Using the well-known Feynmann path integral of quantum mechanics, the author describes a path integral formulation of linear elastostatics based on infinitesmal propagators of local support. In its simplest form, the formulation can be used as a relaxation method of solution by updating displacements up to convergence. This may be advantageous for problems involving a large number of unknowns. On the other hand, by equating the updated displacement field to the actual one, a direct method of solution is obtained which leads to non-symmetric (but sparse and banded) discrete equations. Unlike variational principles, this formulation does not require integration over the whole domain, effectively eliminating the need for a background mesh for integration. Also, the method requires only the continuity of the displacement field on the propagator's support. As a consequence, the formulation lends itself to very flexible meshless implementations. This is demonstrated by a simple numerical method in which displacements around each node are approximated by quadratic bivariate polynomials, which is the simplest approximation technique. The feasibility of the method is assessed through a number of numerical examples and comparisons with analytical solutions and other meshless methods. Finally, it is noted that the path integral formulation described here can be directly applied, with minor changes, to intrinsically diffusive processes such as heat flow in solids, and can easily be adapted to other diffusion-like equations such as Navier-Stokes equations. Its adaptation to other problems as elastodynamics and finite elasticity, may require major modifications.