Acta Mathematica Sinica, New Series
1995, Vol.11, No.3, pp. 247–251
On the Hausdorff Dimension of the Inverse Image of
a Compact Set Under a Levy Process
Liu Luqin
Abstract. In this paper, we obtain some formulas for determining the Hausdorff dimension of the
inverse image of compact sets under general Levy processes in Rn .
Keywords.
Levy process, Hausderff dimension, Inverse image
§1. Introduction
Let X = (Ω, F, Ft , Xt , θt , P x ) be a Lévy process in Rn (n ≥ 1) with exponent ψ(z), i.e., X
is a Hunt process with stationary independent increments and
E 0 (exp(iz, Xt )) = exp(−tψ(z)),
where
1
ψ(z) = ia, z + zSz +
2
ix, z
ν(dx)
1 − exp(ix, z) +
1 + |x|2
with a ∈ Rn , S a non-negative definite symmetric n × n matrix, and ν a Borel measure on Rn
satisfying
|x|2
ν(dx) < ∞.
1 + |x|2
For any compact set K ⊂ Rn , our aim is to obtain the formula for dim X −1 (K) where dim
denotes “Hausdorff dimension” and
X −1 (K) = {t ≥ 0 : Xt ∈ K}.
When K = {x} consists of only one point, the results have been got by Blumenthal and Getoor
for general Markov processes (see [1]) and by Hawkes for Levy processes in R1 (see [2]). When
X is a stable process, Hawkes[3] gives the formula for dim X −1 (B) for any Borel set B.
The notation and terminalogy will be those of [4] and [5]. For any compact K ⊂ Rn , define
M(K) = {µ : µ is a Radon measure concentrated on K},
Received March 20, 1992. Revised November 12, 1992.
Supported by the British Council.
(1)
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Acta Mathematica Sinica, New Series
Vol.11 No.3
and
α
−1
2
β(K) = inf α : 0 < α ≤ 1, Re((1 + ψ (z)) )|µ̂(z)| dz < ∞ for some µ ∈ M(K) ,
(2)
where µ̂(z) is the Fourier transform of µ(inf φ = 1). We use m to denote the Lebesgue measure
on Rn .
If K = {x} consists of only one point, then
µ̂(z) = exp{iz, y}µ(dy) = c exp{iz, x}
for µ ∈ M(K), where µ has the form µ(A) = cIA (x) with c constant. So, in this case we have
α
−1
β(K) = β({x}) = inf α : 0 < α ≤ 1, Re[(1 + ψ (z)) ]dz < ∞ ,
which is just the index 1/b defined by Hawkes[2] when n = 1.
If X is a symmetric stable process in R1 of index α(0 < α ≤ 2) with exponent ψ(z) = c|z|α ,
then by Lemma 2.2, (1.11) and (2.2) of [6] we have for 0 < u ≤ 1,
∃ µ ∈ M(K) such that
Re[(1 + ψ u (z))−1 ]|µ̂(z)|2 dz < ∞ ⇐⇒ C1 (K) > 0,
where C1 (K) denotes the 1-capacity of K with respect to a stable process in R1 of index αu.
Hence by Lemma 2.1 of [2] and the fact that all stable processes of index β, β < 1, in R1 have
the same class of polar sets (see, for example, Lemma 2 of [3]), we have dim K > 1 − αu =⇒
∃µ ∈ M(K) such that Re[(1 + ψ u (z))−1 ]|µ̂(z)|2 dz < ∞ =⇒ dim K ≥ 1 − αu, provided that
u < 1/α. Thus, it is easy to see
β(K) = inf{u : 0 < u ≤ 1, dim K ≥ 1 − αu} = (1 − dim K)/α
provided that dim K > 1 − α. So, our results in this paper contain Hawkes’ results in [3], [2] as
corollaries.
§2. The Recurrent Case
Theorem 1. If X is recurrent, then for any compact K,
P x (dim X −1 (K) = 1 − β(K)) = 1 for m-a.e. x ∈ Rn .
(3)
Proof. First of all, we show
P x (dim X −1 (K) ≥ 1 − β(K)) = 1 for any x ∈ Rn .
(4)
