Intrinsic Ultracontractivity and Probability

Abstract

Let $p_t^D(x,y) = p_t(x,y)$ be the Dirichlet heat kernel for $\frac{1}{2}\Delta$ in a domain D ⊂ R ⁿ, n ≥ 2. In [4] E.B. Davies and B. Simon define the semigroup connected with the Dirichlet Laplacian to be intrinsically ultracontractive if there is a positive (in D) eigenfunction φ ₀ for $\frac{1}{2}\Delta$ in D and if for each t > 0 there are positive constants c _t, C_t depending only on D and t such that

$$ {c_{t}}{\phi _{0}}(x){\phi _{0}}(y) < {p_{t}}(x,y) < {C_{t}}{\phi _{0}}(x){\phi _{0}}(y),x,y \in D $$

((1))

.