Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix, means $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of $\mu^\epsilon$ a probability measure on $\mathbb{R}^d$ such that its second moment matrix $S_{\mu^\epsilon}=S_\mu+\epsilon I_d$ with ($\epsilon >0$) and $$W_2^2(\mu,\mu^\epsilon)\leq \epsilon$$ where $W_2^2(.,.)$ is the 2-Wasserstein distance $$W_2^2(\mu,\nu)=\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^d}\Vert x-y\Vert^2d\gamma(x,y)$$ Where $\Gamma(\mu,\nu)$ is the set of coupling of $\mu$ and $\nu$

Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix: $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of a probability measure $\mu^\epsilon$ on $\mathbb{R}^d$ such that its second moment matrix $S_{\mu^\epsilon}=S_\mu+\epsilon I_d$ with ($\epsilon >0$) and $$W_2^2(\mu,\mu^\epsilon)\leq \epsilon,$$ where $W_2^2(.,.)$ is the 2-Wasserstein distance $$W_2^2(\mu,\nu)=\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^d}\Vert x-y\Vert^2d\gamma(x,y),$$ where $\Gamma(\mu,\nu)$ is the set of coupling of $\mu$ and $\nu$.

Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix, means $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of $\mu^\epsilon$ a probability measure on $\mathbb{R}^d$ such that its second moment matrix $S_{\mu^\epsilon}=S_\mu+\epsilon I_d$ with ($\epsilon >0$) and $$W_2^2(\mu,\mu^\epsilon)\leq \epsilon$$ where $W_2^2(.,.)$ is the 2-Wasserstein distance

Let $\mu$ be a probability measure on $\mathbb{R}^d$ such that $S_\mu$ is its second moment matrix, means $$S_\mu=\int_{\mathbb{R}^d}xx^Td\mu(x)$$ I'm trying to prove the existence of $\mu^\epsilon$ a probability measure on $\mathbb{R}^d$ such that its second moment matrix $S_{\mu^\epsilon}=S_\mu+\epsilon I_d$ with ($\epsilon >0$) and $$W_2^2(\mu,\mu^\epsilon)\leq \epsilon$$ where $W_2^2(.,.)$ is the 2-Wasserstein distance $$W_2^2(\mu,\nu)=\inf_{\gamma\in\Gamma(\mu,\nu)}\int_{\mathbb{R}^d}\Vert x-y\Vert^2d\gamma(x,y)$$ Where $\Gamma(\mu,\nu)$ is the set of coupling of $\mu$ and $\nu$