Intrinsic ultracontractivity of the Feynman-Kac semigroup for relativistic stable processes

Abstract

Let X t X_t be the relativistic α \alpha -stable process in R d \mathbf {R}^d , α ∈ ( 0 , 2 ) \alpha \in (0,2) , d > α d > \alpha , with infinitesimal generator H 0 ( α ) = − ( ( − Δ + m 2 / α ) α / 2 − m ) H_0^{(\alpha )}= - ((-\Delta +m^{2/\alpha })^{\alpha /2}-m) . We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup T t T_t for this process with generator H 0 ( α ) − V H_0^{(\alpha )} - V , V ≥ 0 V \ge 0 , V V locally bounded. We prove that if lim | x | → ∞ V ( x ) = ∞ \lim _{|x| \to \infty } V(x) = \infty , then for every t > 0 t >0 the operator T t T_t is compact. We consider the class V \mathcal {V} of potentials V V such that V ≥ 0 V \ge 0 , lim | x | → ∞ V ( x ) = ∞ \lim _{|x| \to \infty } V(x) = \infty and V V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V V in the class V \mathcal {V} we show that the semigroup T t T_t is IU if and only if lim | x | → ∞ V ( x ) / | x | = ∞ \lim _{|x| \to \infty } V(x)/|x| = \infty . If this condition is satisfied we also obtain sharp estimates of the first eigenfunction ϕ 1 \phi _1 for T t T_t . In particular, when V ( x ) = | x | β V(x) = |x|^{\beta } , β > 0 \beta > 0 , then the semigroup T t T_t is IU if and only if β > 1 \beta >1 . For β > 1 \beta >1 the first eigenfunction ϕ 1 ( x ) \phi _1(x) is comparable to \[ exp ⁡ ( − m 1 / α | x | ) ( | x | + 1 ) ( − d − α − 2 β − 1 ) / 2 . \exp (-m^{1/{\alpha }}|x|) \, (|x| + 1)^{(-d - \alpha - 2 \beta -1 )/2}. \]