arXiv subjects

Upanshu Sharma

Publications and source records attributed to Upanshu Sharma.

14 recordsLinked to original sources

arXiv2026

Realisation of constraints in underdamped Langevin dynamics

This article deals with the realisation of constraints in underdamped Langevin dynamics via soft-constrained dynamics. Specifically, we study systems with a large (or small) parameter that controls the constraint mechanisms, e.g. the strength of confinement forces, mass or friction coefficients, and we derive quantitative convergence results for both the constrained variables and the softly constrained dynamics on the limiting subspace. The latter can be either a spatial or a momentum or velocity subspace, depending on the underlying soft constraint mechanism; in this paper we treat only holonomic constraints, i.e. all momentum- or velocity-level constraints are integrable. We explicitly include the initial conditions so that it is clear whether they must satisfy the constraint or not in order to realise the desired constrained dynamics. We discuss the implications of these results as well as questions related to the sampling of the corresponding conditional probability measures.

math.PR↗

arXiv2026

Non-equilibrium functional inequalities for finite Markov chains

Functional inequalities such as the Poincaré and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications, including multiscale and non-reversible dynamics, require analysing probability measures that are not at equilibrium, where the classical theory tied to steady states no longer applies. We introduce generalised versions of these inequalities for arbitrary positive measures on a finite state space, retaining key structural properties of their classical counterparts. In particular, we prove continuity of the associated constants with respect to the reference measure and establish explicit positive lower bounds. As an application, we derive quantitative coarse-graining error estimates for non-reversible Markov chains, both with and without explicit scale separation, and propose a quantitative criterion for assessing the quality of coarse-graining maps.

math.PR↗

arXiv2025

Affine constraints in non-reversible diffusions with degenerate noise

This paper deals with the realisation of affine constraints on nonreversible stochastic differential equations (SDE) by strong confining forces. We prove that the confined dynamics converges pathwise and on bounded time intervals to the solution of a projected SDE in the limit of infinitely strong confinement, where the projection is explicitly given and depends on the choice of the confinement force. We present results for linear Ornstein-Uhlenbeck (OU) processes, but they straightforwardly generalise to nonlinear SDEs. Moreover, for linear OU processes that admit a unique invariant measure, we discuss conditions under which the limit also preserves the long-term properties of the SDE. More precisely, we discuss choices for the design of the confinement force which in the limit yield a projected dynamics with invariant measure that agrees with the conditional invariant measure of the unconstrained processes for the given constraint. The theoretical findings are illustrated with suitable numerical examples.

math.PR↗

arXiv2023

Variational structures beyond gradient flows: a macroscopic fluctuation-theory perspective

Macroscopic equations arising out of stochastic particle systems in detailed balance (called dissipative systems or gradient flows) have a natural variational structure, which can be derived from the large-deviation rate functional for the density of the particle system. While large deviations can be studied in considerable generality, these variational structures are often restricted to systems in detailed balance. Using insights from macroscopic fluctuation theory, in this work we aim to generalise this variational connection beyond dissipative systems by augmenting densities with fluxes, which encode non-dissipative effects. Our main contribution is an abstract framework, which for a given flux-density cost and a quasipotential, provides a decomposition into dissipative and non-dissipative components and a generalised orthogonality relation between them. We then apply this abstract theory to various stochastic particle systems -- independent copies of jump processes, zero-range processes, chemical-reaction networks in complex balance and lattice-gas models.

math-ph↗

arXiv2023

Untangling Dissipative and Hamiltonian effects in bulk and boundary driven systems

Using the theory of large deviations, macroscopic fluctuation theory provides a framework to understand the behaviour of non-equilibrium dynamics and steady states in diffusive systems. We extend this framework to a minimal model of non-equilibrium non-diffusive system, specifically an open linear network on a finite graph. We explicitly calculate the dissipative bulk and boundary forces that drive the system towards the steady state, and non-dissipative bulk and boundary forces that drives the system in orbits around the steady state. Using the fact that these forces are orthogonal in a certain sense, we provide a decomposition of the large-deviation cost into dissipative and non-dissipative terms. We establish that the purely non-dissipative force turns the dynamics into a Hamiltonian system. These theoretical findings are illustrated by numerical examples.

cond-mat.stat-mech↗

arXiv2023

Quantitative coarse-graining of Markov chains

Coarse-graining techniques play a central role in reducing the complexity of stochastic models, and are typically characterised by a mapping which projects the full state of the system onto a smaller set of variables which captures the essential features of the system. Starting with a continuous-time Markov chain, in this work we propose and analyse an effective dynamics, which approximates the dynamical information in the coarse-grained chain. Without assuming explicit scale-separation, we provide sufficient conditions under which this effective dynamics stays close to the original system and provide quantitative bounds on the approximation error. We also compare the effective dynamics and corresponding error bounds to the averaging literature on Markov chains which involve explicit scale-separation. We demonstrate our findings on an illustrative test example.

math.PR↗

arXiv2022

Fourier-Cattaneo equation: stochastic origin, variational formulation, and asymptotic limits

