The Nonhomogeneous Oblique Derivative Problem for a Diffusion Equation on a Compact Riemannian Domain

목차 (Table of Contents)

• l. Introduction
• 2. A Background of the Subject
• 2.1 Basic Concepts of Stochastic Processes ‥‥‥‥‥‥ 5
• 2.2 Connection between Continuous Markov Processes and Diffusion Equations‥‥9
• 3. Smoothing of Cγ,α Manifold Pairs
• 3.1 Cγ,α Continuity ‥‥‥‥‥‥‥‥‥‥ 17
• 3.2 Cγ,α Manifolds ‥‥‥‥‥‥‥18
• 3.3 Smoothing of Cγ,α Manifold Pairs‥‥‥‥‥ 20
• 3.4 Comments on the Proof of smoothing ‥‥‥‥‥ 21
• 4. Construction of a Fundamental Solution 24
• 4.1 The Operator L(s, x ; ∂s, ∂x) ‥‥‥‥‥‥‥ 24
• 4.2 Fundamental Solution ‥‥‥‥‥‥‥ 25
• 4.3 Three Atlases on D ‥‥‥‥‥‥‥ 28
• 4.4 Rewriting the Boundary Operntor ‥‥‥‥‥ 31
• 4.5 Constnlction of a Funndamental Solution ‥‥‥‥ 32
• 4.6 The Explicit Expression of the First Parametrix‥‥‥ 37
• 4.7 Convergence of Iteration ‥‥‥‥‥ 38
• 5. Representation Theorems for Solutions to the Nonhomogeneous Equation 42
• 5.1 Integral Representation ‥‥‥‥‥‥ 42
• 5.2 Stochastic Representation ‥‥‥‥‥‥‥ 44
• 5.3 Comment on the Proof of Theorem 5.5(i) ‥‥‥‥ 46
• 5.4 Comment on the Proof of Theorem 5.7(ii)‥‥‥ 48
• 5.5 The Strict Positivity and Stabilily of the Fundamental Solution 51