Acta Mathematica Sinica, New Series
1998, July, Vol.14, No.3, pp. 381–384
The Lifetime of Diffusion Processes with
Application to Conditioned Diffusions
Wang Fengyu
(Department of Mathematics, Beijing Normal University, Beijing 100875, China)
(Email: [email protected])
Abstract Let τD denote the lifetime of a diffusion process on domain D ⊂ Rd . This paper presents
a sufficient condition for the exponential moment of τD to be finite. Here, both of the domain and
the diffusion operator are general. As an application, the main result of Gao(1995) for conditioned
diffusions is improved on.
Keywords Lifetime, Diffusion process, Conditioned diffusion
1991MR Subject Classification 60J60
Chinese Library Classification O211.6
1
Introduction
Let D be an open domain in Rd . Consider the operator on D
d
∂2
∂
L=
aij (x)
+
bi (x)
,
∂x
∂x
∂x
i
j
i
i,j=1
i=1
d
where aij , bi ∈ C(D) and a(x) = (aij (x)) is nonnegative definite so that the martingale problem
for L is wellposed up to the lifetime τD = inf{t ≥ 0 : xt ∈
/ D}. Suppose that there are two
continuous functions Λ ≥ λ ≥ 0 such that
λ(x)|ξ|2 ≤ a(x)ξ, ξ ≤ Λ(x)|ξ|2 , x ∈ D,
ξ ∈ Rd .
(1.1)
A number of authors have studied the finiteness of EτD for both of the diffusion processes
and conditioned diffusions. See [1–5] and references therein. All of them are restricted to the
uniformly elliptic case, i.e., inf D λ > 0 and supD Λ < ∞. Recently, Gao[6] presented a sufficient
condition for some diffusions which are degenerate on the boundary. In detail, let D be a
bounded C 2 -domain, define δ(x) = d(x, ∂D), λ(r) = inf λ(x), r > 0. If b(x) and Λ(x) are
δ(x)=r
bounded, λ(r) > 0 for r > 0 and
lim λ(r)/r = ∞,
r→0
Received May 7, 1996, Revised March 13, 1997, Accepted June 28, 1997
Research supported in part by NNSFC (19631060) and Beijing Normal University
(1.2)
382
Acta Mathematica Sinica, New Series
Vol.14 No.3
then there exits c > 0 such that supx∈D Exh τD ≤ c for any positive L-harmonic function h. Here
Exh is the expectation of the h-conditioned L-diffusion process starting from x. See [6; Theorem
4.1].
In this paper, Gao’s result is improved on in the sense that L can be degenerate in D. To
show this, we first study the lifetime of the L-diffusion process on D. The following result is
elementary in our study.
Theorem 1.1 If there exists F ∈ Cb2 (D) such that LF ≥ 1 in D, letting δ(F ) = supD F −
inf D F, then
1
, c ∈ [0, δ(F )−1 ).
(1.3)
sup E x ecτD ≤
1
−
cδ(F
)
x∈D
According to Bãuelos[1] , supD E x ecτD < ∞ for some c > 0 is equivalent to supD E x τD < ∞.
Next, we note that the maximum principle holds for L on D if τD < ∞ a.s. To see this, let
f ∈ C 2 (D) ∩ C(D) with f |∂D ≤ 0 and Lf ≥ 0 in D. Then for any x ∈ D and t > 0, we have
f (x) ≤ E x f (xt∧τD ) which implies f (x) ≤ 0 by letting t → ∞.
From Theorem 1.1 we see that a lot of degenerate diffusions satisfy supx∈D E x ecτD < ∞
for some c > 0. For instance, taking b1 ≡ 1, F (x) = x1 , then LF ≥ 1 and F ∈ Cb2 (D) whenever
D is bounded in the first coordinate.
Furthermore, we have the following refinement of Theorem 1.1 which says that the condition
LF ≥ 1 does not necessarily hold at every point in D.
Theorem 1.2 Suppose that a and b are bounded. Let Br = {x ∈ D : λ(x) ≤ r}, r > 0. If
there exists f ∈ Cb2 (D) and r, ρ > 0 such that Lf ≥ ρ on Br , ∇f 2 ≥ ρ on D \ Br , then there
exists c > 0 such that supD E x ecτD < ∞.
