Heat kernels for reflected diffusions with jumps on inner uniform domains

Abstract

In this paper, we study sharp two-sided heat kernel estimates for a large class of symmetric reflected diffusions with jumps on the closure of an inner uniform domain D D in a length metric space. The length metric is the intrinsic metric of a strongly local Dirichlet form. When D D is an inner uniform domain in the Euclidean space, a prototype for a special case of the processes under consideration is symmetric reflected diffusions with jumps on D D , whose infinitesimal generators are non-local (pseudo-differential) operators L \mathcal {L} on D D of the form \[ L u ( x ) = 1 2 ∑ i , j = 1 d ∂ ∂ x i ( a i j ( x ) ∂ u ( x ) ∂ x j ) + lim ε ↓ 0 ∫ { y ∈ D : ρ D ( y , x ) > ε } ( u ( y ) − u ( x ) ) J ( x , y ) d y \mathcal {L} u(x)\! =\!\frac 12 \!\sum _{i, j=1}^d\! \frac {\partial }{\partial x_i}\! \left (\!\!a_{ij}(x) \frac {\partial u(x)}{\partial x_j}\!\right ) \!+ \lim _{\varepsilon \downarrow 0}\! \int _{\{y\in D: \, \rho _D(y, x)>\varepsilon \}}\!\! (u(y)-u(x)) J(x, y)\, dy \] satisfying “Neumann boundary condition”. Here, ρ D ( x , y ) \rho _D(x,y) is the length metric on D D , A ( x ) = ( a i j ( x ) ) 1 ≤ i , j ≤ d A(x)=(a_{ij}(x))_{1\leq i,j\leq d} is a measurable d × d d\times d matrix-valued function on D D that is uniformly elliptic and bounded, and \[ J ( x , y ) ≔ 1 Φ ( ρ D ( x , y ) ) ∫ [ α 1 , α 2 ] c ( α , x , y ) ρ D ( x , y ) d + α ν ( d α ) , J(x,y)≔\frac {1}{\Phi (\rho _D(x,y))} \int _{[\alpha _1, \alpha _2]} \frac {c(\alpha , x,y)} {\rho _D(x,y)^{d+\alpha }} \,\nu (d\alpha ), \] where ν \nu is a finite measure on [ α 1 , α 2 ] ⊂ ( 0 , 2 ) [\alpha _1, \alpha _2] \subset (0, 2) , Φ \Phi is an increasing function on [ 0 , ∞ ) [ 0, \infty ) with c 1 e c 2 r β ≤ Φ ( r ) ≤ c 3 e c 4 r β c_1e^{c_2r^{\beta }} \le \Phi (r) \le c_3 e^{c_4r^{\beta }} for some β ∈ [ 0 , ∞ ] \beta \in [0,\infty ] , and c ( α , x , y ) c(\alpha , x, y) is a jointly measurable function that is bounded between two positive constants and is symmetric in ( x , y ) (x, y) .