Laplace Principle Research Articles

Large deviation principle for pseudo-monotone evolutionary equation

. In this article, we explore Freidlin-Wentzell’s large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework.

Large deviations for stochastic predator–prey model with Lévy noise

This paper discusses the large deviations for stochastic predator–prey model driven by multiplicative Lévy noise. Using Galerkin approximation, we initially prove the existence and uniqueness of solution. Due to the equivalence between Laplace principle and large deviation principle under a Polish space, the method of weak convergence has been followed in order to establish our results for this coupled system of equations.

Laplace principle for large population games with control interaction

This work investigates continuous time stochastic differential games with a large number of players whose costs and dynamics interact through the empirical distribution of both their states and their controls. The control processes are assumed to be open-loop. We give regularity conditions guaranteeing that if the finite-player game admits a Nash equilibrium, then both the sequence of equilibria and the corresponding state processes satisfy a Sanov-type large deviation principle. The results require existence of a Lipschitz continuous solution of the master equation of the corresponding mean field game, and they carry over to cooperative (i.e. central planner) games. We study a linear-quadratic case of such games in details.

On the Global Convergence of Particle Swarm Optimization Methods

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a continuous-time formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution, which acts as Lyapunov function of the dynamics. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.

Weakly maxitive set functions and their possibility distributions

The Shilkret integral with respect to a completely maxitive capacity is fully determined by a possibility distribution. In this paper, we introduce a weaker topological form of maxitivity and show that under this assumption the Shilkret integral is still determined by its possibility distribution for functions that are sufficiently regular. Motivated by large deviations theory, we provide a Laplace principle for maxitive integrals and characterize the possibility distribution under certain separation and convexity assumptions. Moreover, we show a maxitive integral representation result for weakly maxitive non-linear expectations. The theoretical results are illustrated by providing large deviations bounds for sequences of capacities, and by deriving a monotone analogue of Cramér's theorem.

Convergence analysis of Particle Swarm Optimization in one dimension

This paper is devoted to a rigorous analysis of one dimensional Particle Swarm Optimization (PSO) for non convex objective function. We provide a quantitative bound on the error between the equilibrium point and the global minimizer of the given objective function. Our strategy does not rely on the Laplace principle. We also investigate the underdamped PSO with linear objective function.

Large deviations for the stochastic functional integral equation with nonlocal condition

PurposeThe purpose of this paper is to study large deviations for the solution processes of a stochastic equation incorporated with the effects of nonlocal condition.Design/methodology/approachA weak convergence approach is adopted to establish the Laplace principle, which is same as the large deviation principle in a Polish space. The sufficient condition for any family of solutions to satisfy the Laplace principle formulated by Budhiraja and Dupuis is used in this work.FindingsFreidlin–Wentzell type large deviation principle holds good for the solution processes of the stochastic functional integral equation with nonlocal condition.Originality/valueThe asymptotic exponential decay rate of the solution processes of the considered equation towards its deterministic counterpart can be estimated using the established results.

Local Stability of McKean-Vlasov Equations Arising from Heterogeneous Gibbs Systems Using Limit of Relative Entropies.

A family of heterogeneous mean-field systems with jumps is analyzed. These systems are constructed as a Gibbs measure on block graphs. When the total number of particles goes to infinity, the law of large numbers is shown to hold in a multi-class context, resulting in the weak convergence of the empirical vector towards the solution of a McKean–Vlasov system of equations. We then investigate the local stability of the limiting McKean–Vlasov system through the construction of a local Lyapunov function. We first compute the limit of adequately scaled relative entropy functions associated with the explicit stationary distribution of the N-particles system. Using a Laplace principle for empirical vectors, we show that the limit takes an explicit form. Then we demonstrate that this limit satisfies a descent property, which, combined with some mild assumptions shows that it is indeed a local Lyapunov function.

