Sample Paths of the Solution to the Fractional-colored Stochastic Heat Equation

0.2 MB·English

by  Ciprian A. Tudor

Sample Paths of the Solution to the Fractional-colored
Stochastic Heat Equation
5
1
Ciprian A. Tudor 1,2 and Yimin Xiao 3,
0 ∗ †
2
1 Laboratoire Paul Painlev´e, Universit´e de Lille 1

n
F-59655 Villeneuve d’Ascq, France.
a
J 2 Academy of Economical Studies, Piata Romana nr.6, sector 1

7 Bucharest, Romania.
2
[email protected]

] 3 Department of Statistics and Probability, Michigan State University
R
East Lansing, MI 48824, U.S.A.
P
[email protected]
.
h
t
a
m
January 28, 2015
[

1
v
8
2
Abstract
8
6 Let u = u(t,x),t [0,T],x Rd be the solution to the linear stochastic heat
0 { ∈ ∈ }
equationdrivenbyafractionalnoiseintimewithcorrelatedspatialstructure. Westudy
.
1 various path properties of the process u with respect to the time and space variable,
0 respectively. Inparticular,wederivetheir exactuniformandlocalmoduliofcontinuity
5
and Chung-type laws of the iterated logarithm.
1
:
v
i 2010 AMS Classification Numbers: 60H55, 60H07, 91G70.
X
r Key words: Stochastic heatequation, fractionalBrownianmotion, Gaussiannoise,
a
path regularity, law of the iterated logarithm.
1 Introduction
Stochastic analysis of fractional Brownian motion (fBm) naturally led to the study of
stochastic partial differential equations (SPDEs) driven by this Gaussian process. The
motivation comes from wide applications of fBm. We refer, among others, to [13], [21], [26],
∗Supportedby theCNCS grant PN-II-ID-PCCE-2011-2-0015 (Romania).
†Research of Y. Xiao is partially supported by NSFgrants DMS-1307470 and DMS-1309856.
1
[30] and [34] for theoretical studies of SPDEs driven by fBm. To list only a few examples
of applications of fractional noises in various areas, we mention [15] for biophysics, [5] for
financial time series, [11] for electrical engineering, and [7] for physics.
The purpose of our present paper is to study fine properties of the solution to the
stochastic heat equation driven by a Gaussian noise which is fractional in time and colored
in space. Our work continues, in part, the line of research which concerns SPDEs driven by
thefBmbutatthesametimeitfollows theresearch lineinitiated byDalang[8]whichtreats
equations with white noise in time and non trivial correlation in space. More precisely, we
consider astochastic linear equation driven by a Gaussian noise thatbehaves as afractional
Brownian motion with respect to its time variable and has a correlated spatial structure.
A necessary and sufficient condition for the existence of the solution has been given in [3]
and other results has been given in [35], [27], [1], [2] among others.
To briefly describe the context, consider the stochastic heat equation with additive
noise
∂u 1
= ∆u+W˙ , t [0,T], x Rd, (1)
∂t 2 ∈ ∈
u(0,x) = 0, x Rd,
∈
where T > 0 is a constant and W˙ is usually referred to as a space-time white noise. It
is well-known (see for example the now classical paper [8]) that (1) admits a unique mild
solution if and only if d= 1. This mild solution is defined as
t
u(t,x) = G(t s,x y)W(ds,dy), t [0,T], x R. (2)
Z0 ZRd − − ∈ ∈
In the above, W = W(t,A);t 0, A (Rd) is a zero-mean Gaussian process with
b
{ ≥ ∈ B }
covariance given by
E(W(t,A)W(s,B)) = (s t)λ(A B), (3)
∧ ∩
where λ denotes the Lebesque measure and (Rd) is the collection of all bounded Borel
