We consider the multivariate risk model with common renewal process among the lines of business, and Brownian perturbations. Assuming that the integrated tail distribution of claims is multivariate subexponential, we establish an asymptotic relation for the infinite-time ruin probability. A more explicit expression is given in case of claim distribution from multivariate regular variation. The results indicate the insensitivity of the asymptotic behavior of the ruin probability with respect to Brownian perturbations. Furthermore, we show that a multivariate distribution with finite expectation, that belongs to the class of multivariate dominatedly varying and long-tailed distributions, possesses integrated tail distribution from the class of multivariate subexponential distributions, which makes easier the checking of conditions in the theorem.