Measure Evolution Under Maps And RWRE On 6-regular Planar Graphs

We study the following 3 Questions: 1. Measure evolution in singular cases;2. Measures' images under 'non-measurable' maps; 3. 0-1 laws for RWRE on regularplanar graphs. Where RWRE denotes Random Walk in Random Environments.Question 1 is studied in Chapter 1. Measure-valued diffusion process de-scribing how measures evolve under flows or "imaginary" flows on R^d is constructed.The interest of the process is that on one hand, it can be viewed as a measure-valuedflow; on the other hand, the general stochastic flows of measurable maps or kernelsdo not cover it.Question 2 is looked into in Chapter 2. Let (Ω, F) and (E,ε) be twomeasurable spaces, andμa nonzero measure on (Ω, F). Letμ^* be the outermeasure ofμ, A_μ^*. the set of allμ^*-measurable sets onΩ, and F^μthe completionof F with respect toμ. Supposeφis an arbitrary map fromΩto E. Clearly, ifφ: (Ω, A_μ^*)→(E,ε) is measurable, thenμ^*(φ^-1(·)) is a measure on (E,ε). Onthe contrary, even ifφ:Ω→(E,ε) is not A_μ^*-measurable,μ^*(φ^-1(·)) can also bea measure on (E,ε); furthermore,μ*(φ^-1(·)) is aσ-finite measure on(E,ε) if andonly ifφis F^μ-measurable andμisσ-finite. The result is of theoretic interests formeasure theory.Question 3 is investigated in Chapter 3. For RWRE on 2-dimensional in-tegral lattice Z² (a 4-regular graph), there is a 0-1 law for RWRE tends to infinityalong a fixed direction in uniformly elliptic product random environments. But for6-regular planar graphs obtaining by parting R² by identical 3-regular polygons, the0-1 law for RWRE tends to infinity along a fixed direction holds true for uniformlyelliptic product random environments. The result is a useful complement to RWREon Z².