## 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 $$Af(x)=E_ xf(X_ 1)-f(x)$$.

### MSC:

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