Long time behavior of heat kernels of operators with unbounded drift terms
The authors consider the differential operator: \[ A=- \text{div}(a(x)\nabla)+ b\cdot\nabla \] on \(L^2(\mathbb{R}^d)\), under the following assumptions on the coefficients: \[ a_{kj}\in L^\infty(\mathbb{R}^d),\;\exists\upsilon> 0: \sum^d_{j,k=1} a_{kj}(x) \xi_j\xi_k\geq \upsilon|\xi|^2\;\forall x,\xi\in\mathbb{R}^d, \] \[ b= (b_j)\in C^1(\mathbb{R}^d,\mathbb{R}^d)\text{ with div\,}b\leq w, \] where \(w\) is a constant in \(\mathbb{R}\). By the use of the sesquilinear forms the authors obtain a realization of \(A\), which is (minus) the generator of a strongly continuous semigroup on \(L^2(\mathbb{R}^d)\). The corresponding semigroup is given by an integral kernel. The time behaviour of the integral kernel is studied. The estimates on its decay at infinity are obtained. For symmetric diffusion coefficients a local lower estimate is given. An example with the Ornstein-Uhlenbeck operator is considered.
scientific article; zbMATH DE number 5854867
scientific article