arXiv subjects

Nicolas Privault

Publications and source records attributed to Nicolas Privault.

At least 19 recordsLinked to original sources

arXiv2026

Probabilistic representation and classical solutions of wave equations with complex polynomial nonlinearities

We review the probabilistic representation of solutions of wave equations with polynomial nonlinearities in spatial dimensions d=1,2,3 using stochastic branching processes. Under regularity assumptions on the initial data, we derive conditions ensuring the integrability of the corresponding Monte Carlo estimator, and the existence and smoothness of mild and classical solutions. We also present numerical results and comparisons with grid-based algorithms for the solution of nonlinear wave equations.

math.NA↗

arXiv2026

Generalization error bounds for two-layer neural networks with Lipschitz loss function

We derive generalization error bounds for the training of two-layer neural networks without assuming boundedness of the loss function, using Wasserstein distance estimates on the discrepancy between a probability distribution and its associated empirical measure, together with moment bounds for the associated stochastic gradient method. In the case of independent test data, we obtain a dimension-free rate of order $O(n^{-1/2} )$ on the $n$-sample generalization error, whereas without independence assumption, we derive a bound of order $O(n^{-1 / ( d_{\rm in}+d_{\rm out} )} )$, where $d_{\rm in}$, $d_{\rm out}$ denote input and output dimensions. Our bounds and their coefficients can be explicitly computed prior to the training of the model, and are confirmed by numerical simulations.

stat.ML↗

arXiv2025

Mixing rates for linear operators under infinitely divisible measures on Banach spaces

We derive rates of convergence for the mixing of operators under infinitely divisible measures in the framework of linear dynamics on Banach spaces. Our approach is based on the characterization of mixing in terms of codifference functionals and control measures, and extends previous results obtained in the Gaussian setting via the use of covariance operators. Explicit mixing rates are obtained for weighted shifts under compound Poisson, {\alpha}-stable, and tempered {\alpha}-stable measures.

math.PR↗

arXiv2025

Gaussian fluctuations of generalized $U$-statistics and subgraph counting in the binomial random-connection model

We derive normal approximation bounds for generalized $U$-statistics of the form \begin{equation*} S_{n,k}(f):=\sum_{ 1 \leq \beta (1),\dots,\beta (k) \leq n \atop \beta (i)\ne\beta (j), \ 1\leq i\ne j \leq k} f\big(X_{\beta (1)},\dots,X_{\beta (k)},Y_{\beta (1),\beta (2)},\dots,Y_{\beta (k-1),\beta (k)}\big), \end{equation*} where $\{X_i\}_{i=1}^n$ and $\{Y_{i,j}\}_{1\le i<j\le n}$ are independent sequences of i.i.d. random variables. Our approach relies on moment identities and cumulant bounds that are derived using partition diagram arguments. Normal approximation bounds in the Kolmogorov distance and moderate deviation results are then obtained by the cumulant method. Those results are applied to subgraph counting in the binomial random-connection model, which is a generalization of the Erd\H{o}s-R\'enyi model.

math.PR↗

arXiv2025

Stability analysis of a branching diffusion solver for semilinear heat equations

Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In this paper, we study the stability of a functional branching representation of PDE solutions by deriving sufficient criteria for the integrability of the multiplicative weighted progeny of stochastic branching processes. We also prove the uniqueness of mild solutions under uniform integrability assumptions on random functionals.

math.PR↗

arXiv2025

Probabilistic representation of ODE solutions with quantitative estimates

This paper considers the probabilistic representation of the solutions of ordinary differential equations (ODEs) by the generation of marked random trees in which marks can be interpreted as mutant types in population genetics models. We present sufficient conditions on equation coefficients that ensure the integrability and uniform integrability of the functionals of random trees used in this representation. Those conditions rely on the analysis of a marked branching process that controls the growth of random trees and provide implicit lower bounds on the explosion time of the underlying ODE, thus providing a connection between branching process explosion and the existence and uniqueness of ODE solutions.

