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We study existence, uniqueness and regularity of solutions for ordinarydifferential equations with infinitely many derivatives such as (linearizedversions of) nonlocal field equations of motion appearing in particle physics,nonlocal cosmology and string theory. We develop an appropriate Lorentzianfunctional calculus via Laplace transform which allows us to interpretrigorously 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$f(\partial_t) \phi = J(t)$ (t greater or equal to 0) in the space ofexponentially bounded functions, and we also analyze in full detail thedelicate issue of the initial value problem. In particular, we state conditionsunder which the solution $\phi$ admits a finite number of derivatives, and weprove rigorously that if an a priori data directly connected with ourLorentzian calculus is specified, then the initial value problem is well-posedand it requires only a finite number of initial conditions.

The initial value problem for ordinary differential equations with infinitely many derivatives