Green and Poisson functions with Wentzell boundary conditions
The paper deals with the boundary value problem with a terminal condition \[ \begin{cases}\partial_tu(t,x)+A(t)u(t,x)=f(t,x),\quad \forall t>0,\quad x\in \dot\mathbb R_+^d,\\ B(t)u(t,\widetilde x,0)+g(t,\widetilde x)=0,\quad \forall t>0,\quad \widetilde x\in \mathbb R^{d-1},\\ u(T,x)=\varphi(x),\quad \forall x\in \dot\mathbb R_+^d,\end{cases}\tag{P} \] where \(\dot\mathbb R_+^d=\{x:\,\,x_d>0\}\) and \(A(t)=A_2(t)+I(t)\). Here \(A_2(t)\) is a second order uniformly elliptic differential operator with bounded and Hölder continuous coefficients, of the form \[ \begin{aligned} A_2(t)\varphi(x)&=A_0(t)\varphi(x)+\sum_{i=1}^da_i(t,x)\partial_i \varphi(x)-a_0(t,x)\varphi(x),\\ A_0(t)\varphi(x)&=\frac{1}{2}\sum_{i,j=1}^da_{ij}(t,x) \partial_{ij}\varphi(x),\end{aligned} \] \(I(t)\) is the integro-differential operator \[ I(t)\varphi(x)=\int_{\mathbb R_*^m}[\varphi(x+j(\zeta,t,x))-\varphi(x)-j (\zeta,t,x)\cdot\nabla\varphi(x)]m(\zeta,t,x)\pi(d\zeta), \] and \(B\) is the uniform Wentzell type boundary second order differential operator \[ \begin{aligned} B(t)\varphi(x)&=B_0(t)\varphi(x)+b_d(t,\widetilde x)\partial_d\varphi(x)-\rho(t,\widetilde x)A(t)\varphi(x),\\ B_0(t)&=\frac{1}{2}\sum_{i,j=1}^{d-1}b_{ij}(t,\widetilde x)\partial_{ij}+\sum_{i=1}^{d-1}b_i(t,\widetilde x)\partial_i-b_0(t,\widetilde x).\end{aligned} \] Under some assumptions on the coefficients of the above operators, the authors present the construction (for \(t>0\)) and estimates (for \(t\in [0,T]\), with \(T\) a given positive constant) of the Green and Poisson functions associated to problem (P), in several stages. In the first case (constant coefficients) it is investigated the Dirichlet problem \[ \begin{cases}\partial_tu(t,x)+A_2(t)u(t,x)=f(t,x),\quad \forall t>0,\quad x\in \dot\mathbb R_+^d,\\ u(t,\widetilde x,0)=g(t,\widetilde x),\quad \forall t>0,\quad \widetilde x\in \mathbb R^{d-1},\\ u(0,x)=\varphi(x),\quad \forall x\in \dot\mathbb R_+^d.\end{cases} \] In the second case, it is studied for \(A=\frac{1}{2}\Delta u\) the problem with Wentzell boundary condition \[ \begin{cases}\partial_tu(t,x)=\frac{1}{2}\Delta u(t,x)+f(t,x),\quad \forall t>0,\quad x\in \dot\mathbb R_+^d,\\ Bu(t,x)+g(t,x)=0,\quad \forall t>0,\quad x\in \partial \dot\mathbb R_+^d,\\ u(0,x)=u_0(x),\quad \forall x\in \dot\mathbb R_+^d,\end{cases} \] and then some explicit computations in elastic case and sticky case are given. \noindent In the third case-variable coefficients, a detailed indication for the construction of Green and Poisson functions is finally addressed.
integro-differential second order operator
stochastic differential equations with jumps
scientific article; zbMATH DE number 5161794
scientific article