We introduce a variational structure for the Fourier-Cattaneo (FC) system which is a second-order hyperbolic system. This variational structure is inspired by the large-deviation rate functional for the Kac process which is closely linked to the FC system. Using this variational formulation we introduce appropriate solution concepts for the FC equation and prove an a priori estimate which connects this variational structure to an appropriate Lyapunov function and Fisher information, the so-called FIR inequality. Finally, we use this formulation and estimate to study the diffusive and hyperbolic limits for the FC system.

math.AP↗

arXiv2021

Non-reversible sampling schemes on submanifolds

Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of an ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.

math.NA↗

arXiv2021

Stochastic gradient descent and fast relaxation to thermodynamic equilibrium: a stochastic control approach

We study the convergence to equilibrium of an underdamped Langevin equation that is controlled by a linear feedback force. Specifically, we are interested in sampling the possibly multimodal invariant probability distribution of a Langevin system at small noise (or low temperature), for which the dynamics can easily get trapped inside metastable subsets of the phase space. We follow [Chen et al., J. Math. Phys. 56, 113302, 2015] and consider a Langevin equation that is simulated at a high temperature, with the control playing the role of a friction that balances the additional noise so as to restore the original invariant measure at a lower temperature. We discuss different limits as the temperature ratio goes to infinity and prove convergence to a limit dynamics. It turns out that, depending on whether the lower ("target") or the higher ("simulation") temperature is fixed, the controlled dynamics converges either to the overdamped Langevin equation or to a deterministic gradient flow. This implies that (a) the ergodic limit and the large temperature separation limit do not commute in general, and that (b) it is not possible to accelerate the speed of convergence to the ergodic limit by making the temperature separation larger and larger. We discuss the implications of these observation from the perspective of stochastic optimisation algorithms and enhanced sampling schemes in molecular dynamics.

math.OC↗

arXiv2020

Coarse-graining of non-reversible stochastic differential equations: quantitative results and connections to averaging

This work is concerned with model reduction of stochastic differential equations and builds on the idea of replacing drift and noise coefficients of preselected relevant, e.g. slow variables by their conditional expectations. We extend recent results by Legoll & Lelièvre [Nonlinearity 23, 2131, 2010] and Duong et al. [Nonlinearity 31, 4517, 2018] on effective reversible dynamics by conditional expectations to the setting of general non-reversible processes with non-constant diffusion coefficient. We prove relative entropy and Wasserstein error estimates for the difference between the time marginals of the effective and original dynamics as well as an entropy error bound for the corresponding path space measures. A comparison with the averaging principle for systems with time-scale separation reveals that, unlike in the reversible setting, the effective dynamics for a non-reversible system need not agree with the averaged equations. We present a thorough comparison for the Ornstein-Uhlenbeck process and make a conjecture about necessary and sufficient conditions for when averaged and effective dynamics agree for nonlinear non-reversible processes. The theoretical results are illustrated with suitable numerical examples.

math.AP↗

arXiv2018

An inequality connecting entropy distance, Fisher Information and large deviations

In this paper we introduce a new generalisation of the relative Fisher Information for Markov jump processes on a finite or countable state space, and prove an inequality which connects this object with the relative entropy and a large deviation rate functional. In addition to possessing various favourable properties, we show that this generalised Fisher Information converges to the classical Fisher Information in an appropriate limit. We then use this generalised Fisher Information and the aforementioned inequality to qualitatively study coarse-graining problems for jump processes on discrete spaces.

math.FA↗

arXiv2018

Effective dynamics for non-reversible stochastic differential equations: a quantitative study

Coarse-graining is central to reducing dimensionality in molecular dynamics, and is typically characterized by a mapping which projects the full state of the system to a smaller class of variables. While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. In this article, starting with a non-reversible dynamics, we introduce and study an effective dynamics which approximates the (non-closed) projected dynamics. Under fairly weak conditions on the system, we prove error bounds on the trajectorial error between the projected and the effective dynamics. In addition to extending existing results to the non-reversible setting, our error estimates also indicate that the notion of mean force motivated by this effective dynamics is a good one.

math.PR↗

arXiv2017

Variational approach to coarse-graining of generalized gradient flows

In this paper we present a variational technique that handles coarse-graining and passing to a limit in a unified manner. The technique is based on a duality structure, which is present in many gradient flows and other variational evolutions, and which often arises from a large-deviations principle. It has three main features: (A) a natural interaction between the duality structure and the coarse-graining, (B) application to systems with non-dissipative effects, and (C) application to coarse-graining of approximate solutions which solve the equation only to some error. As examples, we use this technique to solve three limit problems, the overdamped limit of the Vlasov-Fokker-Planck equation and the small-noise limit of randomly perturbed Hamiltonian systems with one and with many degrees of freedom.

math.AP↗

arXiv2014

Coarse-graining and fluctuations: Two birds with one stone

We show how the mathematical structure of large-deviation principles matches well with the concept of coarse-graining. For those systems with a large-deviation principle, this may lead to a general approach to coarse-graining through the variational form of the large-deviation functional.

math.AP↗