Diffusion processes with generators of generalized divergence form

In this paper we shall discuss a c e rta in c la ss of diffusion processes with singular coefficients.Our class contains th e class o f d iv erg en c e fo rm .Hence we call it generalized divergence form (G. D. F . in abbreviation).G. D. F . have some applications to problems arising from fluid mechanics.Indeed, t h e B urgers process ([1 2 ]) and the vortex process are of generalized divergence form . Especially the vortex process is a n example o f generalized divergence form w hich is not of divergence form ([3 ], [1 1 ], [1 3 ]).Since the coefficients of these diffusion processes are singular, the construction is not easy.Our results are useful to construct such diffusion processes a n d also applicable fo r a homogenization problem ([8], [9]).a A differentiable operator A = Via ' is sa id to b e a diveraz i gence form if a i i (t, x) a re measurable functions satisfying (0.1) 2-1Ie12 e=(e1) i, 1.1Fundamental solutions of parabolic equations in a divergence form have been researched by many mathematicians.Especially important theoretical improvements were done by Nash [7] and Moser [ 6 ] , T h e most remarkable result was obtained by Aronson [ 1 ] .He showed the following estimate : L et p(s, x, t, y) be a fundamental solution of 72 -A .