Now, we turn to the study of conditioned diffusions. Let h be a positive L-harmonic function
in D. Then the probability density of the L-diffusion conditioned by h is
pht (x, y) =
h(y)
pt (x, y),
h(x)
where pt (x, y) is the one of the L-diffusion process. See [5]. The following result is a consequence
of Theorem 1.2.
Corollary 1.1 Let D be a bounded C 2 -domain and a, b be bounded in D. If (1.2) holds
and there exists f ∈ Cb2 (D) such that inf D ∇f 2 > 0, inf B0 Lf > 0, then there exist c1 , c2 > 0
such that sup Exh ec1 τD ≤ c2 holds for any positive L-harmonic function h.
x∈D
Obviously, Corollary 1.1 recovers [6; Theorem 4.1] since B0 = ∅ if λ > 0 in D. Next, since
the present operator L can be degenerate in D (see Example 1.1 below), Corollary 1.1 is actually
an improvement on the main result of [6]. Finally, for some special domain, the condition (1.2)
can be replaced by a more relaxed one. For instance, if D is a ball, we have the following result.
Corollary 1.2 Let D = {x ∈ Rd : |x| < R} for some R > 0. Then Corollary 1.1 holds
with (1.2) replaced by
limr→0 r−1 λ(r) > R−1 limδ(x)→0 max{0, −b(x), x}.
(1.2 )
Example 1.1 Take D = {x ∈ Rd : |x| < 1}, a(x) = α(|x|)(1−|x|2 )I and bi (x) = 1−α(x),
i = 1, · · · , d, where α ∈ C 2 ([0, 1]) with 0 ≤ α ≤ 1, α(r) = 0 for r ≤ 1/2 and α(1) = 1. Setting
f (x) = di=1 xi , then ∇f 2 = d, Lf = 1 on B0 and limr→0 r−1 λ(r) = 2. Hence the conditions
of Corollary 1.2 hold.
Wang Fengyu
2
The Lifetime of Diffusion Processes with Application to Conditioned Diffusions
383
Proofs
Proof of Theorem 1.1 The idea of the proof comes from [7] and [8]. Fix c ∈ [0, δ(F )−1 ).
From Itô’s formula we obtain
dect F (xt ) = cect F (xt )dt + ect dF (xt ) = ect dMt + ect [cF (xt ) + LF (xt )]dt
for some (local) martingale Mt . Let α = supD F, β = inf D F. Then
αE x ec(t∧τD )
≥ E x ec(t∧τD ) F (xt∧τD ) ≥ F (x) + E x
≥β+
1+cβ
x c(t∧τD )
c (E e
t∧τD
0
ecs (1 + cF (xs ))ds
− 1).
Hence E x ec(t∧τD ) ≤ (1 − cδ(F ))−1 and the proof is completed by letting t → ∞.
Proof of Theorem 1.2 Since f ∈ Cb2 and a, b are bounded, there exists M > 0 such that
Lf ≥ −M in D. Let F1 (x) = exp[(1 + M )f (x)/(ρr)]. We have
2
(1 + M )
M
LF1 (x) = 1 +
rρ F1 (x)Lf (x) + r2 ρ2 F1 (x)a∇f, ∇f (x)
−1
r (1 + M ) inf F1 ,
if x ∈ Br ,
D
≥
−1
(rρ) (1 + M )M inf F1 , if x ∈ D \ Br .
D
Hence there exists N > 0 such that L(N F1 ) ≥ 1 and the proof is completed by means of
Theorem 1.1.
From now on, we assume that D is a bounded C 2 -domain and a, b are bounded on D.
For r ≥ 0, let Dr = {x ∈ D : δ(x) > r}. It is clear that there exist α, β > 0 such that
δ ∈ C 2 (D \ Dα ), ∇δ2 ≥ β in D \ Dα and λ(r) > 0 for r ∈ (0, α]. See [6] for details. The
following lemma improves on [6; Theorem 3.1] and the proof presented here is much simpler.
Lemma 2.1 Under the conditions of Corollary 1.1, there exists c > 0 such that supx∈D E x
cτD
e
< ∞.
