We show that the initial value problem for Hamilton–Jacobi equations with multiplicative rough time dependence, typically stochastic, and convex Hamiltonians satisfies finite speed of propagation. We prove that in general the range of dependence is bounded by a multiple of the length of the “skeleton” of the path, that is a piecewise linear path obtained by connecting the successive extrema of the original one. When the driving path is a Brownian motion, we prove that its skeleton has almost surely finite length. We also discuss the optimality of the estimate.

中文翻译：

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具有乘法粗糙时间相关性和凸哈密顿量的 Hamilton-Jacobi 方程的传播速度

我们表明，具有乘法粗糙时间相关性（通常是随机和凸哈密顿量）的 Hamilton-Jacobi 方程的初值问题满足有限传播速度。我们证明，一般而言，依赖范围以路径“骨架”长度的倍数为界，即通过连接原始路径的连续极值获得的分段线性路径。当驱动路径是布朗运动时，我们证明它的骨架几乎肯定是有限长度的。我们还讨论了估计的最优性。