具有无穷多个导数的常微分方程的初值问题
The initial value problem for ordinary differential equations with infinitely many derivatives
摘要：
我们研究具有无穷多个导数的常微分方程解的存在性、唯一性和正则性，例如粒子物理学、非局部宇宙学和弦论中出现的非局部场运动方程（的线性化版本）。我们通过拉普拉斯变换开发了一个适当的洛伦兹函数演算，它使我们能够严格解释半直线上形式为$f(\partial_t)$的算子，其中$f$是一个解析函数。我们在指数有界函数空间中找到了方程$f(\partial_t) ϕ= J(t)$（t大于或等于0）的最一般解，并详细分析了初值问题的微妙性.特别是，我们的状态下的解决方案$ϕ$承认有限数量的衍生物的条件下，我们严格证明，如果一个先验数据直接与我们的洛伦兹演算指定，那么初始值问题是适定的，它只需要有限数量的初始条件。
英文摘要：
We study existence, uniqueness and regularity of solutions for ordinary differential equations with infinitely many derivatives such as (linearized versions of) nonlocal field equations of motion appearing in particle physics, nonlocal cosmology and string theory. We develop an appropriate Lorentzian functional calculus via Laplace transform which allows us to interpret rigorously an operator of the form $f(\partial_t)$ on the half line, in which $f$ is an analytic function. We find the most general solution to the equation $f(\partial_t) ϕ= J(t)$ (t greater or equal to 0) in the space of exponentially bounded functions, and we also analyze in full detail the delicate issue of the initial value problem. In particular, we state conditions under which the solution $ϕ$ admits a finite number of derivatives, and we prove rigorously that if an a priori data directly connected with our Lorentzian calculus is specified, then the initial value problem is well-posed and it requires only a finite number of initial conditions.
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