Large Deviation Principle for McKean–Vlasov Quasilinear Stochastic Evolution Equations

This paper is devoted to investigating the Freidlin–Wentzell’s large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

Wentzell–Freidlin Large Deviation Principle for Stochastic Convective Brinkman–Forchheimer Equations

This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.

Large deviations built on max-stability

In this paper, we show that the basic results in large deviations theory hold for general monetary risk measures, which satisfy the crucial property of max-stability. A max-stable monetary risk measure fulfills a lattice homomorphism property, and satisfies under a suitable tightness condition the Laplace Principle (LP), that is, admits a dual representation with affine convex conjugate. By replacing asymptotic concentration of probability by concentration of risk, we formulate a Large Deviation Principle (LDP) for max-stable monetary risk measures, and show its equivalence to the LP. In particular, the special case of the asymptotic entropic risk measure corresponds to the classical Varadhan–Bryc equivalence between the LDP and LP. The main results are illustrated by the asymptotic shortfall risk measure.

Hysteresis of a Tension-Sensitive K+ Channel Revealed by Time-Lapse Tension Measurements.

Various types of channels vary their function by membrane tension changes upon cellular activities, and lipid bilayer methods allow elucidation of direct interaction between channels and the lipid bilayer. However, the dynamic responsiveness of the channel to the membrane tension remains elusive. Here, we established a time-lapse tension measurement system. A bilayer is formed by docking two monolayer-lined water bubbles, and tension is evaluated via measuring intrabubble pressure as low as <100 Pa (Young–Laplace principle). The prototypical KcsA potassium channel is tension-sensitive, and single-channel current recordings showed that the activation gate exhibited distinct tension sensitivity upon stretching and relaxing. The mechanism underlying the hysteresis is discussed in the mode shift regime, in which the channel protein bears short “memory” in their conformational changes.

Reliability evaluation with limited and censored time-to-failure data based on uncertainty distributions

Limited and censored time-to-failure data are common in practical life testing, which may lead to a lack of knowledge, i.e., epistemic uncertainties, and makes it difficult to evaluate product reliability accurately. Under this situation, the large-sample based probability theory is not appropriate anymore. To deal with such problem, in this paper, the uncertainty theory based reliability evaluation is proposed to cope with the limited and censored time-to-failure data, where two types of life testing censoring (type-I and type-II censoring) and two types of time-to-failure data (precise and interval data) are considered. Firstly, the lifetime model and reliability evaluations based on uncertainty theory are presented. Then, the uncertain statistics method is presented for each combination of censoring types and data types in life testing, which includes four steps: order statistics construction, belief degree construction according to the Laplace principle of indifference, parameter estimation based on the uncertain principle of least squares, and uncertain hypothesis testing. Two simulation studies and two practical cases with comprehensive and in-depth discussion are conducted to illustrate the practicability and effectiveness of the proposed method. The results show that the proposed method can quantify epistemic uncertainties properly and provide more stable and accurate mean time to failure results under limited and censored time-to-failure data compared with probability and Bayesian probability methods.

Large deviation principle for a class of SPDE with locally monotone coefficients

This work aims to prove the large deviation principle for a class of stochastic partial differential equations with locally monotone coefficients under the extended variational framework, which generalizes many previous works. Using stochastic control and the weak convergence approach, we prove the Laplace principle, which is equivalent to the large deviation principle in our framework. Instead of assuming compactness of the embedding in the corresponding Gelfand triple or finite dimensional approximation of the diffusion coefficient in some existing works, we only assume some temporal regularity in the diffusion coefficient.

