FUNDAMENTAL SOLUTION FOR SUPER-CRITICAL NON-SYMMETRIC LEVY-TYPE OPERATORS

We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation partial differential t = G kappa for non-symmetric non-local operatorsL kappa f (x) := Rd(f (x + z) - f (x) - 1|z|<1 (z, Vf (x)))kappa(x, z)J(z) dz ,under broad assumptions on kappa and J. Of special interest is the case when the order of the operator G kappa is smaller than or equal to 1. Our approach rests on imposing suitable cancellation conditions on the internal drift coefficient integral(Zz kappa()x, z)J(z)dz , 0 < r < 1 , r <=|z|<1 which allows us to handle the non-symmetry of z 7 -> kappa(x, z)J(z). The results are new even for the 1-stable Le ' vy measure J(z) = |z|-d(-1).