If β(K) = 1 there is nothing to prove. Assume β(K) < 1. Choose β(K) ≤ α < 1, 0 < α so that
∃ µ ∈ M(K) such that
Re((1 + ψ α (z))−1 )|µ̂(z)|2 dz < ∞.
Let Tt (ω ) be a stable subordinator of index α on some probability space (Ω , F , P ) which is
independent of Xt (ω). Thus Yt = XTt is a Levy process on (Ω × Ω , F × F , P x × P ) whose
Liu Luqin
On the Hausdorff Dimension of the Inverse Image
249
exponent is ψ α (z) (see, for example, [3]). Let Kα = {x ∈ Rn : P x × P (∃ t > 0 such that
Yt ∈ K) = 1} and let Kαr be the set of points Yt -regular for K so that Kαr ⊂ Kα . Thus Kαr = ∅
because K is not polar for Yt by Hawkes[9] and Yt satisfies Hunt’s hypothesis (H) (see [7]).
∀y ∈ Kα ,
(5)
P y × P (XTt ∈ K for some t > 0) = 1.
By Fubini Theorem,
P y {ω : P (ω : Tt (ω ) ∈ X −1 (K)(ω) for some t > 0) = 1} = 1,
and hence
P y (ω : X −1 (K)(ω) is non-polar for Tt ) = 1.
By Lemma 2.1 of [2], we have
P y (ω : dim X −1 (K) ≥ 1 − α) = 1
∀y ∈ Kα .
(6)
Again from [7] we know that K − Kαr is polar for Yt as it is semipolar for Yt . Hence K − Kα is
polar for Yt . From (5) we can see
P y × P (XTt ∈ Kα for some t > 0) = 1
∀y ∈ Kα .
Fixing y ∈ Kα , by [4, p.58, 60, 73, 75] we can choose a compact K0 such that
P y × P (XTt ∈ K0 for some t > 0) > 0 and K0 ⊂ Kα .
By Fubini Thm. we have that K0 is non-polar for Xt (note that t → Tt (ω ) is strictly increasing,
see [8]). Since X is recurrent, we obtain (see, for example, [5, p.122])
P x (Xt ∈ K0 for some t > 0) = 1
∀x ∈ Rn .
(7)
Let TK0 = inf{t > 0 : Xt ∈ K0 }. By the strong Markov property and (6) we have
P x (TK0 < ∞) = P x [P X(TK0 ) (dimX −1 (K) ≥ 1 − α); TK0 < ∞]
= P x [dim X −1 (K)(θTK0 ) ≥ 1 − α).
But X −1 (K)(θTK0 ) = {t : XtK0 +t ∈ K}, so TK0 + X −1 (K)(θTK0 ) ⊂ X −1 (K). Hence dim X −1
(K)(θTK0 ) ≤ dim X −1 (K). Thus we have from (7) for any x ∈ Rn that
1 = P x (TK0 < ∞) ≤ P x (dim X −1 (K) ≥ 1 − α).
(8)
Now by letting α decrease to β(K) through a countable sequence we have (4).
Next we prove
P x (dim X −1 (K) ≤ 1 − β(K)) = 1 for m-a.e. x ∈ Rn .
(9)
It is trivial if β(K) = 0. Suppose 0 < α < β(K). So for any µ ∈ M(K), Re((1 + ψ α (z))−1 )
|µ̂(z)|2 dz = ∞. By [9] Thm. 2, noting its proof or the statement (b) of [9, p.136], we can see
that K is essentially polar for Yt , i.e.,
P x × P {(ω, ω ) : XTt (ω ) (ω) ∈ K for some t > 0} = 0 for m-a.e. x ∈ Rn .
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Acta Mathematica Sinica, New Series
Vol.11 No.3
Hence
P x {ω : P (ω : Tt (ω ) ∈ X −1 (K)(ω) for some t > 0) = 0} = 1 m-a.e. x.
By [2],
P x (ω : dim X −1 (K) ≤ 1 − α) = 1 for m-a.e. x ∈ Rn ;
and letting α increase to β(K) gives (9).
Remark. If X has a strong Feller resolvent (this is equivalent to that X has resolvent density
with respect to m, see [10]), one can easily show that Yt = XTt has resolvent density with
respect to m so that every essentially polar set is polar for Yt by [10]. In this case our theorem
can be improved to be:
P x (dim X −1 (K) = 1 − β(K)) = 1 for any x ∈ Rn .
(10)
§3. The Transient Case
For β(K) ≤ α < 1, 0 < α, let Tt , Yt and Kα be defined as before. Let


