## Two-sided heat kernel estimates for censored stable-like processes (2008)

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Citations: | 25 - 17 self |

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@TECHREPORT{Chen08two-sidedheat,

author = {Zhen-qing Chen and Panki Kim and Renming Song},

title = {Two-sided heat kernel estimates for censored stable-like processes},

institution = {},

year = {2008}

}

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### Abstract

In this paper we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

### Citations

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Heat Kernels and Spectral Theory
- Davies
- 1989
(Show Context)
Citation Context ... a fundamental problem in both analysis and probability theory. In fact, two-sided heat kernel estimates for diffusions in Rd have a long history and many beautiful results have been established. See =-=[10, 12]-=- and the references therein. But, due to the complication near the ∗ Research partially supported by NSF Grant DMS-06000206. † Research partially supported by the Korea Science and Engineering Foundat... |

238 |
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Citation Context ... Markovian and hence a regular symmetric Dirichlet form on L 2 (D, dx), and therefore there is an associated symmetric Hunt process X = {Xt, t ≥ 0, Px, x ∈ D} taking values in D (cf. Theorem 3.1.1 of =-=[14]-=-). The process X is called a censored α-stable-like process in D. We fix an arbitrary symmetric measurable extension of C(·, ·) onto R d × R d satisfying (2.2) and we still denote it by C(·, ·). It is... |

88 |
Heat kernel estimates for stable-like processes on d-sets. Stochastic Process
- Chen, Kumagai
(Show Context)
Citation Context ...his paper is the reflected stable-like process X on D, whose subprocess killed upon leaving D is the censored stable-like process X. The reflected α-stable-like processes have been studied in [3] and =-=[6]-=-. In particular, two-sided heat kernel estimates have been obtained in [6] for reflected stable-like processes on open d-sets (including globally Lipschitz open sets) in R d —see (2.4) below. When D i... |

77 |
Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet
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(Show Context)
Citation Context ...0-20010-0). 1sboundary, two-sided estimates for the transition density functions of killed diffusions in a domain D (equivalently, the Dirichlet heat kernels) have been established only recently. See =-=[11, 12, 13]-=- for upper bound estimates and [21] for the lower bound estimates of the Dirichlet heat kernels in bounded C 1,1 domains. Markov processes with discontinuous sample paths constitute an important famil... |

71 | The boundary Harnack principle for the fractional - Bogdan - 1997 |

56 | Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat
- Chen, Kumagai
(Show Context)
Citation Context ...onstant c = c(α, d) > 0, independent of t > 0 and x and y, such that Px � Xt ∈ B � y, 2 −1 t 1/α�� 20 ≥ c td/α+1 . |x − y| d+αsProof. The proof is a simple modification of that of Proposition 4.11 in =-=[7]-=-. For reader’s convenience, we spell out the details here. By Lemma 4.1, starting at z ∈ B(y, 4 −1 t 1/α ), with probability at least c1 = c1(α) > 0 the process X does not move more than 6 −1 t 1/α by... |

52 |
Estimates on Green functions and Poisson kernels for symmetric stable processes
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- 1998
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Citation Context ...n function are very important in understanding deep potential theoretic properties of Markov processes. Such two-sided estimates for the Green functions of symmetric stable processes were obtained in =-=[9, 19]-=-. In [4], sharp two-sided estimates for the Green functions of censored stable processes (i.e. when C(x, y) is a constant) in bounded C 1,1 connected open sets in R d were obtained for d ≥ 2 and α ∈ (... |

41 |
Intrinsic ultracontractivity and conditional gauge for symmetric stable processes
- Chen, Song
- 1997
(Show Context)
Citation Context ...acontractivity gives sharp two-sided estimates of the transition density function for each fixed t > 0. The intrinsic ultracontractivity of semigroups of killed jump processes was first considered in =-=[8]-=-, where it was shown that the semigroup of the killed symmetric α-stable process on a bounded C 1,1 domain is intrinsically ultracontractive. In [19] it was shown that the semigroup of the killed symm... |

39 |
Some theorems on stable processes
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Citation Context .../α δ t −1/α D (t −1/α x) α−1 (3.18) for some positive constant c2 = c2(T, r0, α, Λ0). Here pRd(t, x, y) is the transition density function of the symmetric α-stable process in Rd and it is known (cf. =-=[1, 6]-=-) that � pRd(t, x, y) ≍ t −d/α t ∧ |x − y| d+α � on R+ × R d × R d . (3.19) 18sBy symmetry, the inequality (3.18) for pD(t, x, y) holds with role of x and y interchanged. Using the Chapman-Kolmogorov’... |

36 | Heat kernel estimates for Dirichlet fractional Laplacian
- Chen, Kim, et al.
(Show Context)
Citation Context ... can be written in the form where ∆ α/2 u(x) = lim ε↓0 � {y∈R d : |y−x|>ε} A(d, α) := (u(y) − u(x)) A(d, α) |x − y| d+α dy for u ∈ C∞ c (R d ), α Γ( d+α 2 ) 2 1−α π d/2 Γ(1 − α 2 ). In a recent paper =-=[5]-=-, we succeeded in establishing sharp two-sided estimates for the heat kernel of the fractional Laplacian ∆ α/2 with zero exterior condition on D c (or equivalently, the transition density function of ... |

34 |
Explicit constants for Gaussian upper bounds on heat kernels
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- 1987
(Show Context)
Citation Context ... a fundamental problem in both analysis and probability theory. In fact, two-sided heat kernel estimates for diffusions in Rd have a long history and many beautiful results have been established. See =-=[10, 12]-=- and the references therein. But, due to the complication near the ∗ Research partially supported by NSF Grant DMS-06000206. † Research partially supported by the Korea Science and Engineering Foundat... |

