Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

Assessing the Safety Training Needs of Spanish-Speaking Workers in the Southeastern Logging Industry

Safety in logging operations in the southeastern United States has long been an important issue. Recently, a growing number of Spanish-speaking workers (SSW) have gained employment on logging operations in the region. There is concern that injury and fatality rates could increase due to inexperience, possible lack of proper safety training, and language-barrier problems attributed to SSW. The objectives of this study are to, (1) determine the current percentage of the logging workforce in the southeastern United States comprised of SSW, (2) document the current state of logging safety training as it relates to SSW, and (3) recommend strategies to address the short- and long-term logging safety training needs of SSW. Data regarding the current population of SSW were collected in 2005 through field surveys of 1,890 logging crews operating in the southeastern United States. Additional data were acquired through field interview questionnaires completed in the fall of 2005 with 41 sample logging contractors who employ SSW. As of 2005, SSW represented 3.37% of the logging industry workforce in the southeastern United States. Ten percent of the operations surveyed employed one or more SSW. Of the questionnaire respondents, 90% employed at least one SSW who understood English well enough to effectively interpret instructions to the other SSW. Seventy-three percent of the loggers interviewed believed that “hands-on” demonstration training was the most effective way to present safety training to SSW. A majority of the respondent loggers believed that simply distributing safety training manuals and brochures printed in Spanish was unlikely to be effective, because only about one-half of the SSW they employed were literate. Recommendations were developed, based on the relevant literature and data collected through the questionnaire, to address the safety concerns associated with SSW in the logging industry.

Monotone Stochastic Recursions and their Duals

A duality is presented for real-valued stochastic sequences [Vn] defined by a general recursion of the form Vn+1 = f(Vn, Un), with [Un] a stationary driving sequence and f nonnegative, continuous, and monotone in its first variable. The duality is obtained by defining a dual function g of f, which if used recursively on the time reversal of [Un] defines a dual risk process. As a consequence, we prove that steady-state probabilities for Vn can always be expressed as transient probabilities of the dual risk process. The construction is related to duality of stochastically monotone Markov processes as studied by Siegmund (1976, The equivalence of absorbing and reflecting barrier problems for stochastically monotone Markov processes, Annals of Probability 4: 914–924). Our method of proof involves an elementary sample-path analysis. A variety of examples are given, including random walks with stationary increments and two reflecting barriers, reservoir models, autoregressive processes, and branching processes. Finally, general stability issues of the content process are dealt with by expressing them in terms of the dual risk process.