## INTRINSIC ULTRACONTRACTIVITY OF NON-SYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS (2008)

Venue: | TOHOKU MATH. J. |

Citations: | 20 - 17 self |

### BibTeX

@MISC{Kim08intrinsicultracontractivity,

author = {Panki Kim and Renming Song},

title = { INTRINSIC ULTRACONTRACTIVITY OF NON-SYMMETRIC DIFFUSION SEMIGROUPS IN BOUNDED DOMAINS},

year = {2008}

}

### OpenURL

### Abstract

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.

### Citations

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Citation Context ...with lim|x|→∞ f(x) = 0. We say a Markov process Y in R d has the Feller property if for every g ∈ C0(R d ), Ex[g(Yt)] is in C0(R d ). Any Lévy process in R d has the Feller property (for example, see =-=[4, 21]-=-). For any open set U, we use τU to denote the first exit time of U for X. i.e., τU := inf{t > 0 : Xt /∈ U}. Given an open set U ⊂ R d , we define X U t (ω) = Xt(ω) if t < τU(ω) and X U t (ω) = ∂ if t... |

416 |
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Citation Context ...with lim|x|→∞ f(x) = 0. We say a Markov process Y in R d has the Feller property if for every g ∈ C0(R d ), Ex[g(Yt)] is in C0(R d ). Any Lévy process in R d has the Feller property (for example, see =-=[4, 21]-=-). For any open set U, we use τU to denote the first exit time of U for X. i.e., τU := inf{t > 0 : Xt /∈ U}. Given an open set U ⊂ R d , we define X U t (ω) = Xt(ω) if t < τU(ω) and X U t (ω) = ∂ if t... |

272 | Heat Kernels and Spectral Theory - Davies - 1990 |

230 | Logarithmic Sobolev inequalities - Gross - 1975 |

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Citation Context ...density function p D (t, x, y) for X D t is strictly positive in D × D. Remark 3.7. Even if the Lévy process has a smooth and strictly positive transition density, it is non-trivial to show (A5) (see =-=[2, 10]-=- for the case of killed Brownian motions in a domain, [6] for the case of killed symmetric stable processes in a domain and [24] for the case of killed non-symmetric stable processes in a domain). If ... |

104 |
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Citation Context ... (A3), {P D t } and { ̂ P D t } are compact operators in L2 (D, dx). Moreover pD (t, x, y) is strictly positive in D × D by (A5). Thus it follows from Jentzsch’s Theorem (Theorem V.6.6 on page 337 of =-=[22]-=-) and the strong continuity of {P D t } and { ̂ P D t } that the common value λ0 := sup Re(σ(AD)) = sup Re(σ( ÂD)) < 0 is an eigenvalue of multiplicity 1 for both AD and ÂD, and that an eigenfunction ... |

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Citation Context ...density function p D (t, x, y) for X D t is strictly positive in D × D. Remark 3.7. Even if the Lévy process has a smooth and strictly positive transition density, it is non-trivial to show (A5) (see =-=[2, 10]-=- for the case of killed Brownian motions in a domain, [6] for the case of killed symmetric stable processes in a domain and [24] for the case of killed non-symmetric stable processes in a domain). If ... |

80 |
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Citation Context ...ppose that U is a subdomain of V . For each T > 0 and y ∈ U, (t,x) → r U (t,x,y) and (t,x) → ̂r U (t,x,y) are parabolic in (0,T] × U for Y and ̂ Y , respectively. Proof. See the proof of Lemma 4.5 in =-=[6]-=-. □ Corollary 3.4. Suppose that U is a subdomain of V . For each T > 0 and y ∈ U, and any nonnegative bounded function f on U, the functions ∫ r U (t,x,y)f(y)dy and g(t,x) := Ex[f(Y U t )] = ̂g(t,x) :... |

78 | Brownian motion and Harnack’s inequality for Schrödinger Hamiltonians - Aizenman, Simon - 1982 |

