#### DMCA

## WEAK KAM THEORY TOPICS IN THE STATIONARY ERGODIC SETTING (2009)

Citations: | 7 - 3 self |

### Citations

5412 | Variational Analysis,
- Rockafellar, Wets
- 1998
(Show Context)
Citation Context ...e inequality holds for every ω ∈ T∞ . Hence H is jointly continuous in (ω,p) and the proof is complete. □ We are now in position to prove Therem A.1. Proof of Theorem A.1. We recall (see for instance =-=[20]-=-) that a convex function ψ : R N → R is locally Lipschitz, and its Lipschitz constant in BR can be controlled with the supremum of |ψ| on BR+2, for every R > 0. In particular the Hamiltonian H satisfi... |

1546 |
Introduction to the Modern Theory of Dynamical Systems.
- Katok, Hasselblatt
- 1995
(Show Context)
Citation Context ...0) = H0(x,p) for every (x,p) ∈ R N × R N , 25The proof of Theorem A.1 will require some preliminary work. We start by some classical definitions and results from the theory of dynamical systems, see =-=[13]-=-. A continuous map τ : Ω → Ω defined on a Hausdorff topological space Ω will be said to be minimal if the orbit orb(ω) := {τ n (ω) : n ∈ Z} of every point ω ∈ Ω is dense in Ω. A Borel probability meas... |

374 |
Solutions de viscosite des equations de Hamilton-Jacobi.
- Barles
- 1994
(Show Context)
Citation Context ...the details. Here we just recall the main items. We say that a Lipschitz random function is a solution (resp. subsolution) of (8) if it is a viscosity solution (resp. a.e. subsolution) a.s. in ω (see =-=[3, 4]-=- for the definition of viscosity (sub)solution in the deterministic case). Notice that any such subsolution is almost surely in Lip κa (R n ), where κa := sup{ |p| : H(x,p,ω) ≤ a for some (x,ω) ∈ R N ... |

334 | Generalized solutions of Hamilton Jacobi equations. - Lions - 1982 |

316 |
Homogenization of differential operators and integral functionals,”
- Zhikov, Kozlov, et al.
- 1994
(Show Context)
Citation Context ... the symmetric difference. Given a random variable f : Ω → R, for any fixed ω ∈ Ω the function x ↦→ f(τxω) is said to be a realization of f. The following properties follow from Fubini’s Theorem, see =-=[14]-=-: if f ∈ Lp (Ω), then P–almost all its realizations belong to L p loc (RN); if fn → f in Lp (Ω), then P–almost all realizations of fn converge to the corresponding realization of f in L p loc (RN). Th... |

93 |
Homogenization of Hamilton-Jacobi equations, unpublished,
- Lions, Papanicolaou, et al.
- 1986
(Show Context)
Citation Context ... in R, and H is a continuous Hamiltonian, convex and superlinear in the momentum variable, and stationary with respect to the action of R N . As it is well known, this framework includes the periodic =-=[16]-=-, quasi–periodic [2] and almost–periodic cases [12] as particular instances. A stationary critical value, denoted by c, can be defined in this setting as the minimal value a for which the above equati... |

58 |
Stochastic homogenization of Hamilton-Jacobi equations and some applications,
- Souganidis
- 1999
(Show Context)
Citation Context ...nce of approximate or exact correctors via Lax representation formulae, depending on whether 0 belongs or not to the interior of the flat part of the effective Hamiltonian obtained via homogenization =-=[19, 22]-=-. This permits, among other things, to carry out the homogenization procedures through Evans’ perturbed test function method. Even if in the multidimensional analysis [10] many analogies with the one–... |

37 |
Global minimizers of autonomous Lagrangians.
- Contreras, Iturriaga
- 1999
(Show Context)
Citation Context ... will denote by Af(ω) the collection of points y of RN enjoying the above condition with a = cf. The set Af(ω) is closed for every ω ∈ Ω. We will also use later an equivalent definition of Af(ω), see =-=[6]-=-. For every ω ∈ Ω, let L(x,q,ω) := max {〈p,q〉 − H(x,p,ω)} , p∈RN (x,q) ∈ RN × RN and, for every t > 0, ht(x,y,ω) := inf Then {∫ t 0 } (L(γ, ˙γ,ω) + c) ds : γ(0) = x, γ(t) = y , x, y ∈ RN . Af(ω) = {y ... |

37 |
Correctors for the homogenization of Hamilton-Jacobi equations in the stationary ergodic setting,
- Lions, Souganidis
- 2003
(Show Context)
Citation Context ...tice that the set appearing at the right–hand side of (11) is non void, since it contains the value sup (x,ω) H(x,0,ω), which is finite thanks to (H3). Moreover, the infimum is attained. In fact, see =-=[10, 17]-=- Theorem 3.3. Sc = ∅. It is apparent by the definitions that c ≥ cf. A more precise result, establishing the relation with the effective Hamiltonian obtained via the homogenization [19, 22], will be ... |

37 |
Homogenization for stochastic Hamilton-Jacobi equations.
- Rezakhanlou, Tarver
- 2000
(Show Context)
Citation Context ...nce of approximate or exact correctors via Lax representation formulae, depending on whether 0 belongs or not to the interior of the flat part of the effective Hamiltonian obtained via homogenization =-=[19, 22]-=-. This permits, among other things, to carry out the homogenization procedures through Evans’ perturbed test function method. Even if in the multidimensional analysis [10] many analogies with the one–... |

