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Potential Analysis, ISSN 0926-2601, 2/2012, Volume 36, Issue 2, pp. 317 - 337
... ≤ 2 in proper geodesic metric spaces. By means of a general Hamilton–Jacobi semigroup we prove that these are equivalent, and moreover equivalent to the hypercontractivity of the Hamilton–Jacobi semigroup... 
Geodesic metric space | Hamilton–Jacobi semigroup | Probability Theory and Stochastic Processes | Mathematics | Secondary 36C05 | Geometry | Primary 70H20 | 49L99 | 47D06 | Potential Theory | Functional Analysis | Poincaré inequalities | Logarithmic–Sobolev inequalites | Talagrand inequalites | Metric-measure space | Hamilton-Jacobi semigroup | Logarithmic-Sobolev inequalites | METRIC-MEASURE-SPACES | TRANSPORTATION COST | HOPF-LAX FORMULA | BRASCAMP | MATHEMATICS | MAPS | Poincare inequalities | GEOMETRY
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 09/2016, Volume 261, Issue 5, pp. 2702 - 2737
...–Jacobi equations in one space dimension. Some properties of the effective Hamiltonian arising in the nonconvex case are also discussed. 
Nonconvex Hamilton–Jacobi equation | Dynamical properties of effective Hamiltonians | Metric problem | Stochastic homogenization | Nonconvex Hamilton-Jacobi equation | MATHEMATICS | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs | Mathematics
Journal Article
Journal of Differential Equations, ISSN 0022-0396, 08/2019, Volume 267, Issue 5, pp. 2918 - 2949
...–Jacobi equations. The nonconvex Hamiltonians, which are generally uneven and inseparable, are generated by a sequence of quasiconvex Hamiltonians and a sequence of quasiconcave Hamiltonians through the min-max formula... 
Uneven | Min-max formula/identity | Stochastic homogenization | Hamilton–Jacobi equation | Nonconvex | Stationary ergodic | MATHEMATICS | VISCOSITY SOLUTIONS | Hamilton-Jacobi equation | CORRECTORS
Journal Article
Journal of functional analysis, ISSN 0022-1236, 2018, Volume 275, Issue 8, pp. 2096 - 2161
...) evolution equations in the variational framework. Furthermore, differential games associated to such evolution equations can be investigated following the Krasovskiĭ–Subbotin approach similarly as in finite dimensions. 
Nonlinear evolution equations | Path-dependent PDEs | Minimax solutions | Optimal control | BOUNDARY CONTROL-PROBLEMS | HJB EQUATIONS | VISCOSITY SOLUTIONS | STATE CONSTRAINTS | NONLINEAR 2ND-ORDER EQUATIONS | MATHEMATICS | RISK-SENSITIVE CONTROL | PARTIAL-DIFFERENTIAL-EQUATIONS | BELLMAN EQUATIONS | OPTIMAL STOCHASTIC-CONTROL | SADDLE-POINT
Journal Article
Journal de mathématiques pures et appliquées, ISSN 0021-7824, 09/2018, Volume 117, pp. 221 - 262
...–Jacobi equation in a periodic environment which is perturbed either by medium with increasing period or by a random Bernoulli perturbation with small parameter... 
Homogenization | Weak KAM theory | Hamilton–Jacobi equations | Random media | Viscosity solutions | MATHEMATICS | Hamilton-Jacobi equations | MATHEMATICS, APPLIED | STOCHASTIC HOMOGENIZATION | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | BERNOULLI PERTURBATIONS | Analysis of PDEs | Mathematics | Optimization and Control
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 10/2015, Volume 54, Issue 2, pp. 1507 - 1524
...–Jacobi equations. The new idea is to introduce a family of “sub-equations” and to control solutions of the original equation by the maximal subsolutions of the latter... 
35B27 | Mathematics | Systems Theory, Control | Calculus of Variations and Optimal Control; Optimization | Analysis | Theoretical, Mathematical and Computational Physics | MATHEMATICS | MATHEMATICS, APPLIED | BELLMAN EQUATIONS | MEDIA | Analysis of PDEs
Journal Article
Neural Networks, ISSN 0893-6080, 03/2018, Volume 99, pp. 166 - 177
...–Jacobi–Bellman (HJB) equation, Lyapunov technique, and inverse optimality, and hence guarantees that the chaotic drive network synchronizes with the chaotic response network influenced by uncertain noise signals... 