b
B
subsets of Rd. The integral in (2) is a Wiener integral with respect to the Gaussian process
W and G is the Green kernel of the heat equation given by
G(t,x) = (2πt)−d/2exp −|x2|t2 if t > 0,x ∈ Rd, (4)
( 0 (cid:16) (cid:17) if t 0,x Rd.
≤ ∈
Consequently the mild solution u(t,x),t [0,T],x R is a centered two-parameter
{ ∈ ∈ }
Gaussian process (also called a Gaussian random field). It is well-known (see e.g. [33] or
[35]) that the solution defined by the (2) is well-defined if and only if d = 1 and in this case
the covariance of the solution (2), when x R is fixed, satisfies
∈
1
E(u(t,x)u(s,x)) = √t+s t s , for every s,t [0,T]. (5)
√2π − | − | ∈
(cid:16) p (cid:17)
2
This establishes an interesting connection between the law of the solution (2) and the
so-called bifractional Brownian motion introduced by [14]. Recall that, given constants
H,K
H (0,1) and K (0,1], the bifractional Brownian motion (B ) is a centered
∈ ∈ t t∈[0,T]
Gaussian process with covariance
RH,K(t,s) := R(t,s) = 1 t2H +s2H K t s 2HK , s,t [0,T]. (6)
2K −| − | ∈
(cid:16) (cid:17)
(cid:0) (cid:1)
Relation (5) implies that, when x R is fixed, the Gaussian process u(t,x),t [0,T] is
∈ { ∈ }
a bifractional Brownian motion with parameters H = K = 1 multiplied by the constant
2
2 K 1 . Therefore, many sample path properties of the solution to the heat equation
− √2π
driven by space time white-noise follow from [31, 16, 31, 36].
In order to avoid the restriction to the dimension d = 1, several authors considered
other types of noises, usually more regular than the time-space white noise. An approach is
to consider the white-colored noise, meaning a Gaussian process W = W(t,A);t 0, A
{ ≥ ∈
(Rd) with covariance with zero mean and covariance
b
B }
E(W(t,A)W(s,B)) = (t s) f(z z )dzdz . (7)
′ ′
∧ −
ZAZB
Here the kernel f is the Fourier transform of a tempered non-negative measure µ on Rd,
i.e. µ is a non-negative measure which satisfies:
l
1
µ(dξ) < for some l > 0. (8)
Rd 1+ ξ 2 ∞
Z (cid:18) | | (cid:19)
Recall that the Fourier transform f of µ is defined through
f(x)ϕ(x)dx = ϕ(ξ)µ(dξ), ϕ (Rd),
Rd RdF ∀ ∈ S
Z Z
where (Rd) denotes the Schwarz space on Rd. A useful connection between the kernel f
S
and its associated measure µ is the Parseval inequality: for any ϕ,ψ (Rd),
∈S
ϕ(x)f(x y)ψ(y)dxdy = (2π) d ϕ(ξ) ψ(ξ)µ(dξ). (9)
−
Rd Rd − RdF F
Z Z Z
In this work, we will include the following two following basic examples.
Example 1 The Riesz kernel of order α:
f(x)= R (x) := γ x d+α, (10)
α α,d −
| |
where 0 < α < d, γ = 2d απd/2Γ((d α)/2)/Γ(α/2). In this case, µ(dξ) = ξ αdξ.
α,d − −
− | |
3
Example 2 The Bessel kernel of order α:
f(x) = B (x) := γ ∞w(α d)/2 1e we x2/(4w)dw,
α α′ − − − −| |
Z0
where α > 0, γ = (4π)α/2Γ(α/2). In this case, µ(dξ) = (1+ ξ 2) α/2dξ.
α′ | | −
For any function g L1(Rd) we denote by g the Fourier transform of g, i.e.,
∈ F
( g)(ξ) = e i ξ,x g(x)dx, ξ Rd,
−h i
F Rd ∈
Z
where , denotes the inner product in Rd. Recall that for any t R and x Rd, the
h· ·i ∈ ∈
Fourier transform of the fundamental solution (4) is given by
t ξ 2
G(t,x )(ξ) = exp i x,ξ | | 1 (ξ), ξ Rd. (11)
F −· h i− 2 {t>0} ∈
(cid:18) (cid:19)
Dalang [8] proved that the stochastic heat equation with white-colored noise admits
a unique solution if and only if
1
µ(dξ) < . (12)
Rd 1+ ξ 2 ∞
Z | |
