Linear inverse problems for Markov processes and their regularisation

We study the solutions of the inverse problem $$g(z)=\int f(y) P_T(z,dy)$$ for a given $g$, where $(P_t(\cdot,\cdot))_{t \geq 0}$ is the transition function of a given Markov process, $X$, and $T$ is a fixed deterministic time, which is linked to the solutions of the ill-posed Cauchy problem $$u_t + A u=0, \qquad u(0,\cdot)=g,$$ where $A$ is the generator of $X$. A necessary and sufficient condition ensuring square integrable solutions is given. Moreover, a family of regularisations for the above problems is suggested. We show in particular that these inverse problems have a solution when $X$ is replaced by $\xi X + (1-\xi)J$, where $\xi$ is a Bernoulli random variable, whose probability of success can be chosen arbitrarily close to $1$, and $J$ is a suitably constructed jump process.