29 |
Almost periodic homogenization of Hamilton-Jacobi equations
- Ishii
- 1999
(Show Context)
Citation Context ...d superlinear in the momentum variable, and stationary with respect to the action of R N . As it is well known, this framework includes the periodic [16], quasi–periodic [2] and almost–periodic cases =-=[12]-=- as particular instances. A stationary critical value, denoted by c, can be defined in this setting as the minimal value a for which the above equation possesses admissible subsolutions, that is Lipsc... |

28 |
A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations
- Davini, Siconolfi
(Show Context)
Citation Context ...ula holds for every x ∈ RN and t > 0: { ∫ 0 } u(x) = inf u(γ(−t)) + (L(γ(s), ˙γ(s),ω) + c)ds : γ(0) = x , (28) −t where γ varies in the family of absolutely continuous curves from [−t,0] to R N , see =-=[8]-=-. By standard arguments of the Calculus of Variations [5], a minimizing absolutely continuous curve does exist for any fixed t > 0 thanks to the coercivity and lower semicontinuity properties of L. Mo... |

26 |
One-Dimensional Variational Problems. An Introduction, Oxford
- Buttazzo, Giaquinta, et al.
- 1998
(Show Context)
Citation Context ...u(γ(−t)) + (L(γ(s), ˙γ(s),ω) + c)ds : γ(0) = x , (28) −t where γ varies in the family of absolutely continuous curves from [−t,0] to R N , see [8]. By standard arguments of the Calculus of Variations =-=[5]-=-, a minimizing absolutely continuous curve does exist for any fixed t > 0 thanks to the coercivity and lower semicontinuity properties of L. Moreover such curves turn out to be equi–Lipschitz continuo... |

19 |
Theory of random sets. Probability and its Applications (New
- Molchanov
- 2005
(Show Context)
Citation Context ...bra related to the Fell topology on the family of closed subsets of R N . This, in turn, coincides with the Effros σ–algebra. If X(ω) is measurable in this sense then it is also graph–measurable, see =-=[18]-=- for more details. A closed random set X is called stationary if it, in addition, satisfies (3). Note that in this event the set {ω : X(ω) = ∅ }, which is measurable by the Effros measurability of X,... |

16 |
Fourier analysis on groups. Reprint of the 1962 original. Wiley Classics Library. A Wiley-Interscience Publication
- Rudin
- 1990
(Show Context)
Citation Context ...on R N is said to be almost–periodic if it is bounded, continuous and if it can be approximated, uniformly on R N , by finite linear combinations of functions in the set {e 2πi〈λ, x〉 : λ ∈ R N }, see =-=[1, 21]-=- for instance. This appendix is devoted to show that any almost–periodic Hamiltonian is a specific realization of a stationary ergodic Hamiltonian, with underlying probability space Ω separable in a m... |

10 |
Exact and approximate correctors for stochastic Hamiltonians: the 1-dimensional case
- Davini, Siconolfi
(Show Context)
Citation Context ... 0 and that are almost sure subsolutions either in the viscosity sense or, equivalently, almost everywhere in R N . The condition on the gradient implies almost sure sublinear growth at infinity, see =-=[9, 10]-=-. The stationary critical value is in general distinct from the free critical value cf, i.e. the minimal value a for which the above equation admits subsolutions, without any further qualification. Mo... |

10 | PDE aspects of Aubry-Mather theory for continuous convex Hamiltonians - Fathi |

9 |
Multiscale Young measures in almost periodic homogenization and applications
- Ambrosio, Frid
- 2009
(Show Context)
Citation Context ...measure which is ergodic with respect to the action of R N . This allows to include the almost–periodic case within the stationary ergodic framework, but the problem is that G N is non–separable, see =-=[1]-=- for similar issues. Thus we have to resort to a different construction, exposed in the Appendix, that we believe of independent interest. We basically exploit that any almost–periodic function on R N... |

7 | Bolza problems with discontinuous Lagrangians and Lipschitz-continuity of the value function
- Davini
(Show Context)
Citation Context ...mizing absolutely continuous curve does exist for any fixed t > 0 thanks to the coercivity and lower semicontinuity properties of L. Moreover such curves turn out to be equi–Lipschitz continuous, see =-=[7]-=-. Given an increasing sequence tn with limn tn = +∞, we denote by γn the corresponding minimizers and extend them on the whole interval (−∞,0] by setting γn(t) = γn(−tn) in (−∞, −tn), for any n. Thank... |

4 |
Filtrations of random processes in the light of classification theory. I. A topological zero-one law. Preprint (2001) (ArXiv: math/0107121
- Tsirelson
(Show Context)
Citation Context ...pace, where P is the probability measure and F the σ–algebra of P–measurable sets. Here separable is understood in the measure theoretic sense, meaning that the Hilbert space L2 (Ω) is separable, cf. =-=[23]-=- also for other equivalent definitions. A property will be said to hold almost surely (a.s. for short) on Ω if it holds up to a subset of probability 0. We will indicate by Lp (Ω), p ≥ 1, the usual Le... |

3 | Multiscale homogenization for first-order Hamilton-JacobiBellman equations - Arisawa |

1 | A metric analysis of critical Hamilton–Jacobi equations in the stationary ergodic setting
- Davini, Siconolfi
- 2008
(Show Context)
Citation Context ...ry ergodic Hamiltonian. For every a ∈ R, we are interested in the stochastic Hamilton–Jacobi equation H(x,Dv(x,ω),ω) = a in R N . (8) The material we are about to expose has been already presented in =-=[9, 10]-=-, to which we refer for the details. Here we just recall the main items. We say that a Lipschitz random function is a solution (resp. subsolution) of (8) if it is a viscosity solution (resp. a.e. subs... |