Differential Equations with Infinitely Many Derivatives and the Borel Transform

Differential equations with infinitely many derivatives, sometimes also referred to as “nonlocal” differential equations, appear frequently in branches of modern physics such as string theory, gravitation and cosmology. We properly interpret and solve linear equations in this class with a special focus on a solution method based on the Borel transform. This method is a far-reaching generalization of previous studies of nonlocal equations via Laplace and Fourier transforms, see for instance (Barnaby and Kamran, J High Energy Phys 02:40, 2008; Górka et al., Class Quantum Gravity 29:065017, 2012; Górka et al., Ann Henri Poincaré 14:947–966, 2013). We reconsider “generalized” initial value problems within the present approach and we disprove various conjectures found in modern physics literature. We illustrate various analytic phenomena that can occur with concrete examples, and we also treat efficient implementations of the theory. © 2015, Springer Basel.