We present two new sharp regularity results (regularizing effect and propagation of regularity) for viscosity solutions of uniformly convex homogeneous (space independent) Hamilton–Jacobi equations. The estimates do not depend on the convexity constants of the Hamiltonians. The sharp propagation of regularity result holds in dimension larger than one without additional smoothness assumptions on the data if and only if the Hamiltonians is quadratic in the gradient, a very surprising fact in the theory of Hamilton–Jacobi equations. In turn, the estimates yield new intermittent stochastic regularization results for pathwise (stochastic) viscosity solutions of Hamilton–Jacobi equations with uniformly convex Hamiltonians and rough multiplicative time dependence. Finally, the intermittent estimates allow for the study of the long time behavior of the pathwise (stochastic) viscosity solutions.

中文翻译：

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Hamilton–Jacobi方程的新规律性结果以及路径（随机）黏性溶液的长期行为

对于均匀凸均质（空间无关）Hamilton–Jacobi方程的粘度解，我们给出了两个新的尖锐的正则结果（正则化和正则性传播）。估计值不依赖于哈密顿量的凸度常数。当且仅当哈密顿量在梯度上为二次方时，正则结果的急剧传播才能保持其尺寸大于1，而无需对数据进行额外的平滑度假设，这在哈密顿–雅各比方程式理论中是一个非常令人惊讶的事实。反过来，对于具有一致凸哈密顿量和粗糙乘法时间相关性的Hamilton–Jacobi方程的路径（随机）粘度解，估计又产生了新的间歇性随机正则化结果。最后，