Article
Mathematics
Renming Song, Longjie Xie
Summary: This paper investigates a time-dependent stable-like operator with drift and establishes the well-posedness for the martingale problem associated with it using the Littlewood-Paley theory. It also studies a class of stochastic differential equations driven by generators of the form L-t and proves the pathwise uniqueness of strong solutions for coefficients in certain Besov spaces.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics
S. Menozzi, A. Pesce, X. Zhang
Summary: This study deals with non degenerate Brownian stochastic differential equations with Holder continuous spatial diffusion coefficients and unbounded drift with linear growth. Two-sided bounds for density and pointwise controls of derivatives up to order two are derived under additional spatial Wilder continuity assumptions on the drift. The estimates show the transport of initial conditions by unbounded drift through an auxiliary, possibly regularized, flow.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Charles-Edouard Brehier
Summary: In this paper, we design numerical schemes for a class of slow-fast systems of stochastic differential equations. The proposed scheme exhibits an asymptotic preserving property, where a limiting scheme consistent with the averaged equation is obtained as the time-scale parameter approaches 0. The authors also illustrate the recently proved averaging result for the considered SDE systems and highlight the main differences compared to the standard Wiener case.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2022)
Article
Mathematics, Applied
P. E. Chaudru de Raynal, S. Menozzi, A. Pesce, X. Zhang
Summary: We establish heat kernel and gradient estimates for the density of kinetic degenerate Kolmogorov stochastic differential equations. Our results are established under minimal assumptions that guarantee the SDE is weakly well posed.
BULLETIN DES SCIENCES MATHEMATIQUES
(2023)
Article
Multidisciplinary Sciences
Arnald Puy, Pierfrancesco Beneventano, Simon A. Levin, Samuele Lo Piano, Tommaso Portaluri, Andrea Saltelli
Summary: Mathematical models are becoming more detailed for better predictions and insights, even without validation or training data. However, this practice can lead to fuzzier estimates as it increases the effective dimensions of the model.
Article
Mathematics, Applied
Jian Wang, Bing Yao Wu
Summary: By utilizing the coupling method and localization technique, non-uniform gradient estimates were established for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Levy noises, with coefficients satisfying local monotonicity conditions.
ACTA MATHEMATICA SINICA-ENGLISH SERIES
(2021)
Article
Statistics & Probability
Lucio Galeati, Chengcheng Ling
Summary: In this paper, we prove several stability estimates for comparing solutions driven by different (b(i), sigma(i)) in both Ito and Stratonovich SDEs. These estimates may depend on negative Sobolev norms of the difference b(1) - b(2). We also discuss several applications of these results to McKean-Vlasov SDEs, criteria for strong compactness of solutions, and Wong-Zakai type theorems.
ELECTRONIC JOURNAL OF PROBABILITY
(2023)
Article
Statistics & Probability
Lucio Galeati, Fabian A. Harang, Avi Mayorcas
Summary: This article studies distribution dependent stochastic differential equation driven by a continuous process, without any specification on its law, following the approach initiated in [17]. Several criteria for existence and uniqueness of solutions beyond the classical globally Lipschitz setting are provided. In particular, well-posedness of the equation and almost sure convergence of the associated particle system are shown for drifts satisfying either Osgood-continuity, monotonicity, local Lipschitz or Sobolev differentiability type assumptions.
ELECTRONIC JOURNAL OF PROBABILITY
(2022)
Article
Mathematics
Mathis Fitoussi
Summary: This paper investigates a formal stochastic differential equation with a time-inhomogeneous Besov drift and a symmetric alpha-stable process. We prove that the martingale solution associated with this equation has a special density that satisfies heat kernel bounds and gradient estimates.
POTENTIAL ANALYSIS
(2023)
Article
Mathematics, Applied
Wei Mao, Bo Chen, Surong You
Summary: This paper develops the averaging principle for stochastic differential equations driven by G-Brownian motion with non-Lipschitz coefficients. By proving the convergence properties of the solutions, it demonstrates that the solution of the averaged G-SDEs converges to that of the standard one, and provides two examples for illustration.
