A stochastic Hamilton-Jacobi equation with infinite speed of propagation (Q517495)

Consider the Hamilton-Jacobi equation with rough time dependence \[ \partial_t u = \big( |\partial_x u| - |\partial_y u| \big) \dot{\xi}(t) \quad \text{on } (0, T) \times \mathbb{R}^2 \] for an arbitrary \(\xi \in C([0, T])\), and let \(u : [0, T] \times \mathbb{R}^2 \to \mathbb{R}\) be the viscosity solution with initial condition \[ u(0, x, y) = |x - y| + \Theta(x, y), \] where \(\Theta \geq 0\) is such that \(\Theta(x, y) \geq 1\) if \(\min \{ x, y \} \geq R\). As a consequence of the main result of the present paper, one has \(u(T, 0, 0) > 0\) as soon as \(V_{0, T}(\xi) > R\), where \(V_{0, T}(\xi)\) denotes the total variation of \(\xi\). The proof of this result is based on a differential game associated with the Hamilton-Jacobi equation. This shows that the finite speed of propagation property for Hamilton-Jacobi equations does not transfer to Hamilton-Jacobi equations with rough time dependence. More precisely, it is known that for Hamilton-Jacobi equations \[ \partial_t u = H(D u) \quad \text{on } (0, T) \times \mathbb{R}^N \] with a \(C\)-Lipschitz function \(H : \mathbb{R}^N \to \mathbb{R}\) and two (viscosity) solutions \(u^1\) and \(u^2\) with \(u^1(0,\cdot) = u^2(0,\cdot)\) on \(B(R)\) one has \[ u^1(t,\cdot) = u^2(t,\cdot)\quad \text{ on } B(R - Ct) \quad \text{for all \(t \geq 0\).} \] Choosing a function \(\xi\) of unbounded variation, the main result of the paper shows that this property is no longer true for a Hamilton-Jacobi equation with rough time dependence of the form \[ \partial u_t = H(D u)\dot{\xi}(t) \quad \text{on } (0, T) \times \mathbb{R}^N. \]