## ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS (2004)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Bendikov04onthe,

author = {Alexander Bendikov and Laurent Saloff-coste},

title = {ON THE REGULARITY OF SAMPLE PATHS OF SUB-ELLIPTIC DIFFUSIONS ON MANIFOLDS},

year = {2004}

}

### OpenURL

### Abstract

Using heat kernel Gaussian estimates and related properties, we study the intrinsic regularity of the sample paths of the Hunt process associated to a strictly local regular Dirichlet form. We describe how the results specialize to Riemannian Brownian motion and to sub-elliptic symmetric diffusions. 1.

### Citations

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Citation Context ...lizations of the following celebrated properties of the Brownian sample paths: (i) The Lévy-Khinchine law of the iterated logarithm asserts that, almost surely, lim sup = 1 0 4 log log(1 ) See, e.g., =-=[30, 33, 40]-=-. (ii) Lévy’s result on the modulus of continuity of Brownian paths asserts that, almost surely, lim 0 sup = 1 0 1 4( ) log(1 ( )) See, e.g., [30, 33, 40]. (iii) If 3, the theorem of Dvoretski and Erd... |

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Citation Context ...n particular, ( ) ( ) is a continuous function. • ( ) is a complete metric space. These hypotheses imply that ( ) is a path metric space (i.e., can be defined in terms of “shortest paths”). See e.g., =-=[10, 24]-=- and [46]. Path metric spaces are also called length spaces or inner metric spaces. It also implies that the cut-off functions : sup ( ) 0 = ( ( ) )+ are in D C0( ) and satisfy ( ) . This is a crucial... |

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Citation Context ...ating more general setups are harder to find. The best studied of the three problems discussed here is probably the law of iterated logarithm and its more advanced version due to Strassen (see, e.g., =-=[14, 45]-=-). The work [3] is very much in the spirit of the present paper. There is much less literature on Dvoretzky-Erdös rate of escape [15]. For random walks in R , see [18, 39]. Below we present applicatio... |

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Diusion Processes and their Sample Paths
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Citation Context ...lizations of the following celebrated properties of the Brownian sample paths: (i) The Lévy-Khinchine law of the iterated logarithm asserts that, almost surely, lim sup = 1 0 4 log log(1 ) See, e.g., =-=[30, 33, 40]-=-. (ii) Lévy’s result on the modulus of continuity of Brownian paths asserts that, almost surely, lim 0 sup = 1 0 1 4( ) log(1 ( )) See, e.g., [30, 33, 40]. (iii) If 3, the theorem of Dvoretski and Erd... |

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Citation Context ...ollows). Consider a set = 1 of left invariant vector fields such that the Lie algebra generated by equals the Lie algebra of . In such cases, we say that satisfies the Hörmander condition (see, e.g., =-=[28, 38, 56]-=-). 2 Consider the Dirichlet form E on ( ) obtained as the least extension of 1 2 C 0 ( )716 A. BENDIKOV AND L. SALOFF-COSTE We let ( P ) be the associated diffusion. If is a right Haar measure, then ... |

177 |
On the parabolic kernel of the Schrödinger operator
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- 1986
(Show Context)
Citation Context ...erator = div grad. Then, for any , P -almost surely, lim sup 0 ( ) = 1 4 log log(1 ) Proof. This easily follows from Sections 4.1 and 5.1. The needed Gaussian heat kernel bounds can be extracted form =-=[34]-=- (see also [43, 54]). REMARK. This Theorem extends without changes to manifolds with boundary as long as we consider “interior” starting point (i.e., points that are not on the boundary). In fact, the... |

127 |
and metrics defined by vector fields. I: Basic properties
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Citation Context ...ollows). Consider a set = 1 of left invariant vector fields such that the Lie algebra generated by equals the Lie algebra of . In such cases, we say that satisfies the Hörmander condition (see, e.g., =-=[28, 38, 56]-=-). 2 Consider the Dirichlet form E on ( ) obtained as the least extension of 1 2 C 0 ( )716 A. BENDIKOV AND L. SALOFF-COSTE We let ( P ) be the associated diffusion. If is a right Haar measure, then ... |