ADVANCES IN DIFFERENCE EQUATIONS
(2021)
Article
Mathematics
Mariusz Mirek, Tomasz Z. Szarek, Blazej Wrobel
Summary: We prove that the discrete spherical maximal functions corresponding to the Euclidean spheres in Z(d) with dyadic radii have bounded norms in l(p)(Z(d)) for all p in [2, infinity], independent of the dimensions d >= 5. The asymptotic formula in the Waring problem for the squares with a dimension-free multiplicative error term plays a crucial role in our argument. By introducing new approximating multipliers, we demonstrate how to absorb exponential growth in norms arising from the sampling principle and obtain dimension-free estimates for the discrete spherical maximal functions.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
Article
Statistics & Probability
Xiaocui Ma, Haitao Yue, Fubao Xi
Summary: This work investigates the averaging method for doubly perturbed distribution dependent SDEs and establishes an approximation theorem. By using the fixed point theorem, the well-posedness of doubly perturbed distribution dependent SDEs is proven. Furthermore, it is shown that the solutions of the original equations converge to those of the averaged equations in terms of mean square and probability. An example is also provided to demonstrate the practical applications of the results.
STATISTICS & PROBABILITY LETTERS
(2022)
Article
Statistics & Probability
Zachary William Bezemek, Konstantinos Spiliopoulos
Summary: In this paper, a fully-coupled slow-fast system of McKean-Vlasov stochastic differential equations with full dependence on various components is considered and convergence rates to its homogenized limit are derived. Periodicity assumptions are not made, but conditions on the fast motion are imposed to ensure ergodicity. The proof also yields related ergodic theorems and results on the regularity of Poisson type equations and the associated Cauchy problem on the Wasserstein space which are of independent interest.
STOCHASTICS AND DYNAMICS
(2023)
Article
Mathematics, Applied
Goncalo dos Reis, Greig Smith, Peter Tankov
Summary: This paper presents Monte-Carlo methods for evaluating expectations of functionals of solutions to McKean-Vlasov Stochastic Differential Equations (MV-SDE). Two importance sampling algorithms are proposed to reduce the variance of the Monte-Carlo estimator. The complete measure change algorithm and the decoupling algorithm both show significant reduction in variance compared to the standard Monte Carlo approximation. The methodological approach uses large deviations and Pontryagin principle to solve the variance minimization problem.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics
D. Crisan, P. Dobson, M. Ottobre
Summary: This research introduces a criterion for uniformly in time convergence of weak error in Euler scheme for stochastic differential equations, emphasizing the importance of exponential decay and bounds on moments. Lyapunov-type conditions are neither sufficient nor necessary for the convergence of weak error in the Euler approximation. Further study is needed to understand conditions for the validity of (i) in order to achieve uniform convergence of weak errors.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2021)
Article
Mathematics, Applied
Stefano Marchesani, Stefano Olla, Lu Xu
Summary: The study focuses on the quasi-static limit of scalar one-dimensional hyperbolic equations with strictly concave or convex flux and time dependent boundary conditions. It shows that the quasi-stationary profile evolves with the quasi-static equation, and its entropy solution is determined by the stationary profile corresponding to the boundary data at a given time.
NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
(2021)
Article
Mathematics, Applied
Tony Lelievre, Dorian Le Peutrec, Boris Nectoux
Summary: This work examines the exit point distribution from a bounded domain of a stochastic process, taking into account the influence of initial conditions on the distribution. The proofs rely on analytical results on the dependency of the exit point distribution on the initial condition, as well as large deviation techniques and results on the genericity of Morse functions.
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2022)
Article
Statistics & Probability
Benjamin Jourdain, Tony Lelievre, Pierre-Andre Zitt
Summary: By drawing a parallel between metadynamics and self interacting models for polymers, the study focuses on the longtime convergence of the original metadynamics algorithm in the adiabatic setting, and discusses the bias introduced when the adiabatic assumption does not hold.
ANNALS OF APPLIED PROBABILITY
(2021)
Article
Mathematics, Applied
Tony Lelievre, Lise Maurin, Pierre Monmarche
Summary: The study proposes an investigation into the robustness of the Adaptive Biasing Force method under generic (possibly non-conservative) forces. The researchers ensure the satisfaction of the flat histogram property and establish the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms. Using classical entropy techniques, the study proves the exponential convergence of the biasing force and law over time for both methods.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2022)
Article
Chemistry, Physical
Zineb Belkacemi, Paraskevi Gkeka, Tony Lelievre, Gabriel Stoltz
Summary: Free energy biasing methods are powerful in accelerating molecular conformational changes simulation, but they usually require prior knowledge of collective variables. Machine learning and dimensionality reduction algorithms can be used to identify these collective variables. A new iterative method involving autoencoders, FEBILAE, is introduced in this paper to ensure optimization of the same loss at each iteration and achieve collective variable convergence.
