We investigate the limiting behavior of invariant measures in terms of weak convergence and the Wasserstein metric of probability measures for a stochastic fractional Ginzburg-Landau equation defined on the entire space \(\mathbb {R}^{n}\) , where the growth rate \(2\beta + 1\) and the coefficient \(\varpi \) of the drift term satisfy an admissible condition \(|\varpi | \le \sqrt{2\beta + 1}/\beta \) , as in Okazawa and Yokota [35]. Under the assumption of local Lipschitz continuity of the diffusion term, we show that the family of invariant measures \(\{\mathcal {S}^{\varepsilon }\}_{\varepsilon \in [0,1]}\) is tight. Moreover, we show that every weak limit point of invariant measures \(\mu ^{\varepsilon _n} \in \mathcal {S}^{\varepsilon _n}\) with \(\varepsilon _n \rightarrow \varepsilon _0 \in [0,1]\) must be an invariant measure of the limit equation. When the diffusion term is globally Lipschitz continuous, we discuss a stronger convergence of \(\mu ^{\varepsilon } \in \mathcal {S}^{\varepsilon }\) and \(\mu ^{\varepsilon _0} \in \mathcal {S}^{\varepsilon _0}\) as \(\varepsilon \rightarrow \varepsilon _0 \in [0,1]\) under the Wasserstein metric of probability measures, and prove that the Wasserstein distance between \(\mu ^{\varepsilon }\) and \(\mu ^{\varepsilon _0}\) is bounded by \(c |\varepsilon - \varepsilon _0|\) for some constant \(c > 0\) independent of \(\varepsilon \) . The main difficulty caused by the lack of compact Sobolev embeddings on unbounded domains, which obstructs the proof of tightness for \(\bigcup _{\varepsilon \in [0,1]}\mathcal {S}^\varepsilon \) , is handled by establishing uniform tail estimates for solutions outside sufficiently large balls. The challenge of proving convergence in probability and expectation of the solutions is addressed by carefully analyzing the global monotonicity of the drift term using the condition \(|\varpi | \le \sqrt{2\beta + 1}/\beta \) . Our results also extend to the case where the fractional Laplace operator \((-\varDelta )^{\alpha }\) reduces to the classical Laplacian operator.
