Intrinsic ultracontractivity for stable semigroups on unbounded open sets. (English) Zbl 1180.47031
Let \((X_t)\) denote the isotropic stable process of index \(\alpha<2\), i.e., a subordination of a standard Brownian motion by an \(\alpha/2\)-stable subordinator on \(\mathbb R_+\). Let \(D\subseteq \mathbb R^d\) be an open – here unbounded – domain. Let, furthermore, \(\tau_D\) denote the first exit time from \(D\) and define \((P_t^D)\) to be the transition semigroup of the process killed on exiting \(D\), \(P_t^Df(x)=E_x(\tau_D>t; f(X_t))= \int p_t^D(x,y)f(y)\,dy\). Here \(p_t^D\) denotes the kernel of \(P_t^D\). Throughout it is assumed that the operators \(P_t^D\) are compact operators on \(L^2\) and \((\varphi_n = \varphi_n^D)_{n\geq 1}\) is a complete ONS of eigenfunctions with corresponding eigenvalues \(\lambda_n = \lambda_n^D\). \((P_t^D)\) is called intrinsically ultracontractive (IU) if for all \(t>0\) we have \(p_t^D(x,y)\leq C_{D,t}\varphi_1(x)\varphi_1(y)\).
Previous investigations were often restricted to the case of bounded domains or \(\alpha =2\), i.e., to Brownian motions \((X_t)\). Since for \(\alpha < 2\) the process is a jump-process, the results differ essentially.
The first results are obtained for general open domains, in particular (Theorem 1), \(\varphi_1(x) \approx \left(1+|x|\right)^{-d-\alpha} E_x(\tau_D)\) and (Theorem 2) property (IU) is equivalent to an upper estimate \(p_t^D(x,y)\leq C_{D,t}\left(1+|x|\right)^{-d-\alpha}\left(1+|y|\right)^{-d-\alpha}\).
For horn-shaped domains \(D=D_f= \{x:x_1>0\), \(|(x_2, \dots x_d)|\leq f(x_1)\}\) which are non-degenerate, property (IU) is characterized by the behaviour of \((f(u))^\alpha\log u\) for \(u\to\infty\) (Theorem 5).
The proofs rely on boundary Harnack inequalities, in fact, on a uniform version proved by K.Bogdan, T.Kulczycki and M.Kwaśnicki [Probab.Theory Relat.Fields 140, No.3–4, 345–381 (2008; Zbl 1146.31004)]; see also the literature there.
For the case of Brownian motions and unbounded domains, see, e.g., B.Davies [J. Funct.Anal.100, No.1, 162–180 (1991; Zbl 0766.47026)], P.J.Méndes-Hernándes [Mich.Math.J.47, No.1, 79–99 (2000; Zbl 0978.47032)] and the references therein.
Previous investigations were often restricted to the case of bounded domains or \(\alpha =2\), i.e., to Brownian motions \((X_t)\). Since for \(\alpha < 2\) the process is a jump-process, the results differ essentially.
The first results are obtained for general open domains, in particular (Theorem 1), \(\varphi_1(x) \approx \left(1+|x|\right)^{-d-\alpha} E_x(\tau_D)\) and (Theorem 2) property (IU) is equivalent to an upper estimate \(p_t^D(x,y)\leq C_{D,t}\left(1+|x|\right)^{-d-\alpha}\left(1+|y|\right)^{-d-\alpha}\).
For horn-shaped domains \(D=D_f= \{x:x_1>0\), \(|(x_2, \dots x_d)|\leq f(x_1)\}\) which are non-degenerate, property (IU) is characterized by the behaviour of \((f(u))^\alpha\log u\) for \(u\to\infty\) (Theorem 5).
The proofs rely on boundary Harnack inequalities, in fact, on a uniform version proved by K.Bogdan, T.Kulczycki and M.Kwaśnicki [Probab.Theory Relat.Fields 140, No.3–4, 345–381 (2008; Zbl 1146.31004)]; see also the literature there.
For the case of Brownian motions and unbounded domains, see, e.g., B.Davies [J. Funct.Anal.100, No.1, 162–180 (1991; Zbl 0766.47026)], P.J.Méndes-Hernándes [Mich.Math.J.47, No.1, 79–99 (2000; Zbl 0978.47032)] and the references therein.
Reviewer: Wilfried Hazod (Dortmund)
MSC:
| 47D07 | Markov semigroups and applications to diffusion processes |
| 60G52 | Stable stochastic processes |
| 60J45 | Probabilistic potential theory |
Keywords:
intrinsic ultracontractivity; isotropic stable process; first eigenfunction; first exit time; horn shaped domainReferences:
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