All Questions

Let $\mathcal M_2$ be the space of real $2\times 2$ matrices and $\mathcal S_2\subset \mathcal M_2$ be its subset consisting of positive semidefinite elements, i.e. $A\in \mathcal S_2$ iff $A$ is ...

Let $\alpha=(\alpha_1,\ldots,\alpha_m)\subset\mathbb R^m_+$ and $\beta=(\beta_1,\ldots,\beta_n)\subset\mathbb R^n_+$ be given and satisfy $$\sum_{i=1}^m \alpha_i =1 = \sum_{j=1}^n\beta_j.$$ Define $\...

I learned from the gradient flow theory in Wasserstein space that an equation of gradient flow type $$\partial_t \rho + \nabla \cdot (\rho \nabla \frac{\delta F}{\delta \rho})=0,$$ can be derived as ...

Let $f$ be a probability density function (positive such that $\int_{\mathbb{R}} f(x) \mathrm{d} x = 1$) and $X_0 = \{x_n\}_{1\leq n \leq N}$ be $N$ given real points. We also fix $1 \leq P \leq N$ ...

Let $\mu(dx)=\sum_{i=1}^np_i\delta_{x_i}(dx)$ and $\nu(dy)=\rho(y)dy$ be two probability measures on $\mathbb R^d$. Consider the $2-$Wasserstein distance below: $$W_2(\mu,\nu)^2 \quad := \quad \inf_{\...

Let $c:\mathbb R^2\to\mathbb R$ be a Lipschitz and bounded function (which can be supposed as "nice" as possible). Let $\mu$ and $\nu$ be two probability measures on $\mathbb R$ with finite first ...