SUBORDINATED PROCESSES AND CAUCHY
PROBLEMS
by
ERKAN NANE
DEPARTMENT OF STATISTICS AND PROBABILITY
MICHIGAN STATE UNIVERSITY
OUTLINE
• Introduction and history
• Brownian subordinators
• Cauchy problems on bounded domains
• Other subordinators
• Scaling Limits
• Conclusion and open problems
2
INTRODUCTION AND HISTORY
In recent years, starting with the articles of Burdzy (1993)
and (1994), researchers had interest in iterated processes in
which one changes the time parameter with one-dimensional
Brownian motion.
To define iterated Brownian motion Zt, due to Burdzy (1993),
started at z ∈ R, let Xt+, Xt− and Yt be three independent
one-dimensional Brownian motions, all started at 0. Two-sided
Brownian motion is defined to be
Xt =
Xt+, t ≥ 0
−
X(−t)
, t < 0.
Then iterated Brownian motion started at z ∈ R is
Zt = z + X(Yt), t ≥ 0.
3
BM versus IBM: This process has many properties analogous
to those of Brownian motion; we list a few
(1) Zt has stationary (but not independent) increments, and is
a self-similar process of index 1/4.
(2) Laws of the iterated logarithm (LIL) holds: usual LIL by
Burdzy (1993)
25/4
Z(t)
lim sup 1/4
= 3/4 a.s.
3/4
3
t→∞ t (log log(1/t))
Chung-type LIL by Khoshnevisan and Lewis (1996) and Hu et
al. (1995).
(3) Khoshnevisan and Lewis (1999) extended results of Burdzy
(1994), to develop a stochastic calculus for iterated Brownian
motion.
4
(4) In 1998, Burdzy and Khoshnevisan showed that IBM can be
used to model diffusion in a crack.
(5) Local time of this process was studied by Burdzy and
Khoshnevisan (1995), Csáki, Csörgö, Földes, and Révész
(1996), Shi and Yor (1997), Xiao (1998), and Hu (1999).
(6) Bañuelos and DeBlassie (2006) studied the distribution of
exit place for iterated Brownian motion in cones.
(7) DeBlassie (2004) studied the lifetime asymptotics of iterated
Brownian motion in cones and Bounded domains. Nane
(2006), in a series of papers, extended some of the results
of DeBlassie. He also studied the lifetime asymtotics of iterated
Brownian motion in several unbounded domains(parabolashaped domains, twisted domains...).
5
(8) Khoshnevisan and Lewis (1996) established the modulus of
continuity for iterated Brownian motion: with probability one
|Z(s) − Z(t)|
lim sup
sup
= 1.
1/4
3/4
δ→0 0≤s,t≤1 0≤|s−t|≤δ δ (log(1/δ))
6
0.2
0.15
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
t
–0.05
–0.1
7
Figure 1: Simulations of two Brownian motions
0.3
0.1
0.05
0
0.05
0.1
0.15
0.2
0.25
t
–0.05
8
Figure 2: Simulation of IBM Zt1 = X(|Yt|)
0.3
PDE connection
The classical well-known connection of a PDE and a stochastic
process is the Brownian motion and heat equation connection.
Let Xt ∈ Rd be Brownian motion started at x. Then the function
u(t, x) = Ex[f (Xt)]
solves the Cauchy problem
∂
u(t, x) = ∆u(t, x),
t > 0, x ∈ Rd
∂t
u(0, x) = f (x), x ∈ Rd.
In addition to the above properties of IBM there is an
interesting connection between iterated Brownian motion and
the biharmonic operator ∆2; the function
9
u(t, x) = Ex[f (Zt)]
solves the Cauchy problem (Allouba and Zheng (2001) and
DeBlassie (2004))
∂
∆f (x)
u(t, x) = √
+ ∆2u(t, x);
∂t
πt
u(0, x) = f (x).
(1)
for t > 0 and x ∈ Rd. The non-Markovian property of IBM is
reflected by the appearance of the initial function f (x) in the
PDE.
2
Let q(t, s) = √ 2 exp − s4t be the transition density of
4πt
reflected one-dimensional Brownian motion, |Bt|.