math.PR↗

arXiv2025

Binary Galton-Watson trees with mutations

We consider a multitype Galton-Watson process that allows for the mutation and reversion of individual types in discrete and continuous time. In this setting, we explicitly compute the time evolution of quantities such as the mean and distributions of different types. This allows us in particular to estimate the proportions of different types in the long run, as well as the distribution of the first time of occurrence of a given type as the tree size or time increases. Our approach relies on the recursive computation of the joint distribution of types conditionally to the value of the total progeny. In comparison with the literature on related multitype models, we do not rely on approximations.

q-bio.PE↗

arXiv2024

Normal to Poisson phase transition for subgraph counting in the random-connection model

We consider the limiting behavior of the count of subgraphs isomorphic to a graph $G$ with $m\geq 0$ fixed endpoints (or roots) in the random-connection model, as the intensity $\lambda$ of the underlying Poisson point process tends to infinity. When connection probabilities are of order $\lambda^{-\alpha}$ we identify a phase transition phenomenon depending on a critical decay rate $\alpha^\ast_m (G)>0$ such that normal approximation for subgraph counts holds when $\alpha \in (0,\alpha^\ast_m (G) )$, and a Poisson limit result holds if $\alpha = \alpha^\ast_m (G)$. Our approach relies on cumulant growth rates derived by the convex analysis of planar diagrams that enumerate the partitions involved in cumulant identities. As a result, by the cumulant method we obtain normal approximation results with convergence rates in the Kolmogorov distance, and a Poisson limit theorem, for subgraph counts.

math.PR↗

arXiv2024

On the random generation of Butcher trees

The main goal of this paper is to provide an algorithm for the random sampling of Butcher trees and the probabilistic numerical solution of ordinary differential equations (ODEs). This approach complements and simplifies a recent approach to the probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees is compared to the finite order truncation of Butcher series in numerical experiments.

math.CA↗

arXiv2023

Graph connectivity with fixed endpoints in the random-connection model

We consider the count of subgraphs with an arbitrary configuration of endpoints in the random-connection model based on a Poisson point process on ${\Bbb R}^d$. We present combinatorial expressions for the computation of the cumulants and moments of all orders of such subgraph counts, which allow us to estimate the growth of cumulants as the intensity of the underlying Poisson point process goes to infinity. As a consequence, we obtain a central limit theorem with explicit convergence rates under the Kolmogorov distance, and connectivity bounds. Numerical examples are presented using a computer code in SageMath for the closed-form computation of cumulants of any order, for any type of connected subgraph and for any configuration of endpoints in any dimension $d\geq 1$. In particular, graph connectivity estimates, Gram-Charlier expansions for density estimation, and correlation estimates for joint subgraph counting are obtained.

math.PR↗

arXiv2023

Semilinear fractional elliptic PDEs with gradient nonlinearities on open balls: existence of solutions and probabilistic representation

We provide sufficient conditions for the existence of viscosity solutions of fractional semilinear elliptic PDEs of index $\alpha \in (1,2)$ with polynomial gradient nonlinearities on $d$-dimensional balls, $d\geq 2$. Our approach uses a tree-based probabilistic representation of solutions and their partial derivatives using $\alpha$-stable branching processes, and allows us to take into account gradient nonlinearities not covered by deterministic finite difference methods so far. In comparison with the existing literature on the regularity of solutions, no polynomial order condition is imposed on gradient nonlinearities. Numerical illustrations demonstrate the accuracy of the method in dimension $d=10$, solving a challenge encountered with the use of deterministic finite difference methods in high-dimensional settings.

math.NA↗

arXiv2023

Normal approximation of subgraph counts in the random-connection model

This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Poisson point process. By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by M\"obius inversion. Using the Statulevi\v{c}ius condition, we deduce convergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity. Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.

math.PR↗

arXiv2022

Numerical solution of the incompressible Navier-Stokes equation by a deep branching algorithm

