alpha >0$. Then, we consider random time grids $\Pi (\omega )=\{t_0(\omega )=0<t_{1}(\omega )<...<t_{m}(\omega )=T\}$ such that for all $1\le k\le m$, $t_{k}(\omega )-t_{k-1}(\omega )=h_{l_{k}}$ for some $l_{k}\in \mathbb{N}$, and we associate the approximation discrete semigroup $P_{T}^{\Pi (\omega )}=Q_{l_{n}}...Q_{l_{1}}.$ Our main result is the following: for any approximation order $\nu $, we can construct random grids $\Pi_{i}(\omega )$ and coefficients $c_{i}$, with $i=1,...,r$ such that \[ P_{t}f=\sum_{i=1}^{r}c_{i}\mathbb{E}(P_{t}^{\Pi _{i}(\omega )}f(x))+O(n^{-\nu}) \]% with the expectation concerning the random grids $\Pi _{i}(\omega ).$ Besides, $\text{Card}(\Pi _{i}(\omega ))=O(n)$ and the complexity of the algorithm is of order $n$, for any order of approximation $\nu$. The standard example concerns diffusion processes, using the Euler approximation for~$Q_l$. In this particular case and under suitable conditions, we are able to gather the terms in order to produce an estimator of $P_tf$ with finite variance. However, an important feature of our approach is its universality in the sense that it works for every general semigroup $P_{t}$ and approximations. Besides, approximation schemes sharing the same $\alpha$ lead to the same random grids $\Pi_{i}$ and coefficients $c_{i}$. Numerical illustrations are given for ordinary differential equations, piecewise deterministic Markov processes and diffusions.

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arXiv2018

Regularity and Stability for the Semigroup of Jump Diffusions with State-Dependent Intensity

We consider stochastic differential systems driven by a Brownian motion and a Poisson point measure where the intensity measure of jumps depends on the solution. This behavior is natural for several physical models (such as Boltzmann equation, piecewise deterministic Markov processes, etc). First, we give sufficient conditions guaranteeing that the semigroup associated with such an equation preserves regularity by mapping the space of the of k-times differentiable bounded functions into itself. Furthermore, we give an explicit estimate of the operator norm. This is the key-ingredient in a quantitative Trotter-Kato-type stability result: it allows us to give an explicit estimate of the distance between two semigroups associated with different sets of coefficients in terms of the difference between the corresponding infinitesimal operators. As an application, we present a method allowing to replace " small jumps " by a Brownian motion or by a drift component. The example of the 2D Boltzmann equation is also treated in all detail.

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arXiv2018

Upper bounds for the function solution of the homogenuous 2D Boltzmann equation with hard potential

We deal with $f\_{t}(dv),$ the solution of the homogeneous $2D$ Boltzmannequation without cutoff. The initial condition $f\_{0}(dv)$ may be anyprobability distribution (except a Dirac mass). However, for sufficiently hardpotentials, the semigroup has a regularization property (see \cite{[BF]}):$f\_{t}(dv)=f\_{t}(v)dv$ for every $t>0.$ The aim of this paper is to give upperbounds for $f\_{t}(v),$ the most significant one being of type $f\_{t}(v)\leqCt^{-η}e^{-\left\vert v\right\vert ^λ}$ for some $η,λ>0.$

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arXiv2017

Convergence in distribution norms in the CLT for non identical distributed random variables

We study the convergence in distribution norms in the Central Limit Theorem for non identical distributed random variables that is $$ \varepsilon_{n}(f):={\mathbb{E}}\Big(f\Big(\frac 1{\sqrt n}\sum_{i=1}^{n}Z_{i}\Big)\Big)-{\mathbb{E}}\big(f(G)\big)\rightarrow 0 $$ where $Z_{i}$ are centred independent random variables and $G$ is a Gaussian random variable. We also consider local developments (Edgeworth expansion). This kind of results is well understood in the case of smooth test functions $f$. If one deals with measurable and bounded test functions (convergence in total variation distance), a well known theorem due to Prohorov shows that some regularity condition for the law of the random variables $Z_{i}$, $i\in {\mathbb{N}}$, on hand is needed. Essentially, one needs that the law of $ Z_{i}$ is locally lower bounded by the Lebesgue measure (Doeblin's condition). This topic is also widely discussed in the literature. Our main contribution is to discuss convergence in distribution norms, that is to replace the test function $f$ by some derivative $\partial_{α}f$ and to obtain upper bounds for $\varepsilon_{n}(\partial_{α}f)$ in terms of the infinite norm of $f$. Some applications are also discussed: an invariance principle for the occupation time for random walks, small balls estimates and expected value of the number of roots of trigonometric polynomials with random coefficients.