JOURNAL OF CHEMICAL THEORY AND COMPUTATION
(2022)
Article
Mathematics, Applied
Frederic Legoll, Tony Lelievre, Upanshu Sharma
Summary: The aim of this article is to design parareal algorithms for thermostated molecular dynamics simulations. The traditional parareal algorithm is not suitable for molecular dynamics due to its limitations. This article proposes a modified version of the parareal algorithm that is better suited for molecular dynamics simulations. However, the modified algorithm still has some limitations, including intermediate trajectory blow-up, encounters with undefined values, and no computational advantage in long time horizons. Through numerical experiments, this article demonstrates that the adaptive algorithm overcomes the limitations of the standard algorithm and achieves significant improvements.
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2022)
Article
Statistics & Probability
Tony Lelievre, Mouad Ramil, Julien Reygner
Summary: This study investigates the properties of the Langevin process on a bounded-in-position domain, proving compactness of its semigroup and the existence of a unique quasi-stationary distribution. A spectral interpretation of the QSD is provided, along with exponential convergence of the process towards the QSD under non-absorption conditions. An explicit formula for the first exit point distribution from the domain, starting from the QSD, is given.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2022)
Article
Physics, Multidisciplinary
Tomasz Komorowski, Stefano Olla
Summary: In this paper, an infinite chain of harmonic oscillators coupled with a Poisson thermostat is investigated. The energy density of the chain, described by the Wigner distribution, satisfies a transport equation outside the location of the thermostat. A boundary condition arises at this site, which explains the reflection-transmission-scattering of the wave energy influenced by the thermostat. The coefficients for these processes are obtained. Unlike the Langevin thermostat case studied in Komorowski et al. (Arch. Ration. Mech. Anal. 237, 497-543, 2020), the Poissonian thermostat scattering generates a continuous cloud of waves with frequencies different from the incident wave in the limit.
ANNALES HENRI POINCARE
(2022)
Article
Mathematics, Applied
Tony Lelievre, Mouad Ramil, Julien Reygner
Summary: This article focuses on classical solutions to the kinetic Fokker-Planck equation within a bounded domain O in position, utilizing the Langevin diffusion process with absorbing boundary conditions to obtain probabilistic representations of the solutions. Important results such as the Harnack inequality, maximum principle, and the smooth transition density for the absorbed Langevin process are provided on the phase-space cylindrical domain D = O x R-d. The study also examines the continuity and positivity of the transition density at the boundary of D.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Statistics & Probability
Anna De Masi, Stefano Marchesani, Stefano Olla, Lu Xu
Summary: This study investigates the one-dimensional asymmetric simple exclusion process with boundary creation/annihilation effects. By controlling the entropy-entropy flux pairs and utilizing coupling arguments, we prove that the system evolves quasi-statically at a specific time scale, with a macroscopic density profile determined by the entropy solution of the stationary Burgers equation, which is influenced by the microscopic boundary rates that change with time.
PROBABILITY THEORY AND RELATED FIELDS
(2022)
Article
Mathematics, Applied
Amirali Hannani, Stefano Olla
Summary: We present a stochastic perturbation of the discrete nonlinear Schrodinger equation that conserves mass and models the effect of a heat bath at a specific temperature. We prove that the corresponding canonical Gibbs distribution is the only invariant measure. In the one-dimensional cubic focusing case, we demonstrate that as time approaches infinity, with continuous approximation and low temperature, the solution converges to the steady wave of the continuous equation that minimizes energy for a given mass.
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2023)
Article
Statistics & Probability
Manon Baudel, Arnaud Guyader, Tony Lelievre
Summary: We show how the Hill relation and the concept of quasi-stationary distribution can be applied to analyze biasing errors in various numerical procedures used in molecular dynamics to compute mean reaction times between metastable states for Markov processes. Theoretical findings are demonstrated on different examples, highlighting the precision of biasing error analysis and the applicability of our study to elliptic diffusions.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2023)
Article
Physics, Mathematical
Tomasz Komorowski, Joel L. Lebowitz, Stefano Olla
Summary: We study a harmonic chain in contact with a thermal bath, subjected to a periodic force at one end while undergoing random velocity reversals. This leads to a limited heat conductivity and the system approaches a time periodic state. We compute the heat current, which represents the time averaged work done on the system, and find that it approaches a finite positive value as the chain's length increases. By rescaling space and adjusting the strength and/or period of the force, we observe a macroscopic temperature profile consistent with the stationary solution of a continuum heat equation with Dirichlet-Neumann boundary conditions.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Physics, Mathematical
Tomasz Komorowski, Joel L. Lebowitz, Stefano Olla
Summary: This article investigates the macroscopic heat equation for a pinned harmonic chain under periodic force and in contact with a heat bath. The microscopic dynamics involve a Hamiltonian equation of motion and a velocity reversal for each particle at exponential times, resulting in a finite heat conductivity. By computing the current and local temperature, we derive the heat equation for the macroscopic temperature profile and solve it with specified initial conditions and fixed heat flux.