84 |
A Harnack inequality for parabolic differential equations
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Citation Context ...ction ( ) = ( ) which is a solution in (0 ) . The proof of (3.15) follows from the proof of [43, Theorem 5.2.10]. For the proof of the second statement based on the classical Moser iteration argument =-=[37]-=-, see, e.g., [43, Theorem 5.2.9]. One can easily state a version of Theorem 3.4 for the case where the various properties are considered “up to scale 0.” See [41, 42, 43]. On occasion, we will also co... |

82 |
Brownian motion on the Sierpinski gasket, Probab. Theory Related Fields 79
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(Show Context)
Citation Context ... results described below. Similar techniques have been used by several authors to prove analogs of the law of iterated logarithm and related results in various settings including fractals. See, e.g., =-=[1, 2, 3]-=-. Still, it is important to realize that such results are not entirely universal (compare with Takeda’s inequality stated in 3.8 below) and that hypotheses of some sort are needed for a Hunt process a... |

79 |
The heat equation on non-compact Riemannian manifolds
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(Show Context)
Citation Context ...ation ( + ) = 0 in = ( ) , we have (3.19) sup inf + where + = = 2 4 2 3 2 2 4 2 2 Moser’s iteration technique [37] adapted as in [41, 43] and Theorem 3.1 give the following important result (see also =-=[19]-=- for a different proof). Theorem 3.5. Fix a ball 0 (resp. 0 0). 1. Assume that (3.3) and (3.6) hold around 0 (resp. up to scale 0). Then the parabolic Harnack inequality (3.19) holds around 0 (resp. u... |

76 | Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds - Grigor’yan - 1999 |

75 |
The Poincaré inequality for vector fields satisfying Hormander's condition
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- 1986
(Show Context)
Citation Context ...lity. We say that a (scale-invariant) Poincaré inequality holds around 0 if there exists a constant 0 such that, for any ball 2 0, (3.6) D 2 0 2 2 ( ) where = ( ) 1 and = ( ). It is known (see, e.g., =-=[31]-=- or [43, Corollary 5.3.5]) that (3.3) and (3.6) together imply the stronger inequality (3.7) 2 0 D 2 1 2 ( ) This inequality is equivalent to say that the lowest non-zero Neumann eigenvalue ( ) in the... |

64 | Brownian motion and harmonic analysis on Sierpinski carpets
- Barlow, Bass
- 1999
(Show Context)
Citation Context ... results described below. Similar techniques have been used by several authors to prove analogs of the law of iterated logarithm and related results in various settings including fractals. See, e.g., =-=[1, 2, 3]-=-. Still, it is important to realize that such results are not entirely universal (compare with Takeda’s inequality stated in 3.8 below) and that hypotheses of some sort are needed for a Hunt process a... |

61 |
Aspects of Sobolev-type inequalities
- Saloff-Coste
(Show Context)
Citation Context ...he Poincaré inequality (3.6) together imply the Sobolev inequality (3.8). Theorem 3.1. Fix a ball 0 . Assume that (3.3) and (3.6) holds around 0. Then the Sobolev inequality (3.8) holds around 0. See =-=[36, 43]-=- for proofs that can be adapted to the present setting. For completeness, we recall that (3.8) can be characterized in terms of what is called a Faber-Krahn inequality, i.e., an inequality relating th... |

59 |
A note on Poincaré, Sobolev, and Harnack inequalities
- Saloff-Coste
- 1992
(Show Context)
Citation Context ...f and will play no role in what follows. We say that a local Sobolev inequality holds up to scale 0 if (3.8) holds true for all balls of radius at most 0. A crucial observation that first appeared in =-=[41]-=- is that the doubling property (3.3) and the Poincaré inequality (3.6) together imply the Sobolev inequality (3.8). Theorem 3.1. Fix a ball 0 . Assume that (3.3) and (3.6) holds around 0. Then the Sob... |

