Periodic Homogenization of Deterministic Control Problems via Limit Occupational Measures

Periodic homogenization for weakly elliptic Hamilton-Jacobi-Bellman equations with critical fractional diffusion

“…with H is as in (7). If u 0 ∈ Li p(R d ), then u is Lipschitz continuous with respect to x uniformly on [0, T ], satisfying estimate (10) with a Lipschitz constant depending only on α, u ∞ , and on the constants C f ,C b ,C K , but is independent of β, γ.…”

“…) are respectively a bounded viscosity subsolution and a bounded viscosity supersolution of equation (7), such that u…”

“…Related to this, it would be interesting to describe the effective problem in terms of an associated optimal control problem. This has been addressed in the deterministic case via the so-called limit occupational measures, see [7,37] and references therein.…”

Periodic Homogenization for Weakly Elliptic Hamilton-Jacobi-Bellman Equations with Critical Fractional Diffusion

“…with H is as in (7). If u 0 ∈ Li p(R d ), then u is Lipschitz continuous with respect to x uniformly on [0, T ], satisfying estimate (10) with a Lipschitz constant depending only on α, u ∞ , and on the constants C f ,C b ,C K , but is independent of β, γ.…”

“…Let (K a ) a∈A be a family of kernels satisfying (K 1) and(K 3) with γ ∈ ( 1 2 , 1]. If u ∈ U SC (R d ) and v ∈ LSC (R d) are respectively a bounded viscosity subsolution and a bounded viscosity supersolution of equation(7),such that u ∈ C 0,1 (R d ) or v ∈ C 0,1 (R d ), then u ≤ v on R d .Proof. We argue by contradiction and assume that M := sup x∈R d (u(x) − v(x)) > 0.…”