JOURNAL OF STATISTICAL PHYSICS
(2023)
Article
Materials Science, Multidisciplinary
Mouad Ramil, Tony Lelievre, Julien Reygner
Summary: Molecular dynamics methods are used to study the time evolution of complex molecular systems and their transitions between stable states. The Parallel Replica algorithm is a powerful tool to efficiently sample rare events in these systems. This research letter establishes the existence of a quasi-stationary distribution for the Langevin dynamics involved in the Parallel Replica algorithm and provides insight into the overdamped limit behavior of the dynamics.
MRS COMMUNICATIONS
(2022)
Article
Statistics & Probability
Neda Mohammadi, Leonardo Santoro, Victor M. Panaretos
Summary: This study considers the nonparametric estimation of the drift and diffusion coefficients in a Stochastic Differential Equation (SDE) using functional data analysis methods. The proposed estimators relate local parameters to global parameters through a novel Partial Differential Equation (PDE) and do not require any specific functional form assumptions. The study establishes almost sure uniform asymptotic convergence rates for the estimators, taking into account the impact of different sampling frequencies.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Masafumi Hayashi, Atsushi Takeuchi, Makoto Yamazato
Summary: This article considers subordinators whose Lévy measures are represented as Laplace transforms of measures on (0,infinity), and refers to them as CME-subordinators. The study shows that the transition probabilities of such processes without drifts are absolutely continuous on (0,infinity) with respect to Lebesgue measure. It is also demonstrated that the densities are bounded in space-time and tend to zero as time goes to infinity, with the speed of decrease being closely related to the behavior near the origin of the corresponding Lévy density.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Huiping Chen, Yong Chen, Yong Liu
Summary: This paper characterizes the relation between the real and complex Wiener-Ito integrals, providing explicit expressions for the kernels of their real and imaginary parts, and obtaining a representation formula for a two-dimensional real Wiener-Ito integral through a finite sum of complex Wiener-Ito integrals. The main tools used are a recursion technique and Malliavin derivative operators. As an application, the regularity of the stationary solution of the stochastic heat equation with dispersion is investigated.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Clement Foucart, Matija Vidmar
Summary: This study introduces a class of one-dimensional positive Markov processes that generalize continuous-state branching processes by incorporating random collisions. The study establishes that these processes, known as CB processes with collisions (CBCs), are the only Feller processes without negative jumps that satisfy a Laplace duality relationship with one-dimensional diffusions. The study also explores the relationship between CBCs and CB processes with spectrally positive migration, and provides necessary and sufficient conditions for attracting boundaries and the existence of a limiting distribution.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Jukka Lempa, Ernesto Mordecki, Paavo Salminen
Summary: This paper investigates the characteristics and properties of diffusion spiders and calculates the density of the resolvent kernel. The study of excessive functions leads to the expression of the representing measure for a given excessive function. These results are then applied to solving optimal stopping problems for diffusion spiders.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Mirko D'Ovidio, Francesco Iafrate
Summary: This article explores the connection between elastic drifted Brownian motions and inverses to tempered subordinators, and establishes a link between multiplicative functionals and dynamical boundary conditions. By representing functionals of the drifted Brownian motion as the inverse of a tempered subordinator, the problem is simplified.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
M. Heldman, S. A. Isaacson, J. Ma, K. Spiliopoulos
Summary: This study derives and proves the large-population mean-field limit for particle-based stochastic reaction-diffusion models, and provides the next order fluctuation corrections. Numerical examples demonstrate the importance of fluctuation corrections for accurate estimation of higher order statistics in the underlying model.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
Article
Statistics & Probability
Renan Gross
Summary: This study focuses on the problem of scenery reconstruction on d-dimensional torus. The researchers proved that the criterion on Fourier coefficients for discrete cycles, discovered by Matzinger and Lember in 2006, also applies in continuous spaces. It is shown that with the right drift, Brownian motion can be used to reconstruct any scenery. The injectivity property of an infinite Vandermonde matrix is also proven.
STOCHASTIC PROCESSES AND THEIR APPLICATIONS
(2024)
