Brownian Motion and Nonlinear Boundary Value Problems

我们的研究结果可以概括如下：1.首先从泛函分析的角度研究了概率论中带边界条件的马氏过程的构造问题。我们的方法的特点是广泛使用的思想和技术的特点，在偏微分方程理论的最新发展。我们构造了一个Feller半群，它对应于这样一种扩散现象，即一个马尔可夫粒子在状态空间中既跳跃又连续地运动，直到它到达粒子确定被扩散的集合时死亡.其次，研究了强空间异质性环境中一类具有不定权的扩散Logistic方程正解的存在唯一性问题。我们证明了最有利的情况将发生在一个相对大的有利区域（具有良好的资源和没有拥挤效应）的位置， 关于我们 远离环境的边界。此外，我们还讨论了正平衡态的稳定性.第三，我们研究了化学反应器理论中的半线性椭圆边值问题，这些问题服从简单的Arrhenius速率定律和牛顿冷却。证明了在无因次放热率的某个临界值处，稳态温度剖面中存在着点火和熄火现象.最后给出了相对de Rham上同调群的一个指数公式的解析证明，该公式可以看作是著名的Hodge-科代拉关于绝对de Rham上同调群理论的推广.在推导我们的指数公式时，满足内边界条件的调和形式理论起着基础性的作用。我们的方法有一个很大的优势，直观的解释指数公式布朗运动从概率论的角度来看。我们的结果可以表述如下：布朗运动通过它的Euler-Poincare特征描述了紧致黎曼流形的拓扑。少

Our results may be summarized as follows :1. First we studied from the viewpoint of functional analysis the problem of construction of Markov processes with boundary conditions in probability theory. Our approach is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the theory of partial differential equations. We constructed a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the set where the particle is definitely absorbed.2. Secondly we studied existence and uniqueness problems of positive solutions of diffusive logistic equations with indefinite weights which model population dynamics in environments with strong spatial heterogeneity. We proved that the most favorable situations will occur if there is a relatively large favorable region (with good resources and without crowding effects) loca … More ted some distance away from the boundary of the environment. Moreover we discuss the stability properties for positive steady states.3. Thirdly we studied semilinear elliptic boundary value problems arising in chemical reactor theory which obey the simple Arrhenius rate law and Newtonian cooling. We proved that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless heat evolution rate.4. Finally we gived an analytic proof of an index formula for the relative de Rham cohomology groups which may be considered as a generalization of the celebrated Hodge--Kodaira theory for the absolute de Rham cohomology groups. In deriving our index formula, the theory of harmonic forms satisfying an interior boundary condition plays a fundamental role. Our approach has a great advantage of intuitive interpretation of the index formula in terms of Brownian motion from the point of view of probability theory. Our result may be stated as follows : Brownian motion describes the topology of a compact Riemannian manifold through its Euler--Poincare characteristic. Less