  The identification of the progenitors of Type Ia Supernovae (SNIa) is
extremely important in several astrophysical contexts, ranging from stellar
evolution in close binary systems to evaluating cosmological parameters.
Determining the distribution of the delay times (DTD) of SNIa progenitors can
shed light on their nature. In this paper we investigate on the diagnostic
capabilities on the DTD of the correlation between the SNIa rate and the parent
galaxy properties by examining its systematics with the various parameters at
play: simple stellar population models, the adopted description for the star
formation history in galaxies, and the way in which the masses of the galaxies
are evaluated. We compute models for the correlations of the SNIa rate with the
parent galaxy color and specific star formation rate for a variety of input
ingredients, and for a few astrophysically motivated DTD laws. The models are
compared to the results of three independent observational surveys. We find
that the scaling of the SNIa rate with the properties of the parent galaxy is
sensitive to all input ingredients mentioned above. This is a severe limitation
on the possibility to discriminate alternative DTDs. In addition, current
surveys show some discrepancies for the rate measured in the reddest and bluest
galaxies, likely due to limited statistics and inhomogeneity of the
observations. For galaxies with intermediate colors the rates are in agreement,
leading to a robust determination of the productivity of SNIa from stellar
populations of $\simeq$ 0.8 events per 1000 \msun. Large stastistics of SNIa
events along with accurate measurements of the star formation history in the
galaxies are required to derive firm constraints on the DTD. LSST will achieve
these results by providing the homogeneous, unbiased and vast database on both
SNIa and galaxies.