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arXiv2017

Non universality for the variance of the number of real roots of random trigonometric polynomials

In this article, we consider the following family of random trigonometric polynomials $p_n(t,Y)=\sum_{k=1}^n Y_{k,1} \cos(kt)+Y_{k,2}\sin(kt)$ for a given sequence of i.i.d. random variables $\{Y_{k,1},Y_{k,2}\}_{k\ge 1}$ which are centered and standardized. We set $\mathcal{N}([0,π],Y)$ the number of real roots over $[0,π]$ and $\mathcal{N}([0,π],G)$ the corresponding quantity when the coefficients follow a standard Gaussian distribution. We prove under a Doeblin's condition on the distribution of the coefficients that $$ \lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,π],Y)\right)}{n} =\lim_{n\to\infty}\frac{\text{Var}\left(\mathcal{N}_n([0,π],G)\right)}{n} +\frac{1}{30}\left(\mathbb{E}(Y_{1,1}^4)-3\right). $$ The latter establishes that the behavior of the variance is not universal and depends on the distribution of the underlying coefficients through their kurtosis. Actually, a more general result is proven in this article, which does not require that the coefficients are identically distributed. The proof mixes a recent result regarding Edgeworth's expansions for distribution norms established in arXiv:1606.01629 with the celebrated Kac-Rice formula.

math.PR↗

arXiv2017

Total variation distance between stochastic polynomials and invariance principles

The goal of this paper is to estimate the total variation distance between two general stochastic polynomials. As a consequence one obtains an invariance principle for such polynomials. This generalizes known results concerning the total variation distance between two multiple stochastic integrals on one hand, and invariance principles in Kolmogorov distance for multi-linear stochastic polynomials on the other hand. As an application we first discuss the asymptotic behavior of U-statistics associated to polynomial kernels. Moreover we also give an example of CLT associated to quadratic forms.

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arXiv2016

Diffusions under a local strong Hörmander condition. Part II: tube estimates

We study lower and upper bounds for the probability that a diffusion process in $\mathbb{R}^n$ remains in a tube around a skeleton path up to a fixed time. We assume that the diffusion coefficients $σ_1,\ldots,σ_d$ may degenerate but they satisfy a strong Hörmander condition involving the first order Lie brackets around the skeleton of interest. The tube is written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $δ$, the diffusion process propagates with speed $\sqrtδ$ in the direction of the diffusion vector fields $σ_{j}$ and with speed $δ=\sqrtδ\times \sqrtδ$ in the direction of $[σ_{i},σ_{j}]$. The proof consists in a concatenation technique which strongly uses the lower and upper bounds for the density proved in the part I.

math.PR↗

arXiv2016

Diffusions under a local strong Hörmander condition. Part I: density estimates

We study lower and upper bounds for the density of a diffusion process in ${\mathbb{R}}^n$ in a small (but not asymptotic) time, say $δ$. We assume that the diffusion coefficients $σ_1,\ldots,σ_d$ may degenerate at the starting time $0$ and point $x_0$ but they satisfy a strong Hörmander condition involving the first order Lie brackets. The density estimates are written in terms of a norm which accounts for the non-isotropic structure of the problem: in a small time $δ$, the diffusion process propagates with speed $\sqrtδ$ in the direction of the diffusion vector fields $σ_{j}$ and with speed $δ=\sqrtδ\times \sqrtδ$ in the direction of $[σ_{i},σ_{j}]$. In the second part of this paper, such estimates will be used in order to study lower and upper bounds for the probability that the diffusion process remains in a tube around a skeleton path up to a fixed time.

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