Dirichlet heat kernel estimates for \(\Delta ^{\alpha /2} + \delta ^{\beta /2}\)
Consider a process of the form \(X^a= X+ aY\) in \(\mathbb{R}^d\), \(d\geq 2\), \(X\), \(Y\) independent, \(X\) symmetric \(\alpha\)-stable. \(Y\) symmetric \(\beta\)-stable, \(0< \beta< \alpha< 2\), \(a\geq 0\), the infinitesimal generator of \(X^a\) \(\Delta^{\alpha/2}+ a^\beta\Delta^{\beta/2}\). For any open subset \(D\subset\mathbb{R}^d\), let \(X^{a,D}\) be the subprocess of \(X^a\) identical with \(X^a\) in \(D\) but killed upon leaving \(D\). It is known that \(X^{a,D}\) has a Hölder-continuous transition density \(p^a_D\) respective the Lebesgue measure, see [\textit{Z.-Q. Chen} and \textit{T. Kumagai}, Probab. Theory Relat. Fields 140, No. 1--2, 277--317 (2008; Zbl 1131.60076)]. Suppose \(D\) is a \(C^{1,1}\)-open subset of \(\mathbb{R}^d\), i.e., there are real numbers \(R>0\) and \(\Lambda>0\) such that, for every \(z\in\partial D\), there exists a \(C^{1,1}\)-function \(\phi= \phi_z: \mathbb{R}^{d-1}\to\mathbb{R}\) satisfying \(\phi(0)= 0\), \(\nabla\phi(0)= (0,\dotsc, 9)\), \(\|\nabla\phi \|_\infty\leq\Lambda\), \(|\nabla\phi(x)- \nabla\phi(z)|\leq \Lambda|x- z|\), and an orthonormal coordinate system \(CS_z\) with origin in \(z\) such that \[ B(z,R)\cap D= \{y= (\widetilde y, y_d)\text{ in }CS_z:|y|< R,\,y_d> \phi(\widetilde y)\}, \] where \(B(z,R)\) denotes the open Euclidean ball with radius \(R\) centered at \(z\). Let \(M>0\) and denote by \(\delta_D(x)\) the Euclidean distance between \(x\) and the complement of \(D\). Then, for every \(T> 0\), there is a constant \(C_1= C_1(R,\Lambda, M,\alpha,\beta, T,d)\geq 1\) such that \[ C^{-1}_1 f^a_D(t, x,y)\leq p^a_D(t,x,y)\leq C_1 f^a_D(t,x,y), \] for every \(a\in(0, M]\), where \[ f^a_D(t,x,y):= \Biggl(1\wedge {\delta_D(x)^{\alpha/2}\over\sqrt{t}}\Biggr)\Biggl(1\wedge{\delta_D(y)^{\alpha/2}\over \sqrt{t}}\Biggr)\times \Biggl(t^{-d/\alpha}\wedge \Biggl({t\over|x- y|^{d+\alpha}}+ {a^\beta t\over |x-y|^{d+\beta}}\Biggr)\Biggr). \] If, in addition, \(D\) is bounded, then, for every \(T>0\), there is a constant \(C_2\geq 1\), depending only on \(\operatorname{diam}(D)\), \(R\), \(\Lambda\), \(M\), \(\alpha\), \(\beta\), and \(T\), so that, for every \(a\in(0, M]\) and \((t,x,y)\in[T,\infty)\times D\times D\), \[ C^{-1}_2 e^{-\lambda_1 t}\delta_D(x)^{\alpha/2} \delta_D(y)^{\alpha/2}\leq p^a_D(t, x,y)\leq C_2 e^{-\lambda_1 t}\delta_D(x)^{\alpha/2} \delta_D(y)^{\alpha/2}, \] where \(\lambda_1> 0\) is the smallest eigenvalue of \(-(\Delta^{\alpha/2}+ a^\beta\Delta^{\beta/2})|_D\). Integration of these estimates leads to estimates for the Green function \[ G^a_D(x,y)= \int^\infty_0 p^a_D(t,x,y)\,dt, \] and from these a uniform boundary Harnack principle for \(X^a\) in \(D\) with explicit decay rate is obtained.
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