The essential argument, using integration by parts twice, is that
10
∂
∂t u(t, x)
∞
∂
=
T (s)f (x) q(t, s)ds
∂t
Z0 ∞
∂2
=
T (s)f (x) 2 q(t, s)ds
∂s 0
∂
= q(t, s) [T (s)f (x)] ∂s
s=0
Z ∞ 2
∂
+
[T (s)f (x)] q(t, s)ds
2
∂s
0
Z
Z
∞
= q(t, 0)Lx[T (0)f (x)] +
L2x [T (s)f (x)] q(t, s)ds
Z ∞ 0
1
= √ Lxf (x) + L2x
T (s)f (x) q(t, s)ds
πt
0
11
Fractional Diffusion
Nigmatullin (1986) gave a Physical derivation of fractional
diffusion
∂β
u(t, x) = Lxu(t, x);
β
∂t
u(0, x) = f (x)
(2)
where 0 < β < 1 and Lx is the generator of some continuous
Markov process X0(t) started at x = 0. Here ∂ β g(t)/∂tβ is
the Caputo fractional derivative in time, which can be defined
β
β−1
as the inverse
Laplace
transform
of
s
g̃(s)
−
s
g(0), with
R ∞ −st
g̃(s) = 0 e g(t)dt the usual Laplace transform.
Zaslavsky (1994) used this to model Hamiltonian chaos.
12
Stochastic solution
Baeumer and Meerschaert (2001) and Meerschaert and
Scheffler (2004) shows that, in the case p(t, x) = T (t)f (x)
is a bounded continuous semigroup on a Banach space (with
corresponding process Xt, Et = inf{u : Du > t}, Dt is a
stable subordinator with index β ) , the formula
t
u(t, x) = Ex(f (XEt )) =
β
Z
∞
0
p(s, x)gβ (
t
−1/β−1
)s
ds
1/β
s
yields a solution to the fractional Cauchy problem:
∂β
u(t, x) = Lxu(t, x); u(0, x) = f (x)
(3)
β
∂t
Here gβ (t) is the smooth density of the stable subordinator, such
R ∞ −st
β
that the Laplace transform g̃β (s) = 0 e gβ (t) dt = e−s .
13
BROWNIAN SUBORDINATORS
(Baeumer, Meerschaert and Nane (2007)) show that, taking
β = 1/2 in the time-fractional diffusion yields exactly 1-D
d
distributions X|Bt| = XEt .
If Lx is the generator of the semigroup T (t)f (x) = Ex[(f (Xt))]
on L1(Rd), then for any f ∈ D(Lx)
∂
Lxf (x)
+ Lx2u(t, x);
u(t, x) = √
∂t
πt
u(0, x) = f (x),
(4)
and the fractional Cauchy problem (3) with β = 1/2 have the
same solution
2
u(t, x) = Ex[f (X(|Bt|))] = √
4πt
14
Z
∞
0
2
s
T (s)f (x) exp −
ds.
4t
Fourier-Laplace method
The Lévy process X0(t) has characteristic function
with
E[exp(ik · X0(t))] = exp(tψ(k))
1
ψ(k) = ik·a− k·Qk+
2
Z
y6=0
eik·y − 1 −
ik · y
ν(dy),
2
1 + ||y||
where a ∈ Rd, Q is a nonnegative definite matrix, and ν is a
d
σ -finite Borel measure
on
R
such that
Z
y6=0
min{1, ||y||2}ν(dy) < ∞.
Denote the Fourier transform by
fˆ(k) =
Z
e−ik·xf (x) dx
Rd
15
Meerschaert and Scheffler (2001) shows that Lxf (x) is the
inverse Fourier transform of ψ(k)fˆ(k) for all f ∈ D(Lx), where
D(Lx) = {f ∈ L1(Rd) : ψ(k)fˆ(k) = ĥ(k) ∃ h ∈ L1(Rd)},
and
1
Lxf (x) = a · ∇f (x) + ∇ · Q∇f (x)
2
Z ∇f (x) · y
+
f (x + y) − f (x) −
ν(dy)
2
1+y
y6=0
(5)
for all f ∈ W 2,1(Rd), the Sobolev space of L1-functions whose
first and second partial derivatives are all L1-functions.
We can also write Lx
(∂/∂x1, . . . , ∂/∂xd)′.
