Probabilistic solution of the Dirichlet problem for biharmonic functions in discrete space. (English) Zbl 0543.60079

Let \(X_ n\) be a Markov chain on a discrete state space E and put \[ f(x)=\lim_{t\to \infty} E_ x\sum^{\eta_ t}_{j=0}(-1)^ jM_{j-1}\{(\tau_ j-\tau_{j-1})\phi(X_{\tau_ j})+\sum^{\tau_ j-1}_{m=\tau_{j-1}}(m-\tau_{j-1})\psi(X_ m)\}, \] where \(\eta_ t\) is an independent Poisson process, \(\tau_ j\) is the \(j+1^{st}\) time at which the process is in the complement of a set \(\Gamma\) in E \((\tau_{-1}=-1)\), \(\phi\) and \(\psi\) are real functions defined on E, and \(M_ j=\prod^{j}_{i=0}(\tau_ i-\tau_{i-1}-1)\). It is shown (theorem 4) that f is the unique function which solves the following boundary value problem: \(A^ 2f=\psi\) in \(\Gamma\), \(f=\phi\) in \(\Gamma^ c\), where A is the generator of \(X_ n\). That is \(Af(x)=E_ xf(X_ 1)-f(x)\).


60J45 Probabilistic potential theory
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
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