1,720,969 research outputs found
Large deviations for interacting particle systems: Joint mean-field and small-noise limit
Full text linkWe consider a system of stochastic interacting particles in Rd and we describe large deviation asymptotics in a joint mean-field and small-noise limit. Precisely, a large deviation principle (LDP) is established for the empirical measure and the stochastic current, as the number of particles tends to infinity and the noise vanishes, simultaneously. We give a direct proof of the LDP using tilting and subsequently exploiting the link between entropy and large deviations. To this aim we employ consistency of suitable deterministic control problems associated to the stochastic dynamics
A note on regularity and separation for the stochastic Allen-Cahn equation with logarithmic potential
No full textWe prove refined space-time regularity for the classical stochastic Allen-Cahn equation with logarithmic potential. This allows to establish a random separation property, i.e. that the trajectories of the solution are strictly separated from the potential barriers. The present contribution extends the results obtained in [2], where separation was proved only for p-Laplace operators with p greater than the space-dimension
Singular stochastic Allen–Cahn equations with dynamic boundary conditions
Full text linkWe prove a well-posedness result for stochastic Allen–Cahn type equations in a bounded domain coupled with generic boundary conditions. The (nonlinear) flux at the boundary aims at describing the interactions with the hard walls and is motivated by some recent literature in physics. The singular character of the drift part allows for a large class of maximal monotone operators, generalizing the usual double-well potentials. One of the main novelties of the paper is the absence of any growth condition on the drift term of the evolution, neither on the domain nor on the boundary. A well-posedness result for variational solutions of the system is presented using a priori estimates as well as monotonicity and compactness techniques. A vanishing viscosity argument for the dynamic on the boundary is also presented
A stochastic maximum principle with dissipativity conditions
No full textIn this paper we prove a version of the maximum principle, in the sense of Pontryagin, for the optimal control of a finite dimensional stochastic differential equation, driven by a multidimensional Wiener process. We drop the usual Lipschitz assumption on the drift term and substitute it with dissipativity conditions, allowing polynomial growth. The control enters both the drift and the diffusion term and takes values in a general metric space
Stochastic maximum principle for optimal control of a class of nonlinear spdes with dissipative drift
No full textWe prove a version of the stochastic maximum principle, in the sense of Pontryagin, for the finite horizon optimal control of a stochastic partial differential equation driven by an infinitedimensional additive noise. In particular, we treat the case in which the nonlinear term is of Nemytskii type, dissipative, and with polynomial growth. The performance functional to be optimized is fairly general and may depend on point evaluation of the controlled equation. The results can be applied to a large class of nonlinear parabolic equations such as reaction-diffusion equations
Ergodic Maximum Principle for Stochastic Systems
No full textWe present a version of the stochastic maximum principle (SMP) for ergodic control problems. In particular we give necessary (and sufficient) conditions for optimality for controlled dissipative systems in finite dimensions. The strategy we employ is mainly built on duality techniques. We are able to construct a dual process for all positive times via the analysis of a suitable class of perturbed linearized forward equations. We show that such a process is the unique bounded solution to a backward SDE on infinite horizon from which we can write a version of the SMP
A variational approach to the mean field planning problem
No full textWe investigate a first-order mean field planning problem of the form-partial derivative(t)u + H(x, Du) = f(x,m) in (0,T) x R-d ,partial derivative(t)m - del.(mH(p) (x, Du)) = 0 in (0,T) x R-d ,m(0,center dot) = m(0) , m(T, center dot) = m(T) in R-d,associated to a convex Hamiltonian H with quadratic growth and a monotone interaction term f with polynomial growth. We exploit the variational structure of the system, which encodes the first order optimality condition of a convex dynamic optimal entropy-transport problem with respect to the unknown density m and of its dual, involving the maximization of an integral functional among all the subsolutions u of an Hamilton-Jacobi equation. Combining ideas from optimal transport, convex analysis and renormalized solutions to the continuity equation, we will prove existence and (at least partial) uniqueness of a weak solution (m, u). A crucial step of our approach relies on a careful analysis of distributional subsolutions to Hamilton-Jacobi equations of the form -partial derivative(t)u + H(x, Du) <= alpha, under minimal summability conditions on alpha, and to a measure theoretic description of the optimality via a suitable contact defect measure. Finally, using the superposition principle, we are able to describe the solution to the system by means of a measure on the path space encoding the local behavior of the players. (C) 2019 Elsevier Inc. All rights reserved
Wasserstein stability of porous medium-type equations on manifolds with Ricci curvature bounded below
Full text linkGiven a complete, connected Riemannian manifold Mn with Ricci curvature bounded from below, we discuss the stability of the solutions of a porous medium-type equation with respect to the 2-Wasserstein distance. We produce (sharp) stability estimates under negative curvature bounds, which to some extent generalize well-known results by Sturm [35] and Otto-Westdickenberg [32]. The strategy of the proof mainly relies on a quantitative L1–L∞ smoothing property of the equation considered, combined with the Hamiltonian approach developed by Ambrosio, Mondino and Savaré in a metric-measure setting [4]
Random separation property for stochastic Allen-Cahn-type equations
Full text linkWe study a large class of stochastic p-Laplace Allen-Cahn equations with singular potential. Under suitable assumptions on the (multiplicative-type) noise we first prove existence, uniqueness, and regularity of variational solutions. Then, we show that a random separation property holds, i.e. almost every trajectory is strictly separated in space and time from the potential barriers. The threshold of separation is random, and we further provide exponential estimates on the probability of separation from the barriers. Eventually, we exhibit a convergence-in-probability result for the random separation threshold towards the deterministic one, as the noise vanishes, and we obtain an estimate of the convergence rate
Large Deviations for Kac-Like Walks
Full text linkWe introduce a Kac’s type walk whose rate of binary collisions preserves the total momentum but not the kinetic energy. In the limit of large number of particles we describe the dynamics in terms of empirical measure and flow, proving the corresponding large deviation principle. The associated rate function has an explicit expression. As a byproduct of this analysis, we provide a gradient flow formulation of the Boltzmann-Kac equation
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