48 | Some problems on random walk in space
- Dvoretzky, Erdős
- 1951
(Show Context)
Citation Context ...ii) Lévy’s result on the modulus of continuity of Brownian paths asserts that, almost surely, lim 0 sup = 1 0 1 4( ) log(1 ( )) See, e.g., [30, 33, 40]. (iii) If 3, the theorem of Dvoretski and Erdös =-=[15]-=- concerning the “rate of escape” of Brownian motion asserts that, for any continuous increasing positive function , one has lim inf 0 ( ) = + 0 almost surely iff [ (2 )] 2 converges diverges. The two ... |

47 | Gaussian estimates for Markov chains and random walks on groups
- Hebisch, Saloff-Coste
- 1993
(Show Context)
Citation Context ...ated logarithm and in Lévy’s modulus of continuity should be changed to a factor 2. The techniques used in this paper are robust and apply without essential changes to some other settings. The papers =-=[21, 26, 52]-=- contain some long time results that are closely related in spirit to the short time results described below. Similar techniques have been used by several authors to prove analogs of the law of iterat... |

45 |
Estimates of heat kernels on Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context ...he same for both inequalities if 2. Next we recall the characterization of (3.8) in terms of heat kernel upper bounds. Proofs that can be adapted to the present setting can be found in [43]. See also =-=[20]-=-. Theorem 3.3. Fix a ball 0 . 1. Assume that the scale-invariant local Sobolev inequality (3.8) holds around 0. Then the doubling property (3.3) holds around 0 and there exists a constant such that fo... |

40 | On the relation between elliptic and parabolic Harnack inequalities - Hebisch, Saloff-Coste |

37 | Analysis on Lie groups - Varopoulos - 1996 |

36 | A lower bound for the heat kernel - Cheeger, Yau - 1981 |

35 | Fundamental solutions for second order subelliptic operators - Fefferman, Sánchez-Calle - 1986 |

34 | Non-Gaussian aspects of heat kernel behaviour - Davies - 1997 |

30 |
Some theorems concerning 2-dimensional Brownian motion
- Spitzer
- 1958
(Show Context)
Citation Context ...EXAMPLE (Brownian motion in the plane). Of course, Brownian motion in the plane is recurrent. However, locall escape still occurs albeit at a very slow rate. As mentioned in the introduction, Spitzer =-=[44]-=- proved the result in this case. Applying the last statement of Theorem 6.7 we see that the local rate of escape is slower than any power function. In fact, for planar Brownian motion and ( ) = exp lo... |

26 | Semi-groups of measures on Lie groups - Hunt |

17 |
Subelliptic second order differential operator
- Jerison, Sànchez-Calle
- 1987
(Show Context)
Citation Context ...sical Sobolev space in R . If = R , we 1 say that is uniformly sub-elliptic if (6.16) holds, and the functions and their partial derivatives of any order are bounded functions. We refer the reader to =-=[32]-=- for an excellent exposition concerning such operators and for further references. Note that sums of squares of Hörmander vector fields are sub-elliptic (see [28, 32]). Consider a regular strictly loc... |

16 |
Analyse sur les boules d’un opérateur sous-elliptique
- Maheux, Saloff-Coste
- 1995
(Show Context)
Citation Context ...he Poincaré inequality (3.6) together imply the Sobolev inequality (3.8). Theorem 3.1. Fix a ball 0 . Assume that (3.3) and (3.6) holds around 0. Then the Sobolev inequality (3.8) holds around 0. See =-=[36, 43]-=- for proofs that can be adapted to the present setting. For completeness, we recall that (3.8) can be characterized in terms of what is called a Faber-Krahn inequality, i.e., an inequality relating th... |

