Assume the Dirichlet form $\varepsilon$ adimits a Carre du champ $\Gamma$ and introduce the multilinear form $\Gamma_2$ $$ \Gamma_2 [f,g;\phi]:=\frac12 \int_X (\Gamma (f,g)L\phi -(\Gamma(f,Lg)+\Gamma(g,Lf))\phi) dm \quad (f,g,\phi)\in D(\Gamma_2) $$ where $D(\Gamma_2):=D_V(L)\times D_V(L)\times D_L^\infty(L)$,$V = \{ f:\varepsilon (f) < \infty \} $ and $$ D_V(L)=\{f\in D(L): Lf \in V\},\quad D_L^\infty(L):=\{\phi \in D(L)\cap L^\infty(X,m):L\phi \in L^\infty(X,m)\}. $$ We say that the strongly local Dirichlet form $\varepsilon$ satisfies the $BE(K,\infty)$($BE(K,N)$) condition, if it admits a Carre du Champ $\Gamma$ and $$ \Gamma_2 [f,f;\phi] \ge K\int_X \Gamma(f)\phi dm \quad \mbox{for every}\quad (f,\phi)\in D(\Gamma_2),\quad \phi \ge 0.\qquad (BE(K,\infty)) $$ $$ \Gamma_2 [f,f;\phi] \ge K\int_X (\Gamma(f) +\frac{1}{N} (Lf)^2) \phi dm \quad (f,\phi)\in D(\Gamma_2), \phi \ge 0.\qquad (BE(K,N)) $$
for Alexandrov spaces with curvature bounded below, do they satisfy $BE(K,N)$ and $BE(K,\infty)$?
Since Alexandrov spaces satisy $CD(K,N)$ which is equivalent to RCD(K,N). And $RCD(K,\infty)$ implies $BE(K,\infty)$, so Alexandrov spaces satisfy $BE(K,\infty)$.
But I don't know the relationship between $RCD(K,N)$ and $BE(K,N)$, so I don't know whether satisfy $BE(K,N)$.
