Initial conditions and degrees of freedom of non-local gravity

We prove the equivalence between non-local gravity with an arbitrary form factor and a non-local gravitational system with an extra rank-2 symmetric tensor. Thanks to this reformulation, we use the diffusion-equation method to transform the dynamics of renormalizable non-local gravity with exponential operators into a higher-dimensional system local in spacetime coordinates. This method, first illustrated with a scalar field theory and then applied to gravity, allows one to solve the Cauchy problem and count the number of initial conditions and of non-perturbative degrees of freedom, which is finite. In particular, the non-local scalar and gravitational theories with exponential operators are characterized by, respectively, two and four initial conditions in any dimension and, respectively, by one and eight degrees of freedom in four dimensions. The fully covariant equations of motion are written in a form convenient to find analytic non-perturbative solutions.