Green's function and invariant density for an integro-differential operator of second order
type: Item
instance of: scholarly article
title: Green's function and invariant density for an integro-differential operator of second order
zbMATH Open document ID: 0702.45011
DOI: 10.1007/BF01790348
author: Maria Giovanna Garroni
author: José-Luis Menaldi
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1989-01-01T00:00:00Z
review text: 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 \…
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45P05
zbMATH DE Number: 4151094
zbMATH Keywords: Green function
zbMATH Keywords: integro-differential initial-boundary value problem
zbMATH Keywords: unique invariant density
zbMATH Keywords: diffusion process with jumps
zbMATH Keywords: asymptotic behaviour
reviewed by: Chengbo Wang
MaRDI profile type: Publication
cites work: ON PROCESSES WITH LÉVY GENERATING OPERATOR IN A HALF-SPACE
cites work: ON STOCHASTIC DIFFERENTIAL EQUATIONS WITH BOUNDARY CONDITIONS IN A HALF-PLANE
cites work: Q3953613
cites work: Q4189539
cites work: Q3693914
cites work: Q3774222
cites work: Reflexion discontinue et systèmes stochastiques
cites work: Fundamental solutions of parabolic systems. I
cites work: Q5520962
cites work: Q3736318
cites work: On the asymptotic behaviour of solutions of integro-differential inequalities
cites work: On pababolic oblique derivative problem with holder continuous coefficients
cites work: Q5556832
cites work: Q5630074
cites work: Q5548525
cites work: Markov processes associated with certain integro-differential operators
cites work: Q4114577
cites work: An Ergodic Control Problem for Reflected Diffusion with Jump
cites work: Reflected diffusion processes with jumps
cites work: Q5533735
cites work: Q4049667
cites work: Q5548527
cites work: Diffusion processes associated with L�vy generators
full work available at URL: https://doi.org/10.1007/bf01790348
OpenAlex ID: W2069822092
description: scientific article; zbMATH DE number 4151094