Nishioka, Kunio The first hitting time and place of a half-line by a biharmonic pseudo process. (English) Zbl 0891.60071 Jap. J. Math., New Ser. 23, No. 2, 235-280 (1997). The operator \(-\Delta^2\) is called a biharmonic operator, and the pseudo-Markov process corresponding to (*) \({\partial u\over\partial t}(t,x)=-\Delta^2u(t,x)\), \(t>0\), \(x\in \mathbb{R}\), is called a biharmonic pseudo-process (or BPP). The author discusses the joint distribution \(P_x [\tau_0 \in dt,\;\omega (\tau_0)\in da]\) of the first hitting time \(\tau_0 \equiv \inf\{t>0: \omega (t)\in (-\infty, 0)\}\) and the first hitting place \(\omega (\tau_0)\) by one-dimensional BPP, where \(\omega(t)\), \(t\geq 0\), is a path of BPP with \(\omega (0)>0\). In particular, the author shows that the density \(q(x;t,a)\) of the joint distribution involves Dirac’s delta functions \(\delta(a)\) and \(\delta'(a)\), which implies, roughly, that BPP is composed of particles of two types, monopoles and dipoles. Moreover, the strong Markov property of BPP with respect to the pair \(\tau_0\), \(\omega (\tau_0)\) is proved, and BPP with an absorbing barrier is also studied. On this account, the author proves that an initial value problem of (*) with Dirichlet boundary condition is explicitly solved by employing the joint distribution and the transition probability for BPP with an absorbing barrier. As for other related works on BPP, see K. J. Hochberg [Ann. Probab. 6, 433-458 (1978; Zbl 0378.60030)], K. Nishioka [J. Math. Soc. Japan 39, 209-231 (1987; Zbl 0622.60081)], and K. Burdzy and A. Mądrecki [Ann. Appl. Prob. 6, No. 1, 200-217 (1996; Zbl 0856.60042)]. Reviewer: I.Dôku (Urawa) Cited in 1 ReviewCited in 15 Documents MSC: 60J45 Probabilistic potential theory 60G20 Generalized stochastic processes Keywords:biharmonic operator; biharmonic pseudo process; first hitting time; first hitting place Citations:Zbl 0378.60030; Zbl 0622.60081; Zbl 0856.60042 PDFBibTeX XMLCite \textit{K. Nishioka}, Jpn. J. Math., New Ser. 23, No. 2, 235--280 (1997; Zbl 0891.60071)