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The first hitting time and place of a half-line by a biharmonic pseudo process. (English) Zbl 0891.60071

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)

MSC:

60J45 Probabilistic potential theory
60G20 Generalized stochastic processes
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