具有斜Robin边界条件的Hamilton--Jacobi--Bellman方程的保正性期望格式
A Positivity-Preserving Expectation Scheme for Hamilton--Jacobi--Bellman Equations with Oblique Robin Boundary Conditions
AI总结：
本文针对带混合导数的各向异性扩散问题，构造了一种保正性的Hamilton--Jacobi--Bellman方程数值格式，该格式无需对角占优条件和CFL关系即可保持正性，且适用于斜Robin边界条件。
AI中文摘要：
对于带有混合导数的各向异性扩散，标准的紧致坐标对齐模板以及一些局部有限体积或有限元构造，若不施加合适的系数或网格条件，会丢失非负系数。本文中，非负系数直接由条件期望生成，而非通过代数模板分解。我们为可能退化的Hamilton--Jacobi--Bellman方程构造了保正性格式，该方程带有可控的斜Robin边界条件。在内部节点处，构造源于一步反射Feynman--Kac恒等式：一个m维Rademacher向量生成P=2^m个等概率的弱Euler分支，外部分支关于其斜投影镜像，其过冲的两倍作为离散边界局部时间D。Robin系数仅通过衰减因子e^{-κD}和De^{-κD/2}g进入，保持等分支概率不变。边界节点使用空间偏移ℓ_h≍h的单独同层闭包，可为隐式；均匀斜度与P_1插值给出内部节点上与网格无关的正权重，因此固定线性Robin数据产生稀疏非奇异M矩阵系统，而一般控制集给出单调收缩。该格式在无对角占优条件、无Δt与h之间任何CFL型关系的情况下，对非负数据保持正性。
英文摘要：
For anisotropic diffusion with mixed derivatives, standard compact coordinate-aligned stencils and some local finite-volume or finite-element constructions can lose nonnegative coefficients unless suitable coefficient or mesh conditions are imposed. Here nonnegative coefficients are generated directly from conditional expectation rather than from an algebraic stencil decomposition. We construct a positivity-preserving scheme for possibly degenerate Hamilton--Jacobi--Bellman equations with controlled oblique Robin boundary conditions. At interior nodes the construction stems from a one-step reflected Feynman--Kac identity: an $m$-dimensional Rademacher vector generates $P=2^m$ equally probable weak-Euler branches, an exterior branch is mirrored about its oblique projection, and twice its overshoot serves as the discrete boundary local time $D$. The Robin coefficient enters only through the attenuation factors $e^{-κD}$ and $De^{-κD/2}g$, leaving the equal branch probabilities unchanged. Boundary nodes use a separate same-level closure over a spatial offset $\ell_h\asymp h$, which may be implicit; uniform obliqueness and $\mathbb P_1$ interpolation give a mesh-independent positive weight on interior nodes, so fixed linear Robin data yield a sparse nonsingular $M$-matrix system and the general control set yields a monotone contraction. Positivity is preserved for nonnegative data without a diagonal-dominance condition and without any CFL-type relation between $Δt$ and $h$.