=
16
ψ(−i∇) where ∇
=
For example, if X0(t) is spherically symmetric stable then
ψ(k) = −Dkkkα and Lx = −D(−∆)α/2, a fractional
derivative in space, using the correspondence kj → −i∂/∂xj
for 1 ≤ j ≤ d.
If X0 has independent
marginals, P
then one possible
P stable
form is ψ(k) = D j (ikj )αj and Lx = D j ∂ αj /∂xα
j using
Riemann-Liouville fractional derivatives in each variable. This
form does not coincide with the fractional Laplacian unless all
α j = 2.
17
The proof does not use Theorem 0.1 in Allouba and Zheng
(2001), rather it relies on a Laplace-Fourier transform argument.
We will use the following notation for the Laplace, Fourier, and
Fourier-Laplace transforms (respectively):
ũ(s, x) =
Z
∞
e−stu(t, x)dt;
Z0
e−ik·xu(t, x)dx;
ZR d
Z ∞
e−ik·x
e−stu(t, x)dtdx.
ū(s, k) =
û(t, k) =
Rd
0
18
Let ψ be the characteristic exponent of Xt.
transforms on both sides of (4) to get
Take Fourier
∂ û(t, k)
1
= √ ψ(k)fˆ(k) + ψ(k)2û(t, k)
∂t
πt
using the fact that ψ(k)fˆ(k) is the Fourier transform of Lxf (x).
Then take Laplace transforms on both sides to get
sū(s, k) − û(t = 0, k) = s−1/2ψ(k)fˆ(k) + ψ(k)2ū(s, k),
using the well-known Laplace transform formula
Z
∞
0
t−β
e−stdt = sβ−1, β < 1.
Γ(1 − β)
19
Since û(t = 0, k) = fˆ(k), collecting like terms yields
(1 + s−1/2ψ(k))fˆ(k)
ū(s, k) =
s − ψ(k)2
(6)
for s > 0 sufficiently large.
On the other hand, taking Fourier transforms on both sides of
(3) with β = 1/2 gives
∂ 1/2û(t, k)
= ψ(k)û(t, k)
1/2
∂t
Take Laplace transforms on both sides, using the fact that
sβ g̃(s) − sβ−1g(0) is the Laplace transform of the Caputo
fractional derivative ∂ β g(t)/∂tβ , to get
s1/2ū(s, k) − s−1/2fˆ(k) = ψ(k)ū(s, k)
20
and collect terms to obtain
s−1/2fˆ(k)
ū(s, k) = 1/2
s − ψ(k)
s−1/2fˆ(k) s1/2 + ψ(k)
= 1/2
· 1/2
s − ψ(k) s + ψ(k)
(1 + s−1/2ψ(k))fˆ(k)
=
s − ψ(k)2
(7)
which agrees with (6). For any fixed k ∈ Rd, the two formulae
are well-defined and equal for all s > 0 sufficiently large.
An easy extension of the argument as above shows that, under
the same conditions, for any k = 2, 3, 4, . . . both the Cauchy
21
problem
k−1 1−j/k
X
∂u(t, x)
t
=
Ljxf (x) + Lkxu(t, x);
∂t
Γ(j/k)
j=1
(8)
u(0, x) = f (x)
and the fractional Cauchy problem:
∂ 1/k
u(t, x) = Lxu(t, x);
1/k
∂t
u(0, x) = f (x),
(9)
have the same unique solution given by
u(t, x) =
Z
∞
p((t/s)β , x)gβ (s) ds
0
with β = 1/k . Hence the process Zt = X(Et) is also the
stochastic solution to this higher order Cauchy problem.
22
Orsingher and Benghin (2004) and (2008) show that for β =
1/2n the solution to
1/2n
∂
n u(t, x) = ∆x u(t, x);
1/2
∂t
u(0, x) = f (x),
(10)
is given by running
In(t) = B1(|B2(|B3(| · · · (Bn+1(t)) · · · |)|)|)
Where Bj ’s are independent Brownian motions, i.e., u(t, x) =
Ex(f (In(t))) solves (10), and solves (8) for k = 2n.