34 |
Properties of Green function of symmetric stable processes
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(Show Context)
Citation Context ...ps of killed jump processes was first considered in [8], where it was shown that the semigroup of the killed symmetric α-stable process on a bounded C 1,1 domain is intrinsically ultracontractive. In =-=[19]-=- it was shown that the semigroup of the killed symmetric α-stable process on any bounded open set is intrinsically ultracontractive. In this paper, we show that, when D is an open d-set in R d with fi... |

29 |
Intrinsic ultracontractivity for symmetric stable processes
- Kulczycki
- 1998
(Show Context)
Citation Context ...en for any t > 0 there exists a positive constant ct > 1 such that p U D(t, x, y) ≥ c −1 t φU 1 (x)φ U 1 (y) for x, y ∈ U. (2.8) The proof of the following result is adapted from an argument given in =-=[20]-=-. Theorem 2.2 Suppose that D is an open d-set in Rd and U is a bounded open subset of D satisfying condition (2.6). Then the semigroup {P U t , t > 0} is intrinsically ultracontractive. Moreover, for ... |

26 |
The boundary behavior of heat kernels of Dirichlet Laplacians
- Zhang
(Show Context)
Citation Context ... for the transition density functions of killed diffusions in a domain D (equivalently, the Dirichlet heat kernels) have been established only recently. See [11, 12, 13] for upper bound estimates and =-=[21]-=- for the lower bound estimates of the Dirichlet heat kernels in bounded C 1,1 domains. Markov processes with discontinuous sample paths constitute an important family of stochastic processes in probab... |

21 |
Censored stable processes, Probab. Theory Related Fields
- Bogdan, Burdzy, et al.
- 2003
(Show Context)
Citation Context ...ansition density function of the killed α-stable process) in any C 1,1 open set. Another important family of discontinuous Markov processes is the family of censored α-stablelike processes studied in =-=[3]-=- (see Section 2 for the precise definition). For any open subset D of R d , a censored α-stable-like process X in D is a strong Markov process whose infinitesimal generator is given by L α � Du(x) := ... |

19 | Censored stable processes - Bogdan, Burdzy, et al. - 2003 |

16 |
The equivalence of certain heat kernel and Green function bounds
- Davies
- 1987
(Show Context)
Citation Context ...0-20010-0). 1sboundary, two-sided estimates for the transition density functions of killed diffusions in a domain D (equivalently, the Dirichlet heat kernels) have been established only recently. See =-=[11, 12, 13]-=- for upper bound estimates and [21] for the lower bound estimates of the Dirichlet heat kernels in bounded C 1,1 domains. Markov processes with discontinuous sample paths constitute an important famil... |

14 |
function estimate for censored stable processes, Probab. Theory Related Fields
- Chen, Kim, et al.
- 2002
(Show Context)
Citation Context ... 0 ∧ u (α−1)/2 � 0 � δD(x) α−1δD(y) α−1 |x − y| 2α−2 ∧ δD(x) (α−1)/2δD(y) (α−1)/2 |x − y| α−1 � In the last estimate, we used the fact that D is bounded. This proves the corollary. ✷ Remark 5.1 As in =-=[4]-=-, estimates of the Green functions can be used to show that the Martin boundaries and minimal Martin boundaries of a large class of censored stable-like processes can all be identified with the Euclid... |

13 |
A construction of Markov processes by piecing out
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- 1966
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Citation Context ...Lebesgue measure of A. 2 Censored stable-like process and intrinsic ultracontractivity Censored α-stable-like processes in open subsets of R d were studied by Bogdan, Burdzy and Chen in [3] (see also =-=[18]-=-). Fix an open set D in R d with d ≥ 1. Define a bilinear form E on C ∞ c (D) by E(u, v) := 1 � � (u(x) − u(y))(v(x) − v(y)) 2 D D C(x, y) where C(x, y) is a measurable symmetric function on D × D sat... |

11 | Boundary Harnack inequality for regional fractional Laplacian
- Guan
- 2007
(Show Context)
Citation Context ...e precisely, zy ∈ ∂D is the unique point such that δD(y) = |y − zy|. Put M3 := c + sup x,y∈D,|x−y|<r0 (|∇yψ1(x, y)| + |∇yψ2(x, y)|) (3.5) with c being the constant in (3.3) and (3.4). It is proved in =-=[15]-=- that under the assumptions (3.3) and (3.4), the boundary Harnack principle holds for the censored stable-like process X in a C 1,1 open set D with the characteristics (r0, Λ0) and the comparison cons... |

7 |
Reflected symmetric α-stable processes and regional fractional Laplacian. Probab. Theory Related Fields 134(4
- Guan, Ma
- 2006
(Show Context)
Citation Context ...itesimal generator has zero Dirichlet boundary condition as opposed to zero exterior condition. This indicates that censored processes are natural and important for boundary problems in analysis (cf. =-=[16, 17]-=-). Now we state the main result of this paper. We assume the censored stable-like processes under consideration enjoy the dilation invariant boundary Harnack principle (BHP) (see Section 3 for the pre... |

5 |
Boundary problems for fractional Laplacians
- Guan, Ma
(Show Context)
Citation Context ...itesimal generator has zero Dirichlet boundary condition as opposed to zero exterior condition. This indicates that censored processes are natural and important for boundary problems in analysis (cf. =-=[16, 17]-=-). Now we state the main result of this paper. We assume the censored stable-like processes under consideration enjoy the dilation invariant boundary Harnack principle (BHP) (see Section 3 for the pre... |