Lasry-Lions regularization and a Lemma of Ilmanen
نویسنده
چکیده
inf u 6 Ttu(x) 6 u(x) 6 Ťtu(x) 6 supu for each t > 0 and each x ∈ H. A function u : H −→ R is called k-semi-concave, k > 0, if the function x −→ u(x)−‖x‖2/k is concave. The function u is called k-semi-convex if −u is k-semiconcave. A bounded function u is t-semi-concave if and only if it belongs to the image of the operator Tt, this follows from Lemma 1 and Lemma 3 below. A function is called semi-concave if it is k-semi-concave for some k > 0. A function u is said C1,1 if it is Frechet differentiable and if the gradient of u is Lipschitz. Note that a continuous function is C1,1 if and only if it is semi-concave and semi-convex, see Lemma 6. Let us recall two important results in that language:
منابع مشابه
Probabilistic Analysis of Mean-Field Games
The purpose of this paper is to provide a complete probabilistic analysis of a large class of stochastic differential games with mean field interactions. We implement the Mean-Field Game strategy developed analytically by Lasry and Lions in a purely probabilistic framework, relying on tailor-made forms of the stochastic maximum principle. While we assume that the state dynamics are affine in th...
متن کاملA uniqueness lemma with applications to regularization and incompressible fluid mechanics
In this paper, we extend our previous result from [24]. We prove that transport equations with rough coefficients do possess a uniqueness property, even in the presence of viscosity. Our method relies strongly on duality and bears a strong resemblance with the wellknown DiPerna-Lions theory first developed in [13]. This uniqueness result allows us to reprove the celebrated theorem of J. Serrin ...
متن کاملDiscrete mean field games
In this paper we study a mean field model for discrete time, finite number of states, dynamic games. These models arise in situations that involve a very large number of agents moving from state to state according to certain optimality criteria. The mean field approach for optimal control and differential games (continuous state and time) was introduced by Lasry and Lions [LL06a, LL06b, LL07]. ...
متن کامل