Kα if 0 < β(K) < 1,



 β(K)≤α<1
K ∗ = Rn
if β(K) = 1,



K
if β(K) = 0 .

α

0<α<1
Theorem 2. If X is transient and has positive resolvent density with respect to m, then for
any compact K ⊂ Rn we have:
(i). 1 − β(K) = ess.supP x dim X −1 (K) ∀x ∈ Rn ,
(ii). x ∈ K ∗ =⇒ P x (dim X −1 (K) = 1 − β(K)) = 1,
(iii). K −K ∗ is polar for X =⇒ for any x ∈ Rn P x (dim X −1 (K) = 1−β(K)|TK < ∞) = 1.
Proof. (i). First of all, under the condition of the theorem, we can see by the equilibrium
principle that A is polar for X if and only if P x (TA < ∞) = 0 for some x. Thus, noting that
the unique role of the recurrence in the proof of Theorem 1 is to guarantee (7) and (8), we have
the following (7) , (8) instead of (7), (8) respectively:
P x (Xt ∈ Kα for some t > 0) > 0 ∀x ∈ Rn ,
P x (dim X −1 (K) ≥ 1 − α) > 0
∀x ∈ Rn .
(7)
(8)
It follows from the proof of (9) that 1 − β(K) ≥ dim X −1 (K)P x -a.e. for any x ∈ Rn , so
that to prove (i) it is sufficient to show
P x (dim X −1 (K) ≥ βn ) > 0
(11)
for some sequence {βn } where βn < 1 − β(K) and βn ↑ 1 − β(K). Choose βn = 1 − αn so that
αn > β(K), αn ↓ β(K) and
Re((1 + ψ αn (z))−1 )|µ̂n (z)|2 dz < ∞
for some µn ∈ M(K). From (8) we see that (11) is true.
Liu Luqin
On the Hausdorff Dimension of the Inverse Image
251
(ii) can be proved by the same argument as in Theorem 1, replacing (7) and (8) by
P x (Xt ∈ Kα for some t > 0) = 1
P x (dim X −1 (K) ≥ 1 − α) ≥ 1
for any
for any
x ∈ K∗,
x ∈ K∗.
(7)
(8)
(iii). Since K − K ∗ is polar for X and XTK ∈ K, XTK ∈ K ∗ P x -a.e. for any x ∈ Rn . From
(ii) and the strong Markov property we have
P x (TK < ∞) = P x [P X(TK ) (dim X −1 (K) = 1 − β(K)); TK < ∞]
= P x [dim X −1 (K)(θTK ) = 1 − β(K); TK < ∞]
= P x [dim X −1 (K) = 1 − β(K); TK < ∞].
This proves (iii).
Acknowledgements. The author would like to express his gratitude to Prof. K. D. Elworthy
in Warwick University and Prof. J. Hawkes in University College of Swansea for their assistance
and to their departments for their hospitality.
References
[1] Blumenthal, R. M. and Getoor, R. K., Local time for Markov processes, Z. W., 3(1964), 50–74.
[2] Hawkes, J., Local time and zero sets for processes with infinitely divisible distributions, J. London Math.
Soc., 8(1974), 517–525.
[3] Hawkes, J., On the Hausdorff dimension of the intersection of the range of a stable process with a Borel
set, Z. W., 19(1971), 90–102.
[4] Blumenthal, R. M. and Getoor, R. K., Markov Processes and Potential Theory, Academic Press, 1968.
[5] Chung, K. L., Lectures from Markov Processes to Brownian Motion, Springer-Verlag, 1982.
[6] Rao, M., On polar sets for Levy processes, J. London Math. Soc., 35(1987), 569–576.
[7] Glover, J. and Rao, M., Hunt’s Hypothesis (H) and Getoor’s Conjecture, Ann. Probab., 14(1986), 1085–
1087.
[8] Blumenthal, R. M. and Getoor, R. K., Some theorems on stable processes, Trans. Amer. Math. Soc.,
95(1960), 263–273.
[9] Hawkes, J., Some geometric aspects of potential theory, Lecture Notes in Math., Vol. 1095, 1084.
[10] Hawkes, J., Potential theory of Levy processes, Proc. London Math. Soc., 38(1979), 335–352.
Liu Luqin
Department of Mathematics
Wuhan University
Wuhan, 430072
China