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The stabilization by noise for parabolic equations in perforated domains,i.e. domains with small holes, is investigated. We show that when the holes aresmall enough, one can stabilize the unstable equations using suitablemultiplicative It\^o noise. The results are quantitative, in the sense that wecan explicitly estimate the size of the holes and diffusion coefficients forwhich stabilization by noise takes place. This is done by using the asymptoticbehaviour of the first eigenvalue of the Laplacian as the hole shrinks to apoint.

Stabilization by Multiplicative It\^o Noise for Chafee-Infante Equation in Perforated Domains