Heat kernel estimates for $\mathrm{\Delta }+{\mathrm{\Delta }}^{\alpha \mathrm{/}2}$ under gradient perturbation

For $d\ge 2$, $\alpha \in (0,2)$ and $M>0$, we consider the gradient perturbation of a family of nonlocal operators $\{\mathrm{\Delta }+a^{\alpha }{\mathrm{\Delta }}^{\alpha \mathrm{/}2},a\in (0,M]\}$. We establish the existence and uniqueness of the fundamental solution $p(t,x,y)$ for \begin{equation*} $\mathcal{L}$^{a,b} = \Delta+a^\alpha\Delta^{\alpha/2} + b\cdot \nabla, \end{equation*} where $b$ is in Kato class ${\mathbb{K}}_{d,1}$ on ${\mathbb{R}}^d$. We show that $p(t,x,y)$ is jointly continuous and derive its sharp two-sided estimates. The kernel $p(t,x,y)$ determines a conservative Feller process $X$. We further show that the law of $X$ is the unique solution of the martingale problem for $({\mathcal{L}}^{a,b},C_c^{\mathrm{\infty }}({\mathbb{R}}^d)$ and $X$ can be represented as $$X_t=X_0+Z_t^a+{\int }_0^{t}b(X_s)ds,t\ge 0,$$ where $Z_t^a=B_t+a{Y}_t$ for a Brownian motion $B$ and an independent isotropic $\alpha$-stable process $Y$. Moreover, we prove that the above SDE has a unique weak solution.