Initial conditions of inflation in a Bianchi I Universe

We investigate the initial conditions of inflation in a Bianchi~I universe that is homogeneous but not isotropic. We use the Eisenhart lift to describe such a theory geometrically as geodesics on a field space manifold. We construct the phase-space manifold of the theory by considering the tangent bundle of the field space and equipping it with a natural metric. We find that the total volume of this manifold is finite for a wide class of inflationary models. We therefore take the initial conditions to be uniformly distributed over it in accordance with Laplace's principle of indifference. This results in a normalisable, reparametrisation invariant measure on the set of initial conditions of inflation in a Bianchi~I universe. We find that this measure favours an initial state in which the inflaton field is at or near its minimum, with a mild preference for some initial anisotropy. Since inflation requires an initial field value with a large displacement from its minimum, we therefore conclude that the theory of inflation requires finely tuned initial conditions.

Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

In this work, we establish the unique global solvability of the stochastic two dimensional viscoelastic fluid flow equations, arising from the Oldroyd model for the non-Newtonian fluid flows perturbed by multiplicative Gaussian noise. A local monotonicity property of the linear and nonlinear operators and a stochastic generalization of the Minty–Browder technique are exploited in the proofs. The Laplace principle for the strong solution of the stochastic system is established in a suitable Polish space using a weak convergence approach. The Wentzell–Freidlin large deviation principle is proved using the well known results of Varadhan and Bryc. The large deviations for shot time are also considered. We also establish the existence of a unique ergodic and strongly mixing invariant measure for the stochastic system with additive Gaussian noise, using the exponential stability of strong solutions.

Uniform large deviation principles for Banach space valued stochastic evolution equations

We prove a large deviation principle (LDP) for a general class of Banach space valued stochastic differential equations (SDEs) that is uniform with respect to initial conditions in bounded subsets of the Banach space. A key step in the proof is showing that a uniform LDP over compact sets is implied by a uniform over compact sets Laplace principle. Because bounded subsets of infinite-dimensional Banach spaces are in general not relatively compact in the norm topology, we embed the Banach space into its double dual and utilize the weak- ⋆ \star compactness of closed bounded sets in the double dual space. We prove that a modified version of our SDE satisfies a uniform Laplace principle over weak- ⋆ \star compact sets and consequently a uniform over bounded sets LDP. We then transfer this result back to the original equation using a contraction principle. The main motivation for this uniform LDP is to generalize results of Freidlin and Wentzell concerning the behavior of finite-dimensional SDEs. Here we apply the uniform LDP to study the asymptotics of exit times from bounded sets of Banach space valued small noise SDE, including reaction diffusion equations with multiplicative noise and two-dimensional stochastic Navier–Stokes equations with multiplicative noise.

Extended Laplace principle for empirical measures of a Markov chain

AbstractWe consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative entropy (or Kullback– Leibler divergence) and cumulant generating functional f ↦ ln ʃ exp (f). Following the approach by Lacker (2016) in the independent and identically distributed case, we generalize the Laplace principle to a greater class of convex dual pairs. We present in depth one application arising from this extension, which includes large deviation results and a weak law of large numbers for certain robust Markov chains—similar to Markov set chains—where we model robustness via the first Wasserstein distance. The setting and proof of the extended Laplace principle are based on the weak convergence approach to large deviations by Dupuis and Ellis (2011).

Large deviations for the 2-D derivative Ginzburg–Landau equation with multiplicative noise

This paper proves a Wentzell–Freidlin type large deviation principle for the 2-D derivative Ginzburg–Landau equation perturbed by a small multiplicative noise, based on the Laplace principle and weak convergence approach.

Large Deviation Principle for Stochastic Thermoelastic System Coupled Sine-Gordon Equation Driven by Lévy Noise

This work discusses the application of large deviation principle for a stochastic nonlinear thermoelastic system coupled Sine-Gordon equation driven by Levy noise. This is done using weak convergence approach formulated by Dupuis and Ellis wherein Laplace principle and large deviation principle are considered equivalent under specific space settings. Initially, the compactness of solution of the control system corresponding to the original system is proved. The proof is completed by ensuring that the solutions of the controlled stochastic system weakly converges to that of the controlled deterministic system with the aid of the estimate of the solution obtained prior.