13 |
Parabolic Harnack inequality for divergence-form second-order differential operators
- Saloff-Coste
- 1995
(Show Context)
Citation Context ...he classical Moser iteration argument [37], see, e.g., [43, Theorem 5.2.9]. One can easily state a version of Theorem 3.4 for the case where the various properties are considered “up to scale 0.” See =-=[41, 42, 43]-=-. On occasion, we will also consider a weaker type of mean value inequality. We say that the Dirichlet space (E D 2 ( )) satisfies a -mean value inequality around 0 (resp. up to scale 0) if there exis... |

13 |
On the geometry defined by Dirichlet forms
- Sturm
- 1993
(Show Context)
Citation Context ...’ work on Gaussian heat kernel bounds, see, e.g., [12, Theorem 3.2.7]. Details concerning the intrinsic distance in the case of general regular strictly local Dirichlet spaces are found in [6, 7] and =-=[46, 47, 51]-=-. We now make a couple of crucial hypotheses about the Dirichlet space (E D 2 ( )), in terms of the intrinsic distance . Throughout the paper we assume that the following properties are satisfied. • T... |

13 | On a martingale method for symmetric diffusion process and its applications
- Takeda
- 1989
(Show Context)
Citation Context ...ated logarithm and in Lévy’s modulus of continuity should be changed to a factor 2. The techniques used in this paper are robust and apply without essential changes to some other settings. The papers =-=[21, 26, 52]-=- contain some long time results that are closely related in spirit to the short time results described below. Similar techniques have been used by several authors to prove analogs of the law of iterat... |

9 |
Opérateurs elliptiques dégénérés associés aux axiomatiques de la théorie du potentiel
- Bony
- 1969
(Show Context)
Citation Context ...he proof follows from the general results of the present paper and the theory developed in [56] which provides the necessary parabolic Harnack inequality up to scale 1 (this in fact goes back to Bony =-=[9]-=-). The precise Gaussian heat kernel lower bound needed to obtain the sharp form of the law of iterated logarithm and the modulus of continuity are taken from [55]. 6.3. Sub-elliptic diffusions. Let be... |

8 |
off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces
- Bendikov, Saloff-Coste, et al.
- 2004
(Show Context)
Citation Context ... and satisfies the triangle in-680 A. BENDIKOV AND L. SALOFF-COSTE equality. For = , it might happen that ( ) = 0 (e.g., on fractals) or + (this actually happens in some interesting cases (see e.g., =-=[4]-=-) but we will not be concerned with such cases in this paper. The idea of the intrinsic distance (or at least its usefulness) seems to have emerged in the eighties in connection with E.B. Davies’ work... |

7 |
Periodic metrics
- Burago
- 1992
(Show Context)
Citation Context ...n particular, ( ) ( ) is a continuous function. • ( ) is a complete metric space. These hypotheses imply that ( ) is a path metric space (i.e., can be defined in terms of “shortest paths”). See e.g., =-=[10, 24]-=- and [46]. Path metric spaces are also called length spaces or inner metric spaces. It also implies that the cut-off functions : sup ( ) 0 = ( ( ) )+ are in D C0( ) and satisfy ( ) . This is a crucial... |

7 | Saloff-Coste L., Hitting probabilities for Brownian motion on Riemannian manifolds
- Grigor’yan
(Show Context)
Citation Context ...p 1 ( ) = inf ( ) + 2 : D C 1 The 1-capacity of is related to the equilibrium measure by (3.24) Cap 1 ( ) = ( ) For all of this, see [8]. We will need the following estimate which is in the spirit of =-=[21, 23]-=- and involves the notion introduced above. Theorem 3.10. For any compact , and any , 0, we have 1 ( ) Cap 1 ( ) sup ( ) Proof. By the Markov property, 1 ( ) = 1 ( ) = ( ) 1 ( ) ( ) Using (3.23), (2.2)... |