23
CAUCHY PROBLEMS ON BOUNDED DOMAINS
Since we are working on a bounded domain, the Fourier
transform methods are not useful. Instead we will employ Hilbert
space methods. Hence, given a complete orthonormal basis
{ψn(x)} on L2(D), we will call
ū(t, n) =
Z
ψn(x)u(t, x)dx;
ZD
Z ∞
Z
û(s, n) =
ψn(x)
e−stu(t, x)dtdx =
ψn(x)ũ(s, x)dx.
D
D
0
the ψn, and ψn-Laplace transforms, respectively.
Let D be bounded and every point of ∂D be regular for D C .
The corresponding Markov process is a killed Brownian motion.
We denote the eigenvalues and the eigenfunctions of ∆D by
24
∞
{λn, φn}∞
n=1, where φn ∈ C (D). The corresponding heat
kernel is given by
pD (t, x, y) =
∞
X
e−λntφn(x)φn(y).
n=1
The series converges absolutely and uniformly on [t0, ∞) ×
D × D for all t0 > 0. In this case, the semigroup given by
TD (t)f (x) = Ex[f (Xt)I(t < τD (X))] =
=
∞
X
Z
pD (t, x, y)f (y)dy
D
e−λntφn(x)f¯(n)
n=1
solves the Heat equation in D with Dirichlet boundary
25
conditions:
∂u(t, x)
= ∆u(t, x), x ∈ D, t > 0,
∂t
u(t, x) = 0, x ∈ ∂D,
u(0, x) = f (x), x ∈ D.
26
Fractional Cauchy problems in bounded domains
Let β ∈ (0, 1), D∞ = (0, ∞) × D and define
∂
∂β
H∆(D∞) ≡ u : D∞ → R :
u, β u, ∆u ∈ C(D∞),
∂t ∂t
∂
u(t, x) ≤ g(x)tβ−1, g ∈ L∞(D), t > 0 .
∂t
Let 0 < γ < 1. Let D be a bounded domain with ∂D ∈ C 1,γ ,
and TD (t) be the killed semigroup of Brownian motion {Xt}
in D . Let Et be the process inverse to a stable subordinator
of index β ∈ (0, 1) independent of {Xt}. Let f ∈ D(∆D ) ∩
C 1(D̄) ∩ C 2(D) for which the eigenfunction expansion (of ∆f )
with respect to the complete orthonormal basis {φn : n ∈ N}
27
converges uniformly and absolutely. Then the unique (classical)
solution of
u ∈ H∆(D∞) ∩ Cb(D̄∞) ∩ C 1(D̄)
∂β
u(t, x) = ∆u(t, x); x ∈ D, t > 0
β
∂t
u(t, x) = 0, x ∈ ∂D, t > 0,
u(0, x) = f (x), x ∈ D.
28
(11)
is given by
u(t, x) =
∞
X
f¯(n)φn(x)Eβ (−λntβ )
n=1
= Ex[f (X(Et))I(τD (X) > Et)]
= ExZ[f (X(Et))I(τD (X(E)) > t)]
t ∞
TD (l)f (x)gβ (tl−1/β )l−1/β−1dl
=
β 0
Z
∞
=
TD ((t/l)β )f (x)gβ (l)dl.
0
Joint work with Meerschaert and Vellaisamy (2008).
Analytic solution in intervals (0, M ) ⊂ R was obtained by
Agrawal (2002).
29
Assume that u(t, x) solves (11). Taking φn- transforms in (11)
we obtain
∂β
ū(t, n) = −λnū(t, n).
β
∂t
(12)
taking Laplace transforms on both sides of (12), we get
sβ û(s, n) − sβ−1ū(0, n) = −λnû(s, n)
(13)
f¯(n)sβ−1
û(s, n) = β
.
s + λn
(14)
which leads to
By inverting the above Laplace transform, we obtain
ū(t, n) = f¯(n)Eβ (−λntβ )
30
in terms of the Mittag-Leffler function defined by
Eβ (z) =
∞
X
k=0
zk
.