Obviously, this condition allows x to be in higher dimensional space. For example in the
case of the Riesz kernel, the stochastic heat equation with white-colored noise admits an
unique solution if and only if d < 2+α.
Under (12), the solution of (1) with white-colored noise can still be written as in
(2). One can compute the covariance of the solution with respect to the time variable (see
e.g. [28]). For fixed x Rd, and for every s t it follows from Parseval’s identity and (11)
∈ ≤
s
E u(t,x)u(s,x) = (2π)−d du µ(dξ)e−12(t−u)|ξ|2e−12(s−u)|ξ|2. (13)
Z0 ZRd
(cid:0) (cid:1)
In the case when f is the Riesz kernel (i.e., µ(dξ) = ξ αdξ, we get
−
| |
E u(t,x)u(s,x) = (2π)−d dξ e−21|ξ|2 1 (t+s)1−d−2α (t s)1−d−2α .
ZRd |ξ|α 1− d−2α (cid:16) − − (cid:17)
(cid:0) (cid:1)
Inthis case, the solution coincides, moduloa constant, with abifractional Brownian motion
with parameters H = 1 and K = 1 d α. Thus the sample path properties of this process
2 − −2
can be deduced from [16, 31, 36].
Ourpurposein thispaperis tostudythelinear heatequation drivenbyafractional-
colored Gaussian noise (see section 2 for details). When the structure of the noise with
respect to the time variable changes and the white noise is replaced by a fractional noise,
the solution does not coincide with a bifractional Brownian motion anymore (see [6] or
4
[35]). Some new methods are needed for analyzing the path properties of the solution
with respect to the time and to the space variables. By appealing to general methods for
Gaussian processes and fields (cf. e.g., [20, 39, 40]), we establish the exact uniform and
local moduli of continuity and Chung type laws of the iterated logarithm for the solution
of (1) with a fractional-colored Gaussian noise.
Throughout this paper, we will use C to denote unspecified positive finite con-
stants which may be different in each appearance. More specific constants are numbered as
c , c ,....
1 2
2 The solution to the stochastic heat equation with fractional-
colored noise
We consider a Gaussian field WH(t,A),t [0,T],A (Rd) with covariance
b
∈ ∈ B
(cid:8) (cid:9)
E WH(t,A)WH(s,B) = R (t,s) f(z z )dzdz , (14)
H ′ ′
−
ZAZB
(cid:0) (cid:1)
where R (t,s) := 1(t2H + s2H t s 2H) is the covariance of the fractional Brownian
H 2 − | − |
motion with index H and f is the Fourier transform of a tempered measure µ. This noise
is usually called fractional-colored noise. We will assume throughout the paper that the
Hurst parameter H is contained in the interval (1,1).
2
Consider the linear stochastic heat equation
∂u 1
= ∆u+W˙ H, t [0,T],x Rd (15)
∂t 2 ∈ ∈
withvanishinginitialcondition, where WH(t,x), t [0,T],x Rd isacenteredGaussian
{ ∈ ∈ }
noise with covariance (14). In the following we collect some known facts:
A necessary and sufficient condition for the existence of the mild solution to the
•
fractional-colored heat equation (15) has been given in [3] (see also [28]). Namely,
(15) has a solution u(t,x),t 0,x Rd that satisfies
{ ≥ ∈ }
sup E u(t,x)2 < +
t [0,T],x Rd ∞
∈ ∈ (cid:0) (cid:1)
if and only if
2H
1
µ(dξ) < . (16)
Rd 1+ ξ 2 ∞
Z (cid:18) | | (cid:19)
When (16) holds, the solution to (15) can be written in the mild form as
•
t
u(t,x) = G(t u,x y)WH(du,dy), t [0,T], x Rd. (17)
Z0 ZRd − − ∈ ∈
5
Itfollowsfrom[3]or[28]that,forx Rd fixed,thecovariancefunctionoftheGaussian
∈
process u(t,x),t [0,T] can be written as
{ ∈ }