7 |
Random thoughts on reversible potential theory
- Lyons
(Show Context)
Citation Context ... ) + sup for all ( ) ( ) ( 3 2 4 2 4) (1 2) . 3.5. Takeda’s inequality. We will make use of the following inequality due to Takeda [52]. The precise form of this inequality stated below is taken from =-=[35]-=-. For any set and 0, set ( ) = inf ( ): = : ( ) Theorem 3.8 ([35, 52]). Let be a compact set. Then P sup (0 ) ( ) ( ) 16 ( ) exp 4 2 The remarkable feature of this inequality is that it holds without ... |

6 |
On the sample paths of Brownian motions on compact infinite dimensional groups
- Bendikov, Saloff-Coste
(Show Context)
Citation Context ... of regular strictly local Dirichlet forms and their associated Hunt processes under some additional assumptions. Without such assumptions, one cannot hope to obtain the results we will describe, see =-=[5]-=-. Our goal is to cover such cases as Brownian motions on Riemannian manifolds and left-invariant symmetric sub-elliptic diffusions on Lie groups. On R , any translation invariant, symmetric, non-degen... |

6 |
Formes de Dirichlet et estimations structurelles dans les milieux discontinus
- Biroli, Mosco
- 1991
(Show Context)
Citation Context ...E.B. Davies’ work on Gaussian heat kernel bounds, see, e.g., [12, Theorem 3.2.7]. Details concerning the intrinsic distance in the case of general regular strictly local Dirichlet spaces are found in =-=[6, 7]-=- and [46, 47, 51]. We now make a couple of crucial hypotheses about the Dirichlet space (E D 2 ( )), in terms of the intrinsic distance . Throughout the paper we assume that the following properties a... |

6 | Sharp estimates for capacities and applications to symmetric diffusions - Sturm - 1995 |

4 |
Laws of the iterated logarithm for some symmetric diffusion processes
- Bass, Kumagai
(Show Context)
Citation Context ... results described below. Similar techniques have been used by several authors to prove analogs of the law of iterated logarithm and related results in various settings including fractals. See, e.g., =-=[1, 2, 3]-=-. Still, it is important to realize that such results are not entirely universal (compare with Takeda’s inequality stated in 3.8 below) and that hypotheses of some sort are needed for a Hunt process a... |

4 |
A Saint-Venant principle for Dirichlet forms on discontinuous
- Biroli, Mosco
- 1995
(Show Context)
Citation Context ...E.B. Davies’ work on Gaussian heat kernel bounds, see, e.g., [12, Theorem 3.2.7]. Details concerning the intrinsic distance in the case of general regular strictly local Dirichlet spaces are found in =-=[6, 7]-=- and [46, 47, 51]. We now make a couple of crucial hypotheses about the Dirichlet space (E D 2 ( )), in terms of the intrinsic distance . Throughout the paper we assume that the following properties a... |

4 |
The geometric aspect of Dirichlet forms
- Sturm
- 1998
(Show Context)
Citation Context ...’ work on Gaussian heat kernel bounds, see, e.g., [12, Theorem 3.2.7]. Details concerning the intrinsic distance in the case of general regular strictly local Dirichlet spaces are found in [6, 7] and =-=[46, 47, 51]-=-. We now make a couple of crucial hypotheses about the Dirichlet space (E D 2 ( )), in terms of the intrinsic distance . Throughout the paper we assume that the following properties are satisfied. • T... |

3 |
Escape rate of Brownian motion on Riemannian manifolds
- Grigor’yan
- 1999
(Show Context)
Citation Context ...ated logarithm and in Lévy’s modulus of continuity should be changed to a factor 2. The techniques used in this paper are robust and apply without essential changes to some other settings. The papers =-=[21, 26, 52]-=- contain some long time results that are closely related in spirit to the short time results described below. Similar techniques have been used by several authors to prove analogs of the law of iterat... |