69 |
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Citation Context ...r B(z,R) ⊂ V (see Theorem 4.4(2) in [17]). In the remainder of this paper, t0 and t1 will always stand for the constants above. With the density estimates (3.1) available, one can follow the ideas in =-=[13]-=- (see also [15, 27]) to prove the parabolic Harnack inequality. For this reason, the proofs of this section will be somewhat sketchy. Lemma 3.1. For each 0 < δ, u < 1, there exists ε = ε(d,δ,u,t1) > 0... |

68 |
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Citation Context ..., with respect to a certain reference measure, Y has a dual process which is a continuous Hunt process satisfying the strong Feller property. The notion of intrinsic ultracontractivity, introduced in =-=[11]-=- for symmetric semigroups, is a very important concept and has been studied extensively. In [18], the concept of intrinsic ultracontractivity was extended to nonsymmetric semigroups and it was proven ... |

60 | Logarithmic Sobolev inequalities and contractivity properties of semigroups - Gross - 1993 |

53 |
Boundary Behavior of Harmonic Functions
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Citation Context ...ins a ball B(Ar(Q), κr) for some Ar(Q) ∈ D. The pair (R, κ) is called the characteristics of the κ-fat open set D. 6Note that every Lipschitz domain and every non-tangentially accessible domain (see =-=[15]-=- for the definition of non-tangentially accessible domains) are κ-fat. Moreover, every John domain is κ-fat (see Lemma 6.3 in [20]). The boundary of a κ-fat open set can be highly nonrectifiable and, ... |

46 |
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Citation Context ... Many results in this section are stated for both X D and its dual ̂ X D . Since the proofs for the two processes are similar, we only present the proofs for X. The following definition is taken from =-=[23]-=-. Definition 3.1. Let κ ∈ (0, 1/2]. We say that an open set D in R d is κ-fat if there exists R > 0 such that for each Q ∈ ∂D and r ∈ (0, R], D ∩ B(Q, r) contains a ball B(Ar(Q), κr) for some Ar(Q) ∈ ... |

38 |
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Citation Context ...intrinsic ultracontractive for a large class of non-smooth domains (see, for instance [1, 3]). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in =-=[6, 7, 19]-=-. After obtaining the main results of this paper, we found out from [13] that the intrinsic ultracontractivity for some large classes of symmetric Lévy processes was studied in [12]. Very recently in ... |

32 |
On some relations between the harmonic measure and the Lévy measure for a certain class of Markov processes
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Citation Context ...y < ∞, x ∈ U and GU(x, ·) is well-defined a.e. U. Also (2.5) implies that for every open set U ⊂ D and A ⊂ Uc with dist(A, U) > 0, we have ∫ ∈ A) = GU(x, y)ν(y − A)dy. (2.6) Px (XτU (for example, see =-=[14]-=-). Similarly the Green function ̂ GU(x, y) of X in U is defined as ∫ ∞ ̂GU(x, y) := ̂p U (t, y, x)dt, (x, y) ∈ U × U, 0 U which is well-defined a.e. U. For every A ⊂ Uc with dist(A, U) > 0, we have ( ... |

29 |
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Citation Context ...ian in a domain D (equivalently, the corresponding process is a killed Brownian motion), the semigroup {e Ht } is intrinsic ultracontractive for a large class of non-smooth domains (see, for instance =-=[1, 3]-=-). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in [6, 7, 19]. After obtaining the main results of this paper, we found out from [13] that the ... |

27 |
Intrinsic ultracontractivity for symmetric stable processes
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Citation Context ...intrinsic ultracontractive for a large class of non-smooth domains (see, for instance [1, 3]). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in =-=[6, 7, 19]-=-. After obtaining the main results of this paper, we found out from [13] that the intrinsic ultracontractivity for some large classes of symmetric Lévy processes was studied in [12]. Very recently in ... |

23 | Ryznar: Estimates of Green functions for some perturbations of fractional Laplacian
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Citation Context ...nstance [1, 3]). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in [6, 7, 19]. After obtaining the main results of this paper, we found out from =-=[13]-=- that the intrinsic ultracontractivity for some large classes of symmetric Lévy processes was studied in [12]. Very recently in [17], we extended the concept of intrinsic ultracontractivity to nonsymm... |

