On Diffusion Semigroups Generated by Semi-Elliptic Differential Operators in Infinite Dimensions

Abstract

We want to study some continuity properties of operator semigroups, generated by a semi-elliptic differential operator on a real separable Hilbert space ℍ. To this end, let us begin by writing the finite-dimensional semi-elliptic differential operator

$$\text{Lu(x) = }\frac{\text{1}}{\text{2}}\sum\limits_{i,j = 1}^n {a_{ij} } (x)\frac{{\partial ^2 u}}{{\partial x_i \partial x_j }}(x) + \sum\limits_{i = 1}^n {b_i } (x)\frac{{\partial u}}{{\partial x_i }}(x)$$

((1))

on ℍ = ℝⁿ in coordinate-free form as

$$ Lu(x) = \frac{1}{2} tr u''(x)(a(x)\cdot ,\cdot ) + u'(x)(b(x))$$

((1a))