Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators. (English) Zbl 0907.47040

Quastel, J.; Varadhan, S. R. S.

Commun. Pure Appl. Math. 50, No. 7, 667-706 (1997).

The authors study existence and uniqueness problems for semigroups and processes generated by differential operators of the form \[ L_t u=\nabla\cdot a\Delta u+\sigma c\cdot\nabla u. \] Here \(a\) is a symmetric and nonnegative definite matrix, \(\sigma\) the square root of \(a\) and \(c\) is a function. The main aim of the paper is to focus cases where \(a\) and \(c\) have minimal regularity, and \(a\) is allowed to degenerate. Weak and very weak solutions of the parabolic equations \[ \partial_tu+ L_tu= 0 \] are studied.

Reviewer: R.Salvi (Milano)

MSC:

47D06	One-parameter semigroups and linear evolution equations
47F05	General theory of partial differential operators
35K10	Second-order parabolic equations

Keywords:

diffusion semigroup; degenerate divergence form operator; weak solutions; existence and uniqueness; semigroups; differential operators; minimal regularity; parabolic equations