Γ(1 + βk)
Inverting now the φn-transform, we get an L2-convergent
solution of Equation (11) as (for each t ≥ 0 )
u(t, x) =
∞
X
f¯(n)φn(x)Eβ (−λntβ )
n=1
31
(15)
IBM in bounded domains
Let Zt = X(|Yt|) be the iterated Brownian motion, D∞ =
(0, ∞) × D and define
∂
u, ∆2u ∈ C(D∞), ∆u ∈ C 1(D̄) ,
H∆2 (D∞) ≡ u : D∞ → R :
∂t
∂
u(t, x) ≤ g(x)t−1/2, g ∈ L∞(D), t > 0 .
∂t
Let D be a domain with ∂D ∈ C 1,γ , 0 < γ < 1. Let
{Xt} be Brownian motion in Rd, and {Yt} be an independent
Brownian motion in R. Let {Et} be the process inverse
to a stable subordinator of index β = 1/2 independent of
{Xt}. Let f ∈ D(∆D ) ∩ C 1(D̄) ∩ C 2(D)(⊂ L2(D)) be
32
such that the eigenfunction expansion of ∆f with respect to
{φn : n ≥ 1} converges absolutely and uniformly. Then the
(classical) solution of
u ∈ H∆2 (D∞) ∩ Cb(D̄∞) ∩ C 1(D̄);
∂
∆f (x)
u(t, x) = √
+ ∆2u(t, x), x ∈ D, t > 0; (16)
∂t
πt
u(t, x) = ∆u(t, x) = 0, t ≥ 0, x ∈ ∂D;
u(0, x) = f (x), x ∈ D
33
is given by
u(t, x) = Ex[f (Zt)I(τD (X) > |Yt|)]
= Ex[f (X(Et))I(τD (X) > Et)]
= EZx[f (X(Et))I(τD (X(E)) > t)]
∞
=2
TD (l)f (x)p(t, l)dl,
(17)
0
where TD (l) is the heat semigroup in D, and p(t, l) is the
transition density of one-dimensional Brownian motion {Yt}.
Proof uses again equivalence with fractional Cauchy problem
for β = 1/2.
34
OTHER SUBORDINATORS
A Lévy process S = {St, t ≥ 0} with values in R is called
strictly stable of index α ∈ (0, 2] if its characteristic function is
given by
πα
α 1 + iνsgn(ξ) tan( 2 )
E exp(iξSt) = exp −t|ξ|
, (18)
χ
where −1 ≤ ν ≤ 1 and χ > 0 are constants. When α = 2
and χ = 2, S is Brownian motion.
35
For any Borel set I ⊆ R, the occupation measure of S on I is
defined by
µI (A) = λ1{t ∈ I : St ∈ A}
(19)
for all Borel sets A ⊆ R, where λ1 is the one-dimensional
Lebesgue measure. If µI is absolutely continuous with respect
to the Lebesgue measure λ1 on R, we say that S has a local time
on I and define its local time L(x, I) to be the Radon-Nikodým
derivative of µI with respect to λ1, i.e.,
dµI
L(x, I) =
(x),
dλ1
∀x ∈ R.
In the above, x is the so-called space variable, and I is the time
variable of the local time. If I = [0, t], we will write L(x, I) as
L(x, t). Moreover, if x = 0 then we will simply write L(0, t) as
Lt .
36
Local time as a subordinator
It is well-known (see, e.g. Bertoin (1996)) that a strictly stable
Lévy process S has a local time if and only if α ∈ (1, 2].
It is well-known (see, e.g. Bertoin (1996)) that the inverse of a
local time Lt of S is a stable subordinator:
Gt = inf{u : Lu > t}, then Gt = ρDt, where Dt is a stable
subordinator of index β = 1 − 1/α.
ρ = π −1Γ(1+1/α)Γ(1−1/α)χ1/αRe{[1+iν tan(πα/2)]−1/α}.
Hence Lt = Et/ρ where β = 1 − 1/α for some c > 0.
For β = 1 − 1/α, c > 0, u(t, x) = Ex[f (X(Lt))] solves
∂β
u(t, x) = cLxu(t, x);
β
∂t
37
u(0, x) = f (x)
(20)
Symmetric stable subordinators
α-time process is a Markov process subordinated to the
absolute value of an independent one-dimensional symmetric
α-stable process:
Zt = X(|St|), where Xt is a Markov process and St is an
independent symmetric α-stable process both started at 0.
This process is self similar with index 1/2α when the outer
process X is a Brownian motion. In this case Nane (2006)
defined the Local time of this process and obtained Laws of the
iterated logarithm for the local time for large time.
