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Closed elliptic space-time jets test the parabolic diagonal HJB inequality

Statement

Let $u$ be a USC viscosity subsolution in the sense of Second-order parabolic viscosity subsolutions and supersolutions, and write the local space-time variable as $z = (t, x)$ . If $((a, p), H) \in {\overline{J}}^{2, +} u (t, x)$ is a closed elliptic jet in all $d + 1$ variables (Second-order elliptic semijets and their value-sensitive closures), and $X$ is the spatial principal block of $H$ , then $a - \max_{c \in A} {\sum_{j} (α_{j}^{c} (x, t) X_{j j} + b_{j}^{c} (x, t) p_{j}) + f^{c} (x, t)} \leq 0.$ For an LSC supersolution and a closed elliptic subjet the reverse inequality holds. All points are interior in time; local periodic lifts are understood.

Facts & Assumptions

Given: The stated viscosity subsolution or supersolution and jet.

[F1]

An ordinary elliptic superjet $(q, H)$ means an upper expansion by $u (z) + q \cdot h + \frac{1}{2} h^{T} H h + o (∣ h ∣^{2})$ . A closed jet is the limit of ordinary jets at points $z_{n} \to z$ , with function values and jet entries converging. Subjets have the reversed expansion (Second-order elliptic semijets and their value-sensitive closures).

[F2]

Every $C^{2}$ space-time polynomial is a $C^{1, 2}$ admissible test; the diagonal HJB residual is continuous in the point, time derivative, spatial gradient and spatial Hessian (Second-order parabolic viscosity subsolutions and supersolutions, Periodic diagonal controlled-diffusion HJB Cauchy problem).

Proof

1.1F1F2givenalgebra

First let the superjet be ordinary. For each $θ > 0$ , the definition of the remainder in [F1] gives a neighborhood where the polynomial $u (z) + q \cdot h + \frac{1}{2} h^{T} (H + θ I) h$ lies above $u (z + h)$ and agrees at $h = 0$ . It is an admissible upper test by [F2]. Its tested derivatives are $a, p, X + θ I_{d}$ , so the subsolution inequality holds for these entries. Continuity in [F2] and $θ ↓ 0$ give the stated inequality for $a, p, X$ .

2.1F1F2step 1.1algebra∎

For a closed superjet, apply step 1.1 at every interior point in its witnessing sequence from [F1]. All sufficiently late points are interior; coefficient continuity and convergence of the finite jet entries let the inequalities pass to the limit by [F2]. For an ordinary subjet, the polynomial with $H - θ I$ lies below the function locally; use the lower-test inequality and let $θ ↓ 0$ . Passing along the closed-subjet witnessing sequence proves the supersolution assertion.

Depends on

Used by

Comparison for periodic diagonal controlled-diffusion HJB equations Theorem

Dependency tree · two levels

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Sources

Crandall, Ishii and Lions, User's guide to viscosity solutions (standard reference, not scraped)