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91
Optimal control of switching diffusions with applications to flexible manufacturing systems
 SIAM J. Control Optim
"... Abstract. A controlled switching diffusion model is developed to study the hierarchical control of flexible manufacturing systems. The existence of a homogeneous Markov nonrandomized optimal policy is established by a convex analytic method. Using the existence of such a policy, the existence of a ..."
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Cited by 52 (4 self)
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Abstract. A controlled switching diffusion model is developed to study the hierarchical control of flexible manufacturing systems. The existence of a homogeneous Markov nonrandomized optimal policy is established by a convex analytic method. Using the existence of such a policy, the existence of a unique solution in a certain class to the associated HamiltonJacobiBellman equations is established and the optimal policy is characterized as a minimizing selector of an appropriate Hamiltonian.
A quantization algorithm for solving multidimensional Optimal Stopping problems
 Bernoulli
, 2001
"... A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its valu ..."
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Cited by 37 (3 self)
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A new grid method for computing the Snell envelop of a function of a R valued Markov chain (X k ) 0#k#n is proposed. (This problem is typically non linear and cannot be solved by the standard Monte Carlo method.) Every X k is replaced by a "quantized approximation" X k taking its values in a grid # k of size N k . The n grids and their transition probability matrices make up a discrete tree on which a pseudoSnell envelop is devised by mimicking the regular dynamic programming formula. We show, using Quantization Theory of probability distributions the existence of a set of optimal grids, given the total number N of elementary R valued vector quantizers. A recursive stochastic algorithm, based on some simulations of (X k ) 0#k#n , yields the optimal grids and their transition probability matrices. Some a priori error estimates based on the quantization errors are established. These results are applied to the computation of the Snell envelop of a di#usion (assuming that it can be directly simulated or using its Euler scheme). We show how this approach yields a discretization method for Reflected Backward Stochastic Di#erential Equation. Finally, some first numerical tests are carried out on a 2dimensional American option pricing problem.
Weighted norm inequalities, offdiagonal estimates and elliptic operators, Part II: Offdiagonal estimates on spaces of homogeneous type
, 2005
"... Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincar ..."
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Cited by 27 (10 self)
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Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the LaplaceBeltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and
Stability of parabolic Harnack inequalities
 Transactions of the American Mathematical Society
, 2004
"... Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. ..."
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Cited by 24 (4 self)
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Abstract. Let (G;E) be a graph with weights faxyg for which a parabolic Harnack inequality holds with spacetime scaling exponent 2. Suppose fa0xyg is another set of weights that are comparable to faxyg. We prove that this parabolic Harnack inequality also holds for (G;E) with the weights fa0xyg. We also give stable necessary and sucient conditions for this parabolic Harnack inequality to hold.
Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov type operators in nondivergence form
 Advances in Differential Equations
, 2006
"... We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound i ..."
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Cited by 16 (4 self)
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We prove some Schauder type estimates and an invariant Harnack inequality for a class of degenerate evolution operators of Kolmogorov type. We also prove a Gaussian lower bound for the fundamental solution of the operator and a uniqueness result for the Cauchy problem. The proof of the lower bound is obtained by solving a suitable optimal control problem and using the invariant Harnack inequality.
Probabilistic interpretation and random walk on spheres algorithms for the PoissonBoltzmann equation in molecular dynamics
, 2010
"... Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators ..."
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Cited by 14 (7 self)
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Abstract. Motivated by the development of efficient Monte Carlo methods for PDE models in molecular dynamics, we establish a new probabilistic interpretation of a family of divergence form operators with discontinuous coefficients at the interface of two open subsets of R d. This family of operators includes the case of the linearized PoissonBoltzmann equation used to compute the electrostatic free energy of a molecule. More precisely, we explicitly construct a Markov process whose infinitesimal generator belongs to this family, as the solution of a SDE including a non standard local time term related to the interface of discontinuity. We then prove an extended FeynmanKac formula for the PoissonBoltzmann equation. This formula allows us to justify various probabilistic numerical methods to approximate the free energy of a molecule. We analyse the convergence rate of these simulation procedures and numerically compare them on idealized molecules models. Résumé. Motivés par le développement de méthodes de MonteCarlo efficaces pour des équations aux dérivées partielles en dynamique moléculaire, nous établissons une nouvelle interprétation probabiliste d’une famille d’opérateurs sous forme divergence et à coefficients discontinus le long de l’interface de deux ouverts de R d. Cette famille d’opérateurs inclut le cas de l’équation de PoissonBoltzmann
SOME PARABOLIC PDEs WHOSE DRIFT IS AN IRREGULAR RANDOM NOISE IN SPACE
, 2006
"... We consider a new class of random partial differential equation of parabolic type where the stochastic term is constituted by an irregular noisy drift, not necessarily Gaussian. We provide a suitable interpretation and we study existence. After freezing a realization of the drift (stochastic proces ..."
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Cited by 14 (11 self)
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We consider a new class of random partial differential equation of parabolic type where the stochastic term is constituted by an irregular noisy drift, not necessarily Gaussian. We provide a suitable interpretation and we study existence. After freezing a realization of the drift (stochastic process), we study existence and uniqueness (in some suitable sense) of the associated parabolic equation and we investigate probabilistic interpretation.
Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics
 In preparation
"... We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the correcto ..."
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Cited by 14 (9 self)
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We study the effective largescale behavior of discrete elliptic equations on the lattice Z d with random coefficients. The theory of stochastic homogenization relates the random but stationary field of coefficients with a deterministic matrix of effective coefficients. This is done via the corrector problem, which can be viewed as a highly degenerate elliptic equation on the infinitedimensional space of admissible coefficient fields. In this contribution we develop quantitative methods for the corrector problem assuming that the ensemble of coefficient fields satisfies a spectral gap estimate w. r. t. a Glauber dynamics. As a main result we prove an optimal estimate for the decay in time of the parabolic equation associated to the corrector problem (i. e. for the “random environment as seen from a random walker”). As a corollary we obtain existence and moment bounds for stationary correctors (in dimension d> 2) and optimal estimates for regularized versions of the corrector (in dimensions d ≥ 2). We also give a selfcontained proof for a new estimate on the gradient of the parabolic, variablecoefficient Green’s function, which is a crucial analytic ingredient in our method. As an application, we study the approximation of the homogenized coefficients via a representative volume element. The approximation introduces two types of errors. Based on our quantitative
SaloffCoste L., Stability results for Harnack inequalities
"... We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Har ..."
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Cited by 13 (2 self)
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We develop new techniques for proving uniform elliptic and parabolic Harnack inequalities on weighted Riemannian manifolds. In particular, we prove the stability of the Harnack inequalities under certain nonuniform changes of the weight. We also prove necessary and sufficient conditions for the Harnack inequalities to hold on complete noncompact manifolds having nonnegative Ricci curvature outside a compact set and a finite first Betti number or just having asymptotically nonnegative sectional curvature. Contents
Note on the equivalence of parabolic Harnack inequalities and heat kernel estimates. http://www.math.uconn.edu/∼bass/papers/phidfapp.pdf
"... We prove the equivalence of parabolic Harnack inequalities and subGaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form. Key Words and Phrases. Harnack inequality, heat kernel estimate, caloric function, metric measure space, volume doubling, Dirichlet ..."
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Cited by 11 (5 self)
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We prove the equivalence of parabolic Harnack inequalities and subGaussian heat kernel estimates in a general metric measure space with a local regular Dirichlet form. Key Words and Phrases. Harnack inequality, heat kernel estimate, caloric function, metric measure space, volume doubling, Dirichlet space