22 | Two-sided estimates on the density of Brownian motion with singular drift
- Kim, Song
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Citation Context ...and ν is a (nonnegative) measure belonging to the Kato class Kd,2 (see below for the definitions of Kd,1 and Kd,2). The existence and uniqueness of this process Y were proven in Bass and Chen [3]. In =-=[15, 16, 17, 19]-=-, we have studied properties of diffusions with measure-valued drifts in bounded domains. Using results in [15, 16, 17, 19], we will prove that, with respect to a certain reference measure, Y has a du... |

22 | Potential theory of truncated stable processes - Kim, Song |

21 | Gaugeability and conditional Gaugeability
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Citation Context ...B(w,r), we have E z x[exp(−A V τX )] ≥ exp(−E B(w,r) z x[A V τX ]) ≥ e B(w,r) −c1 > 0. Combining the identity GB(w,r)(x,z) = G X B(w,r) (x,z)Ezx[exp(−A V τX )], x,z ∈ B(w,r), B(w,r) (Lemma 3.5 (1) of =-=[5]-=-) with (2.9), we arrive at the following result. Proposition 2.7. There exist positive constants c and r1 := r1(d,µ,ν) such that for all r ∈ (0,r1] and B(w,r) ∈ V , we have c −1 G 0 B(w,r) (x,y) ≤ G B... |

17 | Brownian motion with singular drift - Bass, Chen - 2003 |

16 |
Markov processes, Brownian motion, and time symmetry, volume 249 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences
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Citation Context ... S < ∞) = 0. ✷ By the separation property for Feller processes, there exists t0 such that inf Py(τB2 > t) ≥ y∈C1 1 2 and inf Py(̂τB2 > t) ≥ y∈C1 1 2 (3.2) for any t ≤ t0 (see Exercise 2 on page 73 of =-=[9]-=-). 9Lemma 3.5. If (A1)-(A4) are true, then there exists c > 0 such that for any t ≤ t0, ∫ Px(Xt ∈ B2, τD > t) ≥ c GD(x, y)dy, x ∈ D D\B2 and Px( ̂ Xt ∈ B2, ̂τD > t) ≥ c Proof. Note that by Lemma 3.4,... |

15 | Intrinsic ultracontractivity, conditional lifetimes and conditional gauge for symmetric stable processes on rough domains
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(Show Context)
Citation Context ...intrinsic ultracontractive for a large class of non-smooth domains (see, for instance [1, 3]). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in =-=[6, 7, 19]-=-. After obtaining the main results of this paper, we found out from [13] that the intrinsic ultracontractivity for some large classes of symmetric Lévy processes was studied in [12]. Very recently in ... |

15 |
cubes, p-capacity, and Minkowski content, Exposition
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Citation Context ...ry Lipschitz domain and every non-tangentially accessible domain (see [15] for the definition of non-tangentially accessible domains) are κ-fat. Moreover, every John domain is κ-fat (see Lemma 6.3 in =-=[20]-=-). The boundary of a κ-fat open set can be highly nonrectifiable and, in general, no regularity of its boundary can be inferred. Bounded κ-fat open sets may be disconnected. Depending on whether (A1)(... |

14 | First eigenvalues and Comparison of Green’s functions for elliptic operators on manifolds or domains - Ancona - 1997 |

14 | Boundary Harnack principle for Brownian motions with measure-valued drifts in bounded Lipschitz domains
- Kim, Song
- 2007
(Show Context)
Citation Context ...and ν is a (nonnegative) measure belonging to the Kato class Kd,2 (see below for the definitions of Kd,1 and Kd,2). The existence and uniqueness of this process Y were proven in Bass and Chen [3]. In =-=[15, 16, 17, 19]-=-, we have studied properties of diffusions with measure-valued drifts in bounded domains. Using results in [15, 16, 17, 19], we will prove that, with respect to a certain reference measure, Y has a du... |

