  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.
