Another related area of application of the probabilistic methods where some significant progress has recently made is in the field of matrix functions. In fact, in the specific case of the action of a matrix exponential over a vector, we proposed recently a probabilistic method based on a multilevel Monte Carlo method, which as an important feature improves notably the typical slow convergence rate of any Monte Carlo method. The method is based on generating random paths, which evolve through the indices of the matrix, governed now by a suitable continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional.     The multilevel method for the matrix exponential we showed that can be even competitive against the classical deterministic methods based on Krylov subspace methods. Specifically, for large scale problems and extremely large number of processors, the Monte Carlo method clearly outperforms the deterministic method for solving problems consisting in large matrices, not only in terms of computational time, but also in terms of memory requirements. The main advantages of the probabilistic methods are mainly due to its privileged computational features, such as simplicity to code and parallelize. This in practice allows us to develop parallel codes with extremely low communication overhead among processors, having a positive impact in parallel features such as scalability and fault-tolerance. Furthermore, there is also another distinguishing aspect of the method, which is the capability of computing the solution of the problem at specific chosen points, without the need for solving globally the entire problem.