13 | Relative Fatou’s theorem for (−∆) α/2 -harmonic function in κ-fat open set
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(Show Context)
Citation Context ...r hand, since 1 is excessive for ̂ Y D , it is easy to see that (1/h2( ̂ Y D ), Ft) is a supermartingale with respect to Ph2 x , where Ft is the natural filtration of { ̂ Y D } (see, e.g., page 83 in =-=[14]-=-). Thus, with the same proof, one can see that the first inequality in equation (8) on page 179 of [8] is true. Thus, there exists c1 independent of h2 and δ such that (5.2) sup E x∈D h2 x [̂τDδ ] ≤ c... |

13 | Estimates on Green functions and Schrödingertype equations for non-symmetric diffusions with measure-valued drifts
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(Show Context)
Citation Context ...h other with respect to ξ. For any domain U ⊂ V , we define ̂r U (t,x,y) := rU (t,y,x)H(y) . H(x) Since H is strictly positive and continuous, by the joint continuity of r U (t,x,y) (see Section 4 of =-=[17]-=- and the references therein), ̂r U (t,x,y) is jointly continuous on U ×U. Thus, ̂r U (t,x,y) is the transition density of ̂ Y U with respect to the Lebesgue measure and (2.7) ̂GU(x,y) := GU(y,x)H(y) H... |

12 |
Intrinsic ultracontractivity and the Dirichlet Laplacian
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Citation Context ...l are intrinsically ultracontractive. The fact that the parabolic boundary Harnack principle implies the intrinsic ultracontractivity in the symmetric diffusion case was used and discussed in [2] and =-=[12]-=-. As a consequence of the intrinsic ultracontractivity, we have that the supremum of the expected conditional lifetimes of Y D is finite if D is one of the domains above. Many results in this paper ar... |

11 | On dual processes of non-symmetric diffusions with measure-valued drifts
- Kim, Song
- 2008
(Show Context)
Citation Context ...and ν is a (nonnegative) measure belonging to the Kato class Kd,2 (see below for the definitions of Kd,1 and Kd,2). The existence and uniqueness of this process Y were proven in Bass and Chen [3]. In =-=[15, 16, 17, 19]-=-, we have studied properties of diffusions with measure-valued drifts in bounded domains. Using results in [15, 16, 17, 19], we will prove that, with respect to a certain reference measure, Y has a du... |

9 | Elements of Functional Analysis - Brown, Page - 1970 |

9 |
Riesz representation and duality of Markov processes
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Citation Context ... results in Sections 2–3 of [19] are true for Y . In particular, using the same arguments in the proofs of Theorems 2.4–2.5 in [19], it is easy to check that the conditions (i)–(vii) and (70)–(71) in =-=[20]-=- (also, see the Remark on page 391 in [21]) are satisfied. Thus, with respect to the reference measure ξ, Y has a nice dual process. For more detailed arguments, we refer readers to [19].8 P. KIM AND... |

7 | Semismall perturbations in the Martin theory for elliptic equations - Murata - 1997 |

7 | Intrinsic ultracontractivity for Lévy processes
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- 2008
(Show Context)
Citation Context ...discussed in [6, 7, 19]. After obtaining the main results of this paper, we found out from [13] that the intrinsic ultracontractivity for some large classes of symmetric Lévy processes was studied in =-=[12]-=-. Very recently in [17], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups and, by using an analytic method, we proved there that the semigroup of a killed diffusion p... |

7 |
Lifetimes of conditioned diffusions
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(Show Context)
Citation Context ... its dual are intrinsically ultracontractive. The fact that the parabolic boundary Harnack principle implies the intrinsic ultracontractivity in the symmetric diffusion case was used and discussed in =-=[2]-=- and [12]. As a consequence of the intrinsic ultracontractivity, we have that the supremum of the expected conditional lifetimes of Y D is finite if D is one of the domains above. Many results in this... |

6 | Supercontractivity and ultracontractivity for (non-symmetric) diffusion semigroups on manifolds - Röckner, Wang |