38
PDE-connection:
Theorem 3 [Nane (2008)]
Let T (s)f (x) = Ex[f (X(s))] be the semigroup of the
continuous Markov process X(t) and let Lx be its generator.
Let α = 1. Let f be a bounded measurable function in the
domain of Lx, with Dij f bounded and Hölder continuous for all
1 ≤ i, j ≤ n. Then u(t, x) = Ex[f (Zt)] solves
∂2
2Lxf (x)
2
u(t,
x)
=
−
−
L
xu(t, x);
2
∂t
πt
u(0, x) = f (x).
39
For α = l/m 6= 1 rational: the PDE is more complicated since
kernels of symmetric α-stable processes satisfy a higher order
PDE:
2m
∂
∂ 2l
l+1
α
+
(−1)
p
t (0, s) = 0.
2l
2m
∂s
∂t
We also have to assume that we can integrate under the integral
as much as we need in the case where the outer process is BM
(or in general we can take the operator out of the integral). This
is valid for α = 1/m, m = 2, 3, · · · by a Lemma in Nane
(2008).
40
Theorem 4 [Nane (2008)]
Let α ∈ (0, 2) be rational α = l/m, where l and m are relatively
prime. Let T (s)f (x) = Ex[f (X(s))] be the semigroup of the
continuous Markov process X(t) and let Lx be its generator.
Let f be a bounded measurable function in the domain of Lx,
with D γ f bounded and Hölder continuous for all multi index γ
such that |γ| = 2l. Then u(t, x) = Ex[f (Zt)] solves
l+1
(−1)
2m
∂
u(t, x) = −2
2m
∂t
l 2l−2i
X
∂
α
2i−1
p
(0,
s)|
L
f (x)
s=0
t
x
2l−2i
∂s
i=1
L2l
x u(t, x);
−
u(0, x) = f (x).
41
SCALING LIMITS
If Sn = X1 + X2 + · · · + Xn is particle location at time n then
the scaling limit is r −1/αS[rt] =⇒ Xt.
The limit process Xt is called an α-stable Lévy motion.
Another random walk Jn = T1 + T2 + · · · + Tn records the jump
times.
If P [Tn > t] ≈ t−β for 0 < β < 1 then r −1/β J[rt] =⇒ Dt.
Dt is an increasing β -stable Lévy motion(a stable subordinator).
Subordinator can give random time change: X(t) → X(Dt).
42
Inverse Stable subordinator
Waiting time random walk has scaling limit r −1/β J[rt] =⇒ Dt.
The number of jumps by time t is Nt = max{n > 0 : Jn ≤ t}.
Renewal process inverse to random walk {Nt ≥ n} = {Jn ≤
t}
Inverse process has inverse scaling limit r −β N[rt] =⇒ Et.
Inverse stable process {Et ≤ u} = {Du ≥ t} yields the
density formula:
d
d
p(u, t) = P [Et ≤ u] = P [Du ≥ t]
du
du
43
CTRW scaling limits
Particle jump random walk has scaling limit r −1/αS[rt] =⇒ Xt.
Number of jumps has scaling limit r −β N[rt] =⇒ Et.
CTRW is a random walk subordinated to a renewal process
SNt = X1 + X2 + · · · + XNt
CTRW scaling limit is a subordinated process:
r−β/αSNct = (rβ )−1/αSrβ ·r−β Nct ≈ (rβ )−1/αSrβ Et =⇒ XEt
CTRW scaling limit is not Markov, increments are not stationary.
Meerschaert and collaborators (2004).
44
OPEN PROBLEMS
Question 1. For β = 1/2, the inverse stable subordinator
process Et and the process |Yt|, where Yt is a one-dimensional
Brownian motion have the same transition density. Is there
a similar correspondence between Et for β 6= 1/2 and other
symmetric α-stable process Yt for 1 < α < 2.
Question 2. Looking at the governing PDE for subordinators
other than Brownian motion, are there any fractional in time PDE
which has the same solution as the higher order pde?
Question 3. Are there PDE connections of the iterated
processes in bounded domain as the PDE connection of
Brownian motion in bounded domains?
45
THANK YOU
46
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