3 |
Analysis on local Dirichlet spaces-II . Upper Gaussian estimates for the fundamental solutions of parabolic equations
- Sturm
- 1995
(Show Context)
Citation Context ... that (3.10) holds for all , 0 satisfying ( ) 2 0. Then the scale-invariant local Sobolev inequality (3.8) holds around 0. These three theorems admit “up-to-scale- 0” versions. See [20], [41, 43] and =-=[49, 50]-=-. 3.4. Harnack and mean value inequalities. Fix an open set . We say that a function belongs to D loc if for any relatively compact open set with , there exists a function D such that = almost everywh... |

2 |
On a class of hypoelliptic pseudodifferential operators degenerate on a submanifold
- Grushin
- 1971
(Show Context)
Citation Context ...g run. Spitzer [44] notes that this disproves a conjecture of Lévy. EXAMPLE (Grushin’s diffusion). operator on R2 defined by The simplest sub-elliptic operator is the Grushin = ( 2 2 + 2 ) See, e.g., =-=[25, 32]-=-. For this operator, the parabolic Harnack inequality (3.19) holds globally. The volume of small ball centred at a point is quadratic (i.e., = 2 in the notation of Theorem 6.7) if = (0 0) and cubic (i... |

2 |
The rate of escape of random
- Pruitt
- 1990
(Show Context)
Citation Context ... due to Strassen (see, e.g., [14, 45]). The work [3] is very much in the spirit of the present paper. There is much less literature on Dvoretzky-Erdös rate of escape [15]. For random walks in R , see =-=[18, 39]-=-. Below we present applications of the results proved in the previous sections to different natural settings such as Riemannian manifolds and sub-elliptic symmetric diffusions. There are overlaps betw... |

1 |
An integral test for the rate of escape of -dimensional random
- Griffin
- 1983
(Show Context)
Citation Context ... due to Strassen (see, e.g., [14, 45]). The work [3] is very much in the spirit of the present paper. There is much less literature on Dvoretzky-Erdös rate of escape [15]. For random walks in R , see =-=[18, 39]-=-. Below we present applications of the results proved in the previous sections to different natural settings such as Riemannian manifolds and sub-elliptic symmetric diffusions. There are overlaps betw... |

1 |
Probability Theory, An Anlytic View
- Stroock
- 1993
(Show Context)
Citation Context ...ating more general setups are harder to find. The best studied of the three problems discussed here is probably the law of iterated logarithm and its more advanced version due to Strassen (see, e.g., =-=[14, 45]-=-). The work [3] is very much in the spirit of the present paper. There is much less literature on Dvoretzky-Erdös rate of escape [15]. For random walks in R , see [18, 39]. Below we present applicatio... |

1 |
Analysis on local Dirichlet spaces—I, Recurrence, conservativeness and - Liouville properties
- Sturm
- 1994
(Show Context)
Citation Context ...’ work on Gaussian heat kernel bounds, see, e.g., [12, Theorem 3.2.7]. Details concerning the intrinsic distance in the case of general regular strictly local Dirichlet spaces are found in [6, 7] and =-=[46, 47, 51]-=-. We now make a couple of crucial hypotheses about the Dirichlet space (E D 2 ( )), in terms of the intrinsic distance . Throughout the paper we assume that the following properties are satisfied. • T... |

1 |
Analysis on local Dirichlet spaces—III, The parabolic Harnack inequality
- Sturm
- 1996
(Show Context)
Citation Context ... that (3.10) holds for all , 0 satisfying ( ) 2 0. Then the scale-invariant local Sobolev inequality (3.8) holds around 0. These three theorems admit “up-to-scale- 0” versions. See [20], [41, 43] and =-=[49, 50]-=-. 3.4. Harnack and mean value inequalities. Fix an open set . We say that a function belongs to D loc if for any relatively compact open set with , there exists a function D such that = almost everywh... |