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Let (S_n,n≥0) be a random walk evolving on the real line and introduce the first hitting time of the half-line (a,+∞) for any real a: \tau_a=min{n≥1:S_n>a}. The classical Spitzer identity (1960) supplies an expression for the generating function of the couple (\tau_0,S_{\tau_0}). In 1998, Nakajima [Joint distribution of the first hitting time and first hitting place for a random walk. Kodai Math. J. 21 (1998) 192-200.] derived a relationship between the generating functions of the random couples (\tau_0,S_{\tau_0}) and (\tau_a,S_{\tau_a}) for any positive number a. In this note, we propose a new and shorter proof for this relationship and complement this analysis by considering the case of an increasing random walk. We especially investigate the Erlangian case and provide an explicit expression for the joint distribution of (\tau_a,S_{\tau_a}) in this situation.

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... We now propose another proof of (4.1) based on a suitable form of the Spitzer identity due to Nakajima [9] (see also [8]), which we recall below for the convenience of the reader. ...

Consider the high-order heat-type equation ∂^Nu/∂t^N=(-1)^{1+N/2} ∂^Nu/∂x^N for an even integer N>2, and introduce the related Markov pseudo-process (X(t),t≥0). Let us define the drifted pseudo-process (X_b(t),t≥0) by X_b(t)=X(t)+bt. In this paper, we study the following functionals related to (X_b(t),t≥0) the maximum M_b(t) up to time t; the first hitting time of the half line (a,+∞); and the hitting place at this time. We provide explicit expressions for the Laplace-Fourier transforms of the distributions of the vectors (X_b(t),M_b(t)) and , from which we deduce explicit expressions for the distribution of as well as for the escape pseudo-probability: . We also provide some boundary value problems satisfied by these distributions.

We consider the equation
\[ \frac{\partial u}{\partial t}(t,x)=-\Delta^{2}u(t,x) \]
for the biharmonic operator $-\Delta^2$.
We define the pseudo process corresponding to this equation as Nishioka's sense.
We obtain the Laplace-Fourier transform of the joint distribution of the first hitting time $\tau(\omega)=\inf\{t>0:\omega(t)0, \alpha\in\mathbf{R})$ and the first hitting place $\omega(\tau)$, where each path $\omega(t)$ starts from 0 at $t=0$.

Cisplatin and irinotecan are reported to act synergistically. The authors have conducted a phase II trial combining cisplatin and irinotecan in patients with refractory lung cancer to evaluate the activity and safety of the regimen. Twenty-one patients, who had not responded to prior platinum-based chemotherapy, were entered into the study. Both cisplatin (30 mg/m2) and irinotecan (60 mg/m2) were administrated on days 1, 8, and 15, and the regimen was repeated every 28 days. There were six partial responses, and the response rate was 29% (95% confidence interval, 11.3%-52.2%). The median survival time of all patients was 32 weeks, and 1-year and 2-year survival rates were 43% and 11%, respectively. Major toxicities were hematologic; leukopenia of grades 3 and 4 developed in 43% patients, anemia developed in 38%, and thrombocytopenia developed in 19%. One notable nonhematologic toxicity was diarrhea; which was grade 3 in 38%. The weekly chemotherapy combining cisplatin and irinotecan was active against lung cancer, which is refractory to platinum-based chemotherapy. However, skips of drug administration or dose reduction were necessary in 76% patients during two courses of planned administration, though the ratio of actual dose to scheduled dose was 81%. Dose modification would be necessary to yield better results by this weekly chemotherapy.

Consider the high-order heat-type equation ∂^Nu/∂t^N=(-1)^{1+N/2} ∂^Nu/∂x^N for an even integer N>2, and introduce the related Markov pseudo-process (X(t),t≥0). Let us define the drifted pseudo-process (X_b(t),t≥0) by X_b(t)=X(t)+bt. In this paper, we study the following functionals related to (X_b(t),t≥0) the maximum M_b(t) up to time t; the first hitting time of the half line (a,+∞); and the hitting place at this time. We provide explicit expressions for the Laplace-Fourier transforms of the distributions of the vectors (X_b(t),M_b(t)) and , from which we deduce explicit expressions for the distribution of as well as for the escape pseudo-probability: . We also provide some boundary value problems satisfied by these distributions.

The material of this book is the outgrowth of a course of lectures which I have given from time to time at Harvard University on the subject of Dirichlet series and Laplace integrals. It is designed for students who have such a knowledge of analysis as might be obtained by reading the more fundamental parts of the familiar text of E. C. Titchmarsh on the theory of functions. I have taken pains to include proofs of results which such a student might not know even though they might be available elsewhere. For example, the first chapter is devoted largely to the study of the Riemann-Stieltjes integral. Although this material is in constant use by analysts it seems not to have been collected in convenient form. There are only a few instances where I have had to depart from this aim of having the book complete in itself. If he desires, the student may omit such parts without losing the fundamental ideas.

A random walk on the real line starting from 0 is considered. A representation of the Lapalace-Founer transform of the joint distribution of the first hitting time and the first hitting place of the set (-∞, -a) (a > 0) is obtained, which gives a relation with the joint distribution of those of the set (-∞, 0). The leading idea is Wiener-Hopf's factorization theorem. © 1998, Department of Mathematics, Tokyo Institute of Technology. All rights reserved.