## 作用素論セミナー: L_1-analysis of degenerate elliptic operators

2011年 　 9月 20日 14時 45分 ～ 2011年 　 9月 20日 15時 45分

Derek W. Robinson (ANU)

We discuss the existence and uniqueness of solutions of the parabolic evolution equation $\partial\varphi/\partial t+H\varphi=0$
where $H$ is a second-order degenerate elliptic operator. Our interest is in solutions of the evolution equation which have a probabilistic interpretation. Specifically, we discuss $L_1$-solutions of the equation since the positive normalized $L_1$-solutions can be interpreted as
probability distributions. Our analysis is based on methods of functional analysis and potential theory.

Specifically let $\Omega$ be an open subset of $\Ri^d$ and $H=-\sum^d_{i,j=1}\partial_i\,c_{ij}\,\partial_j$ a partial differential operator defined on $C_c^\infty(\Omega)$ where the coefficients $c_{ij}\in W^{1,\infty}_{\rm loc}(\overline\Omega)$ are real symmetric
and $C=(c_{ij})$ is a strictly positive-definite matrix over $\Omega$. The corresponding evolution equation describes a diffusion process.

The assumptions allow two types of degeneracy. One can have $c_{ij}(x)\to0$ as $x\to\partial\Omega$, the boundary of $\Omega$,
or $c_{ij}(x)\to\infty$ as $|x|\to\infty$. Under mild growth conditions on the coefficients we argue that $H$ has a unique extension which generates a continuous semigroup on $L_1(\Omega)$ if and only if it has a unique submarkovian extension, i.e.\ a self-adjoint extension on $L_2(\Omega)$ which generates a submarkovian semigroup. Somewhat surprisingly these uniqueness properties are equivalent to the condition that the capacity of the boundary $\partial\Omega$ of $\Omega$, measured with respect to $H$ is zero. The capacity condition will be explained.

Finally we give examples and illustrations of the uniqueness result and describe the complications that occur if $H$ has lower order terms $\sum_{i=1}^dc_i\partial_i +c_0$.