This paper address sensibility theory for dynamic models, linking correlated inputs to observed outputs. Usual estimation methods of Sobol indices are based on the fact that the input variables are independent. We present in this paper a method to overpass this constraint for Gaussian processes of high dimension in a time related framework. A general method exists with very weak hypothesis but computations are quite impossible in high dimension. Our proposition leads to a natural generalization of Sobol indices for time dependent, causal and correlated inputs. The method of estimation is a modification of the pick-freeze scheme. After having detailed the scheme for the general Gaussian case we detailed the case of high dimensional autoregressive model, which can be also associated with state models. We then apply the results to the case of a building model.