Brownian Motion and Nonlinear Boundary Value Problems

我们的研究结果可以总结如下：1。首先从泛函分析的角度研究了概率论中具有边界条件的马尔可夫过程的构造问题。我们的方法的特点是广泛使用了偏微分方程理论中最近发展的思想和技术。我们构造了一个Feller半群，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吸收的集合时死亡。其次，研究了具有强空间异质性的种群动态模型的不定权扩散logistic方程正解的存在唯一性问题。我们证明，如果在离环境边界一定距离的地方有一个相对较大的有利区域（资源良好，没有拥挤效应），就会出现最有利的情况。此外，我们还讨论了正稳态的稳定性性质。第三，研究了化学反应器理论中的半线性椭圆边值问题，该问题遵循简单的阿伦尼乌斯速率定律和牛顿冷却理论。我们证明了在无量纲热演化率的某些临界值下，在稳定的稳定温度分布中会发生点火和熄灭现象。最后，我们给出了相对de Rham上同群的一个指标公式的解析证明，该指标公式可以看作是对著名的绝对de Rham上同群的Hodge—Kodaira理论的推广。在推导指标公式时，满足内边界条件的调和形式理论起着基本的作用。我们的方法有一个很大的优点，即从概率论的角度直观地解释布朗运动的指标公式。我们的结果可以表述如下：布朗运动通过紧致黎曼流形的欧拉—庞加莱特征来描述其拓扑结构。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