We present an algorithm for the numerical solution of systems of fully nonlinear PDEs using stochastic coded branching trees. This approach covers functional nonlinearities involving gradient terms of arbitrary orders, and it requires only a boundary condition over space at a given terminal time $T$ instead of Dirichlet or Neumann boundary conditions at all times as in standard solvers. Its implementation relies on Monte Carlo estimation, and uses neural networks that perform a meshfree functional estimation on a space-time domain. The algorithm is applied to the numerical solution of the Navier-Stokes equation and is benchmarked to other implementations in the cases of the Taylor-Green vortex and Arnold-Beltrami-Childress flow.

math.NA↗

arXiv2022

Wasserstein distance estimates for jump-diffusion processes

We derive Wasserstein distance bounds between the probability distributions of a stochastic integral (It\^o) process with jumps $(X_t)_{t\in [0,T]}$ and a jump-diffusion process $(X^\ast_t)_{t\in [0,T]}$. Our bounds are expressed using the stochastic characteristics of $(X_t)_{t\in [0,T]}$ and the jump-diffusion coefficients of $(X^\ast_t)_{t\in [0,T]}$ evaluated in $X_t$, and apply in particular to the case of different jump characteristics. Our approach uses stochastic calculus arguments and $L^p$ integrability results for the flow of stochastic differential equations with jumps, without relying on the Stein equation.

math.PR↗

arXiv2022

Closed-form modeling of neuronal spike train statistics using multivariate Hawkes cumulants

We derive exact analytical expressions for the cumulants of any orders of neuronal membrane potentials driven by spike trains in a multivariate Hawkes process model with excitation and inhibition. Such expressions can be used for the prediction and sensitivity analysis of the statistical behavior of the model over time, and to estimate the probability densities of neuronal membrane potentials using Gram-Charlier expansions. Our results are shown to provide a better alternative to Monte Carlo estimates via stochastic simulations, and computer codes based on combinatorial recursions are included.

q-bio.NC↗

arXiv2022

A deep learning approach to the probabilistic numerical solution of path-dependent partial differential equations

Recent work on Path-Dependent Partial Differential Equations (PPDEs) has shown that PPDE solutions can be approximated by a probabilistic representation, implemented in the literature by the estimation of conditional expectations using regression. However, a limitation of this approach is to require the selection of a basis in a function space. In this paper, we overcome this limitation by the use of deep learning methods, and we show that this setting allows for the derivation of error bounds on the approximation of conditional expectations. Numerical examples based on a two-person zero-sum game, as well as on Asian and barrier option pricing, are presented. In comparison with other deep learning approaches, our algorithm appears to be more accurate, especially in large dimensions.

cs.LG↗

arXiv2022

Normal approximation of compound Hawkes functionals

We derive quantitative bounds in the Wasserstein distance for the approximation of stochastic integrals with respect to Hawkes processes by a normally distributed random variable. In the case of deterministic and non-negative integrands, our estimates involve only the third moment of integrand in addition to a variance term using a square norm of the integrand. As a consequence, we are able to observe a "third moment phenomenon" in which the vanishing of the first cumulant can lead to faster convergence rates. Our results are also applied to compound Hawkes processes, and improve on the current literature where estimates may not converge to zero in large time, or have been obtained only for specific kernels such as the exponential or Erlang kernels.

math.PR↗

arXiv2022

Asymptotic analysis of k-hop connectivity in the 1D unit disk random graph model

We propose an algorithm for the closed-form recursive computation of joint moments and cumulants of all orders for k-hop counts in the 1D unit disk random graph model with Poisson distributed vertices. Our approach uses decompositions of k-hop counts into multiple Poisson stochastic integrals. As a consequence, using the Stein method we derive Berry-Esseen bounds for the asymptotic convergence of renormalized k-hop path counts to the normal distribution as the density of Poisson vertices tends to infinity. Computer codes for the recursive symbolic computation of moments and cumulants are provided in appendix.

math.PR↗