Ricerc@Sapienza

Stochastic modelling of fatigue (and other material's deterioration), as well as of cumulative damage in risk theory, are often based on compound sums of independent random variables, where the number of addends is represented by an independent counting process. We consider here a cumulative model where, instead of a renewal process (as in the Poisson case), a linear birth (or Yule) process is used. This corresponds to the assumption that the frequency of “damage” increments accelerates according to the increasing number of “damages”.

On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator
$\partial_t-\triangle+\int_0^t K(x,t,r)\triangle\ dr$
where the integral kernel has a behaviour like a weakly singular one.
In the proof we consider the integral term as a perturbation. The crucial point is
a special choice of the time factor of the weight function.