6 |
Lifetimes of conditioned diffusions. Probab. Theory Relet. Fields 91
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(Show Context)
Citation Context ...ian in a domain D (equivalently, the corresponding process is a killed Brownian motion), the semigroup {e Ht } is intrinsic ultracontractive for a large class of non-smooth domains (see, for instance =-=[1, 3]-=-). For symmetric α-stable processes with α ∈ (0, 2), the intrinsic ultracontractivity has been discussed in [6, 7, 19]. After obtaining the main results of this paper, we found out from [13] that the ... |

6 |
Greenian bounds for Markov processes
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(Show Context)
Citation Context ...∈ U) < 1. Thus U A θ := sup x∈Rd Px(τU > t1) ≤ sup x∈Rd Px(Xt1 ∈ U) < 1. By the Markov property and an induction argument, sup x∈Rd Px(τU > nt1) ≤ θ n . Thus supEx[τU] ≤ x∈U 5 t1 1 − θ < ∞ (2.5)(see =-=[8]-=- for the details). For any bounded open subset U ⊂ D, we will use GU(x, y) to denote the Green function of X in U. i.e., ∫ ∞ GU(x, y) := p U (t, x, y)dt, (x, y) ∈ U × U. By (2.5), 0 ∫ Ex[τU] = U GU(x,... |

6 |
Doubly-Feller process with multiplicative functional
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(Show Context)
Citation Context ...every f ∈ L ∞ (V ), Ex[f(Yt)] is bounded and continuous in V . In particular, the above proposition implies that for any domain U ⊂ V , Y U is Hunt process with the strong Feller property (see, e.g., =-=[7]-=-). We will use G(x,y) to denote the Green function of Y . For any domain U ⊂ V , we will use GU(x,y) to denote the Green function of Y U . Thus, ∫ ∞ ∫ ζ ∫ Ex f(Yt)dt = Ex f(Yt)dt = G(x,y)f(y)dy 0 V 0... |

6 | The lifetime of conditional Brownian motion in the
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- 1984
(Show Context)
Citation Context ...e with respect to Ph2 x , where Ft is the natural filtration of { ̂ Y D } (see, e.g., page 83 in [14]). Thus, with the same proof, one can see that the first inequality in equation (8) on page 179 of =-=[8]-=- is true. Thus, there exists c1 independent of h2 and δ such that (5.2) sup E x∈D h2 x [̂τDδ ] ≤ c1 k0 ∑ sup Ex[̂τ V δ k k=−∞ x∈D Combining (5.1)–(5.2), we have that for each s > 0, there exists δ > 0... |

6 |
A new setting for potential theory
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(Show Context)
Citation Context ... (See Lemma 6.4 in [19] and its proof.) Now, using the above properties and (2.4), we see that Y is a transient diffusion with its Green function G(x,y) with respect to ξ satisfying the conditions in =-=[9]-=- and [23] (see (A1)–(A4) in [19]). In fact, one can follow the arguments in [19] and check that all the results in Sections 2–3 of [19] are true for Y . In particular, using the same arguments in the ... |

4 | On the quasi-regularity of semi-Dirichlet forms’, Potential Analysis 15 - Fitzsimmons - 2001 |

3 | R.G.,The lifetimes of conditioned diffusion processes - Pinsky - 1990 |

1 | Hypercontractivity estimates for nonselfadjoint diffusion semigroups - PINSKY - 1988 |

1 | Boundary behavior of harmnoic functions for truncated stable processes - Kim, Song |

1 |
Basic potential theory of certain nonsymmetric strictly α-stable processes
- Zoran
(Show Context)
Citation Context ...tely continuous with respect to the Lévy measure of X, the semigroup of X D in any bounded open set D is intrinsic ultracontractive. In particular, for the non-symmetric stable process X discussed in =-=[24]-=-, the semigroup of X D is intrinsic ultracontractive for any bounded set D. Using the intrinsic ultracontractivity, we show that the parabolic boundary Harnack principle is true for those processes. M... |