Scuola Normale Superiore

Pisa, Italy

**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.