LemmaStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Initial trace for the explicit diagonal-diffusion HJB scheme

Statement

Let $u_{0}$ be continuous and periodic, and use the explicit diagonal-diffusion scheme of Explicit monotone diagonal-diffusion Bellman grid update with mesh widths $h_{k, j} \to 0$ , time steps $τ_{k} \to 0$ , and its diffusion-dominance and CFL inequalities at every row. Then for every sequence of integers $n_{k} \geq 0$ with $t_{k} = n_{k} τ_{k} \to 0$ and $t_{k}$ within the computed horizon, $\sup_{i} ∣ U_{i}^{n_{k}} - u_{0} (x_{i}) ∣ ⟶ 0.$ The estimate is uniform over all finite control choices generated by the maximization in the scheme.

Facts & Assumptions

Given: The grid scheme, bounded coefficients of Periodic diagonal controlled-diffusion HJB Cauchy problem, continuous periodic initial data, and row conditions from the statement. For any finite transition tree, expectation means the finite sum of endpoint values weighted by products of row coefficients.

[F1]

The update expands using the central second difference and sign-selected one-sided drift; its row conditions are $∣ b_{j}^{a} ∣ h_{j} \leq α_{j}^{a}$ and $τ_{k} \sum_{j} (2 α_{j}^{a} / h_{j}^{2} + ∣ b_{j}^{a} ∣ / h_{j}) \leq 1$ (Explicit monotone diagonal-diffusion Bellman grid update).

[F2]

There are finite constants $B$ , $F$ , and $A_{*}$ with $∣ b^{a} (x, t) ∣ \leq B$ , $∣ f^{a} (x, t) ∣ \leq F$ , and $α_{j}^{a} (x, t) \leq A_{*}$ for every control and point; $u_{0}$ is continuous and periodic (Periodic diagonal controlled-diffusion HJB Cauchy problem).

[F3]

The periodic lift of $u_{0}$ is uniformly continuous on every compact cube; finitely many adjacent periodic lifts therefore share a modulus $ω_{0} (δ) \to 0$ (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line, Heine-Cantor: a continuous map from a compact metric space to any metric space is uniformly continuous, Periodic Cartesian grids and tensor-product multilinear interpolation).

[F4]

The nodes and coordinate neighbors use the periodic Cartesian convention of Periodic Cartesian grids and tensor-product multilinear interpolation.

Proof

1.1F1F2F4givenalgebra

Fix one control row and abbreviate $α = α_{j}^{a}$ , $b = b_{j}^{a}$ , $h = h_{j}$ . If $b \geq 0$ , the two neighbor transition weights in coordinate $j$ are $τ (α / h^{2} - b / h)$ at $x_{i} - h e_{j}$ and $τ α / h^{2}$ at $x_{i} + h e_{j}$ . If $b < 0$ , they are $τ α / h^{2}$ at $x_{i} - h e_{j}$ and $τ (α / h^{2} - ∣ b ∣ / h)$ at $x_{i} + h e_{j}$ . Their sum is $τ (2 α / h^{2} - ∣ b ∣ / h)$ and their signed first moment is $τ b$ . The diffusion-dominance inequality in [F1] makes both neighbor coefficients nonnegative. The center coefficient is $1 - τ \sum_{j} (2 α_{j}^{a} / h_{j}^{2} - ∣ b_{j}^{a} ∣ / h_{j})$ , which is nonnegative by the CFL inequality in [F1]. Including the center, the row weights sum to one. Summing over coordinates gives mean displacement $τ b^{a} (x_{i}, t_{n})$ . The raw squared displacement is at most $τ \sum_{j} (2 α_{j}^{a} (x_{i}, t_{n}) - ∣ b_{j}^{a} (x_{i}, t_{n}) ∣ h_{j}) \leq 2 τ d A_{*} .$ Subtracting the squared mean cannot increase the second moment.

2.1F1F2step 1.1algebra

Unroll $n_{k}$ updates as a finite transition tree. At a node of the tree, the maximum in the scheme chooses one of finitely many control rows; that row is a finite probability distribution by step 1.1. The terminal value is thus a maximum of finite weighted averages of $u_{0} (X_{n_{k}})$ plus accumulated rewards. On every branch the sum of the drift means has magnitude at most $B t_{k}$ . The centered increments have conditional weighted mean zero. For two different times, the weighted cross term vanishes by first summing the later centered increment over all branches extending a fixed earlier path prefix. Expanding the square and using step 1.1 gives, uniformly over all finite policies, $E ∣ X_{n_{k}} - x_{i} ∣^{2} \leq 2 B^{2} t_{k}^{2} + 4 d A_{*} t_{k} .$ The accumulated reward has magnitude at most $F t_{k}$ .

3.1F2F3step 2.1algebra∎

Put $M_{0} = ∥ u_{0} ∥_{\infty}$ . For any $0 < δ < 1$ , split each finite endpoint average according to $∣ X_{n_{k}} - x_{i} ∣ \leq δ$ . On that part the change in $u_{0}$ is at most $ω_{0} (δ)$ by [F3]; on the complement it is at most $2 M_{0}$ . The complement weight is at most $E ∣ X_{n_{k}} - x_{i} ∣^{2} / δ^{2}$ . Therefore every policy average differs from $u_{0} (x_{i})$ by at most $ω_{0} (δ) + \frac{2 M_{0}}{δ^{2}} (2 B^{2} t_{k}^{2} + 4 d A_{*} t_{k}) + F t_{k} .$ This bound is independent of $i$ and of the policy; taking a finite maximum preserves it. First let $k \to \infty$ and then $δ ↓ 0$ to obtain the claimed uniform initial trace.

Depends on

Used by

Half-relaxed limits of the monotone diagonal-diffusion HJB scheme Theorem

Dependency tree · two levels

41 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Michael G. Crandall, Hitoshi Ishii, and Pierre-Louis Lions, User's guide to viscosity solutions of second order partial differential equations (standard reference, not scraped)
Guy Barles and Panagiotis E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations (standard reference, not scraped)