Green's function and invariant density for an integro-differential operator of second order
The authors construct the Green function for an integro-differential initial-boundary value problem, prove the existence of a unique invariant density associated with the corresponding reflected diffusion process with jumps and study the asymptotic behaviour of the solution. The main and useful results are as follows: 1. Let the hypotheses \([H\phi]_{k,\alpha}\leq C([\phi]_{k+1,\alpha}+[\nabla \phi]_{k,\alpha})\), for \(k=-1,0,1,...\), and constant \(C=C_ H\) hold and \(G_ 0\) belongs to \({\mathcal G}_{\alpha}\), i.e. \([G_ 0]_{0,\alpha}<\infty\), \([\nabla G_ 0]_{-1,\alpha}<\infty\). Then the problem: Find G(x,t,y,\(\tau\)) in \({\mathcal G}_{\alpha}\) such that \(G(x,t,y,\tau)=G_ 0(x,t,y,\tau)+\int^{t}_{\tau}ds\int_{\Omega}G_ 0(x,t,z,s)HG(z,s,y,\tau)dz,\) for every x, y in \({\bar \Omega}\), \(0\leq \tau <t\leq 1\), possesses one and only one solution G. Moreover, one has \(G(x,t,y,\tau)=\sum^{\infty}_{k=0}G_ k(x,t,y,\tau),\) where the convergence is uniform over compact sets of the domain of definition and \(G_{k+1}(x,t,y,\tau)=\int^{t}_{\tau}ds\int_{\Omega}G_ 0(x,t,z,s)HG_ k(z,s,y,\tau)dz,\) for \(k=0,1,... \). In particular \(G=G_ 0+G^ 1\), \(G^ 1\) belongs to \({\mathcal G}'_{\alpha}\), where H is a linear operator. 2. Supposing the operators A and B are \(A\phi (x)=- \sum^{d}_{i,j=1}a_{i,j}(x)\partial_{ij}\phi (x)+\sum^{d}_{i=1}a_ i(x)\partial_ i\phi (x)+\int_{E}[\phi (x+\gamma (x,\xi)-\phi (x))]\beta (x,\xi)\pi (d\xi),\) \(B\phi (x)=- \sum^{d}_{i=1}b_ i(x)\partial_ i\phi (x),\) where the coefficients and the Radon measure satisfy for some exponent \(0<\alpha <1\) six given assumptions. Then the functions \(u(x,t)=\int^{t}_{0}d\tau \int_{\Omega}G(x,t-\tau,y)f(y,\tau)dy\) and \(u_{\lambda}(x)=\int^{\infty}_{0}\exp (-\lambda t)\int_{\Omega}G(x,t,y)g(y)dy,\quad 0<\lambda \leq 1,\) are the solutions of the equations \(\partial_ tu+Au=f\) in \(\Omega\times (0,\infty)\), \(u(\cdot,0)=0\) on \({\bar \Omega}\), \(Bu=0\) on \(\partial \Omega \times [0,\infty)\) and \(Au+\lambda u=g\) in \(\Omega\), \(Bu=0\) on \(\partial \Omega\), respectively.
integro-differential initial-boundary value problem
scientific article; zbMATH DE number 4151094
scientific article