Abstract: I'll introduce a metric Hopf-Lax formula w.r.t. generalized distances, looking in particular to the Carnot-Carathéodory case. Then, starting from lower semicontinuous functions, I'll study some properties for the relative Hopf-Lax function, showing in particular the convergence to the original function, as , some locally lipschitz properties and a classic link with minimum problems of calculus of variation.

Moreover, solving a suitable generalized eikonal equation, it is possible to use the Hopf-Lax function to give an existence result for the associated Hamilton-Jacobi-Cauchy problem.

A particular interesting case of the metric Hopf-lax function is the Carnot-Carathéodory inf-convolution. At last I'll show convergence result for C-C inf-convolutions applying the Large Deviation Principle to some hypoelliptic operators.