Lyapunov technique | Coupled stochastic neural networks | Chaotic synchronization | Noise attenuation | Nonlinear optimal control | Hamilton–Jacobi–Bellman (HJB) equation | CRITERIA | FEEDBACK-CONTROL | SLIDING MODE CONTROL | TIME-VARYING DELAYS | NEUROSCIENCES | COMPUTER SCIENCE, ARTIFICIAL INTELLIGENCE | EXPONENTIAL LAG SYNCHRONIZATION | Hamilton-Jacobi-Bellman (HJB) equation | SYSTEMS | UNKNOWN-PARAMETERS | Neural networks
Journal Article
Calculus of Variations and Partial Differential Equations, ISSN 0944-2669, 4/2016, Volume 55, Issue 2, pp. 1 - 39
Journal Article
The Annals of probability, ISSN 0091-1798, 7/2015, Volume 43, Issue 4, pp. 1823 - 1865
Journal Article
SIAM JOURNAL ON CONTROL AND OPTIMIZATION, ISSN 0363-0129, 2019, Volume 57, Issue 1, pp. 23 - 52
In this paper we study the ergodic problem for viscous Hamilton-Jacobi equations with superlinear Hamiltonian and inward drift. We investigate... 
MATHEMATICS, APPLIED | stochastic ergodic control | LARGE TIME BEHAVIOR | ergodic problem | generalized principal eigenvalue | BELLMAN EQUATIONS | viscous Hamilton-Jacobi equation | AUTOMATION & CONTROL SYSTEMS | Analysis of PDEs | Mathematics
Journal Article
Journal de mathématiques pures et appliquées, ISSN 0021-7824, 11/2017, Volume 108, Issue 5, pp. 751 - 782
We continue the study of the homogenization of coercive non-convex Hamilton–Jacobi equations in random media identifying two general classes of Hamiltonians with very distinct behavior... 
Stochastic homogenization | Hamilton–Jacobi equations | Viscosity solutions | MATHEMATICS | Hamilton-Jacobi equations | MATHEMATICS, APPLIED
Journal Article
SIAM Journal on Control and Optimization, ISSN 0363-0129, 2013, Volume 51, Issue 6, pp. 4274 - 4294
We show that the value function of a stochastic control problem is the unique solution of the associated Hamilton-Jacobi-Bellman equation, completely avoiding the proof of the so-called dynamic programming principle (DPP... 
Comparison principle | Stochastic Perron's method | Nonsmooth verification | Viscosity solutions | SUPER-REPLICATION | MATHEMATICS, APPLIED | stochastic Perron's method | viscosity solutions | nonsmooth verification | AUTOMATION & CONTROL SYSTEMS | comparison principle | VERIFICATION | Construction
Journal Article
Stochastic processes and their applications, ISSN 0304-4149, 2017, Volume 127, Issue 6, pp. 1926 - 1959
This paper is concerned with a class of stochastic Hamilton–Jacobi–Bellman equations with controlled leading coefficients, which are fully nonlinear backward stochastic partial differential equations (BSPDEs for short... 
Non-Markovian control | Weak solution | Potential | Stochastic Hamilton–Jacobi–Bellman equation | Backward stochastic partial differential equation | Stochastic Hamilton-Jacobi-Bellman equation | VISCOSITY SOLUTIONS | PARTIAL-DIFFERENTIAL-EQUATIONS | COEFFICIENTS | BACKWARD SPDES | SDES | STATISTICS & PROBABILITY | REGULARITY THEORY | Differential equations
Journal Article
Journal de mathématiques pures et appliquées, ISSN 0021-7824, 02/2018, Volume 110, pp. 1 - 31
Journal Article
Journal of the European Mathematical Society, ISSN 1435-9855, 2018, Volume 20, Issue 4, pp. 797 - 864
We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient... 
Mean curvature equation | Hamilton-Jacobi equation | Stochastic homogenization | Error estimate | error estimate | MATHEMATICS | VISCOSITY SOLUTIONS | MATHEMATICS, APPLIED | PARTIAL-DIFFERENTIAL-EQUATIONS | MEAN-CURVATURE FLOW | PERIODIC MEDIA | mean curvature equation | PLANE-LIKE MINIMIZERS | SURFACES
Journal Article
SIAM journal on control and optimization, ISSN 1095-7138, 2008, Volume 47, Issue 1, pp. 444 - 475
Journal Article
Communications in Mathematical Sciences, ISSN 1539-6746, 2010, Volume 8, Issue 2, pp. 627 - 637
In this note we revisit the homogenization theory of Hamilton-Jacobi and "viscous"-Hamilton-Jacobi partial differential equations with convex nonlinearities in stationary ergodic environments... 
Hamilton-Jacobi equations | Stochastic homogenization | Viscosity solutions | MATHEMATICS, APPLIED | viscosity solutions
Journal Article