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The westward expansion of human millet consumption from north China has important implications for understanding early interactions between the East and West. However, few studies have focused on the Xinjiang Uyghur Autonomous Region, the vast geographical area directly linking the ancient cultures of the Eurasian Steppe and the Gansu Corridor of China. In this study, we present the largest isotopic investigation of Bronze Age China (n = 110) on material from the key site of Tianshanbeilu, in eastern Xinjiang. The large range of δ13C values (–17.6‰ to –7.2‰; –15.5 ± 1.2‰) provides direct evidence of unique dietary diversity and consumption of significant C4 resources (millets). The high δ15N results (10.3‰ to 16.7‰; 14.7 ± 0.8‰) likely reflect sheep/goat and wild game consumption and the arid climate of the Taklamakan Desert. Radiocarbon dates from four individuals indicate Tianshanbeilu was in use between 1940 and 1215 cal bc. The Tianshanbeilu results are then analysed with respect to 52 Bronze Age sites from across Eurasia, to investigate the spread and chronology of significant human millet consumption and human migration. This isotopic survey finds novel evidence that the second millennium bc was a dynamic period, with significant dietary interconnectivity occurring between north China, Central Asia and Siberia. Further, we argue that this ‘Isotopic Millet Road’ extended all the way to the Mediterranean and Central Europe, and conclude that these C4 dietary signatures of millet consumption reflect early links (migration and/or resource transfer) between the Bronze Age inhabitants of modern-day China and Europe.

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In this paper, we study one kind of stochastic recursive optimal control problem with the obstacle constraint for the cost functional described by the solution of a reflected backward stochastic differential equation. We give the dynamic programming principle for this kind of optimal control problem and show that the value function is the unique viscosity solution of the obstacle problem for the corresponding Hamilton-Jacobi-Bellman equation. Introduction.The nonlinear backward stochastic differential equation (BSDE) was been introduced by Pardoux and Peng [12]. Independently, Duffie and Epstein [6] introduced BSDE from economic background. In [6] they presented a stochastic differential recursive utility which is an extension of the standard additive utility with the instantaneous utility depending not only on the instantaneous consumption rate but also on the future utility. Actually, it corresponds to the solution of a particular BSDE whose generator does not depend on the variable z. From a mathematical point of view the results in [12] are more general. Then El Karoui, Peng, and Quenez [11] gave some important properties of BSDEs such as comparison theorem and applications in mathematical finance and optimal control theory. And also in the same paper, the authors gave the formulation of recursive utilities and their properties from the BSDE point of view. The recursive optimal control problem is presented as a kind of optimal control problem whose cost functional is described by the solution of BSDE. In 1992, Peng [13] got the Bellman's dynamic programming principle for this kind of problem and proved that the value function is a viscosity solution of one kind of quasi-linear second order partial differential equation (PDE) which is the well known Hamilton-Jacobi-Bellman equation. Later in 1997, he virtually generalized these results to a much more general situation, under Markvian and even non-Markvian framework (see [14, Chapter 2]). In the Chinese version, Peng used the backward semigroup property of BSDE to prove Bellman's dynamic programming principle for the recursive optimal problem introduced by a BSDE under Markovian and non-Markovian framework. He also proved that the value function is a viscosity solution of a generalized Hamilton-Jacobi-Bellman equation.

A notion of L p -exact controllability is introduced for linear controlled (forward) stochastic differential equations, for which several sufficient conditions are established. Further, it is proved that the L p -exact controllability, the validity of an observability inequality for the adjoint equation, the solvability of an optimization problem, and the solvability of an L p -type norm optimal control problem are all equivalent.