We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on $$\mathbb {R}^n$$. We prove well-posedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To these aims, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions.

中文翻译：

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$$\mathbb {R}^n$$Rn 上一类弱双曲算子的确定性和随机柯西问题

我们研究一类双曲柯西问题，与线性算子和具有多项式有界系数、可变多重性和对合特征的系统相关，全局定义在 $$\mathbb {R}^n$$ 上。我们证明了 Sobolev-Kato 空间中的适定性，在无穷远处失去平滑和衰减。我们还获得了关于奇点传播的结果，就描述温带分布的平滑性和衰减奇点的演变的波前集而言。此外，我们可以证明相关随机柯西问题的随机场解的存在。为了这些目的，我们首先讨论与特征根相关的合适的参数相关傅立叶积分算子族的迭代积分的代数性质，它们参与了基本解决方案的构建。特别是，我们表明，对于这个运算符类，特征的对合性意味着此类表达式的交换属性。