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Summary form only given. We consider a self-defocusing model for a Kerr medium in a planar resonator, which takes into account the vectorial character of the radiation field. For an intensity of the pump beam larger than a given threshold, a y polarized transverse stripe pattern appears. We analyze the spatial behavior of quantum fluctuations around this inhomogeneous steady state, using a Langevin treatment based on Wigner representation. The model is continuous, so that we avoid any restriction to a reduced number of spatial modes. The spatial distribution of the quantum fluctuations around the roll-pattern is dominated by the neutral (or Goldstone) mode, corresponding to rigid spatial displacements of the pattern. The spatial configuration of the field immediately outside the cavity input/output mirror depends on the time window over which fluctuations are averaged: only when the time window is on the order of the cavity lifetime the output field fluctuations are qualitatively similar to that of the intracavity field. We show the existence of a strong anticorrelation between the quantum fluctuations of the intensity of the x-polarized pump and the y-polarized field. We check the possibility to use a Kerr cavity like a quantum non-demolition device, which uses the tilted waves, corresponding in the far field to the y-polarized stripe pattern, as a meter to measure the intensity fluctuation of the pump beam.