Proof a) Choose γ ≥ 0 such that Lδ ≤ γ in D \ Dα . Next, choose α1 ∈ (0, α) and
nonnegative H ∈ C([0, α]) such that H|[0,α1 ] = 1 and H(α) = 0. Define
α
α
α
α
γdu
H(u)
γdt
ds
exp −
du.
g(r) = I[0,α] (r)
exp
r
s βλ(u)
s βλ(u)
u βλ(t)
Then g ∈ Cb2 ([0, ∞)). Since (1.2) holds, g(0) < ∞. Next, we have
Lg ◦ δ = g (δ)Lδ + g (δ)a∇δ, ∇δ ≥ g (δ)γ + g (δ)βλ(δ) ≥ H(δ)I[0,α] (δ).
(2.1)
b) Note that B0 ⊂ Dα and λ, Lf are continuous. Then there exist ρ > 0 and r > 0 such
that Lf ≥ ρ in Br ∩ Dα1 and ∇f 2 ≥ ρ. The proof of Theorem 1.2 shows that there exists
F ∈ Cb2 (D) such that LF ≥ 1 in Dα1 . Finally, choose γ1 ≥ 0 such that LF ≥ −γ1 in D, define
G(x) = F (x) + (γ1 + 1)g ◦ δ(x),
x ∈ D.
Then
LG(x) = LF (x) + (1 + γ1 )Lg ◦ δ(x) ≥ 1,
Now, the lemma follows from Theorem 1.1.
x ∈ D.
384
Acta Mathematica Sinica, New Series
Vol.14 No.3
Proof of Corollary 1.1 Let α and α1 be chosen as in the proof of Lemma 2.1. We claim
that for any positive L-harmonic function h, there holds
sup h = sup h,
Dα1
∂Dα1
inf h = inf h.
Dα1
(2.2)
∂Dα1
Actually, for any x ∈ Dα1 , we have inf h ≤ h(x) = E x h(xτDα1 ) ≤ sup h. Here we have used
∂Dα1
∂Dα1
the fact that τDα1 < ∞. Since L is uniformly elliptic on Dα1 /2 \ Dα , the classical Harnack
inequality yields supDα1 \Dα h ≤ C inf Dα1 \Dα h for some C > 0 independent of h. See [9;
Corollary 9.25] or [10; Theorem 1.1]. From this and (2.2) we obtain
sup h ≤ C inf h.
Dα1
Dα1
(2.3)
On the other hand, our operator is just the one discussed in [6] if it is restricted to D \ Dα .
Combining this with Lemma 2.1, we see that the proof of [6; Theorem 4.1] also works for the
present L.
Proof of Corollary 1.2 Noting that δ(x) = R − |x|, we have
Lδ(x) = −
d
d
1
1 1 1
tra(x) + 3
max{b(x), x, 0}.
aij (x)xi xj −
bi (x)xi ≤
|x|
|x| i=1
|x|
|x| i,j=1
Then there exists α ∈ (0, R) and γ ≥ 0, ε > 0 such that Lδ(x) ≤ γ for |x| ∈ [R − α, R) and
λ(r) > (γ + ε)r for r ∈ (0, α]. Let α1 and H be chosen as in the proof of Lemma 2.1, set
α
α
α
α
γdu
H(u)
γdt
ds
exp −
du.
exp
g(r) =
λ(u)
r
s λ(u)
s
u λ(t)
Then g is well defined on [0, α]. Letting g(r) = 0 for r > α, then g ∈ C 2 ([0, ∞)) and Lg ◦ δ(x) ≥
I[0,α1 ] (δ(x)). From the proof of Lemma 2.1 we see that supx∈D E x τD < ∞. Now, the remainder
of the proof is similar to that of [6; Theorem 4.1], the only difference being that we have to use
“λ(δ(x)/3) > εδ(x)/3” instead of “λ(δ(x)/3) > δ(x)/3” for small δ(x). See [6; p.334] for details.
Acknowledgement The author is grateful to Prof. Mu-Fa Chen for useful discussions
and to a referee for pointing out some errors in the first version of the paper.
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