E(u(t,x)u(s,x)) = αH(2π)−dZ0tZ0s |u−duvd|2v−2H ZRdµ(dξ)e−(t−u2)|ξ|2e−(s−v2)|ξ|2 (18)
with α = H(2H 1). In the particular case where the spatial covariance is given
H
−
by the Riesz kernel, the process t u(t,x) is self-similar with parameter H d α.
7→ − −4
However, for H (1,1), u(t,x),t 0 is no longer a bifractional Brownian motion.
∈ 2 { ≥ }
For 0 <α < d, the notation
µ(dξ) ξ αdξ, (19)
−
≍ | |
means that for every non-negative function h there exist two positive and finite constants
C and C , which may depend on h, such that
′
C h(ξ)ξ αdξ h(ξ)µ(dξ) C h(ξ)ξ αdξ. (20)
′ − −
Rd | | ≤ Rd ≤ Rd | |
Z Z Z
It has been proven in [35] that, under condition (19), there exist two strictly positive
constants c ,c such that for all t,s [0,1] and for all x Rd,
1 2
∈ ∈
c1 t s 2H−d−2α E u(t,x) u(s,x)2 c2 t s 2H−d−2α. (21)
| − | ≤ | − | ≤ | − |
(cid:0) (cid:1)
Remark 1 The Riesz kernel obviously satisfies (19). The Bessel kernel satisfies (19) and
the constants in (20) are C = 1 and C > 0 depending on h.
′
Under Condition (19), the condition (16) is equivalent to
d< 4H +α. (22)
In the sequel (e.g., Theorem 1 below), we will also use the notation f(x) g(x)
≍
which means that there exists two positive and finite constants C and C such that C
′ ′
≤
f(x)/g(x) C for all x in the domain of f and g. In the rest of the paper, we will assume
≤
that (19) is satisfied.
2.1 Sharp regularity in time
For any fixed x Rd, to further study the regularity properties of the sample function
∈
t u(t,x), we decompose the solution u(t,x),t 0 in (17) into the sum of a Gaussian
7→ { ≥ }
process U = U(t),t 0 , which has stationary increments, and another Gaussian process
{ ≥ }
whose sample functions are continuously differentiable on (0, ). In particular, when the
∞
spatial covariance of noise WH is given by the Riesz kernel (10), then U = U(t),t 0
{ ≥ }
becomes fractional Brownian motion with Hurst parameter γ = H d α. We will apply
− −4
this decomposition to obtain the regularity in time of the solution u(t,x),t 0 .
{ ≥ }
6
Motivated by [24] (see also [25, 37] for related results), we introduce the pinned
string process in time U(t),t 0 defined by
{ ≥ }
0
U(t) = G(t u,x y) G( u,x y) WH(du,dy)
Rd − − − − −
Z−∞Z
(cid:0) t (cid:1)
+ G(t u,x u)WH(du,du).
Z0 ZRd − −
Note that U(0) = 0 and U(t) can be expressed as
U(t) = G((t u) ,x y) G(( u) ,x y) WH(du,dy). (23)
+ +
R Rd − − − − −
Z Z
(cid:0) (cid:1)
In the above, a =max(a,0). The following theorem shows that U(t),t 0 has station-
+
{ ≥ }
ary increments and identifies its spectral measure. This information is useful for appleying
the methods in [20, 39, 40] to study sample path properties of U(t),t 0 .
{ ≥ }
Theorem 1 The Gaussian process U(t),t 0 given by (23) has stationary increments
{ ≥ }
and its spectral density is given by
2(2π) dα µ(dξ)
− H
f (τ) = . (24)
U |τ|2H−1 ZRd τ2+ |ξ4|4
Moreover, under condition (19), we have
1
f (τ) . (25)
U ≍ τ 2H−d−2α+1
| |
Proof : For every 0 s < t, by Parseval’s identity (9) and relation (11), we can write
≤
E (U(t) U(s))2
−
2
=(cid:2)E (G((t(cid:3) u) ,x y) G((s u) ,x y))WH(du,dy)
+ +
R Rd − − − − −
(cid:18)Z Z (cid:19)
dudv
= (2π)−dαHZRdµ(dξ)ZRZR |u−v|2−2H (cid:16)e−12(t−u)|ξ|21{t>u}−e−21(s−u)|ξ|21{s>u}(cid:17)
e−12(t−v)|ξ|21 t>v e−21(s−v)|ξ|21 s>v .
× { } − { }
(cid:16) (cid:17)
Let ϕ(u) = e−21(t−u)|ξ|21 t>u e−12(s−u)|ξ|21 s>u . Then its Fourier transform is
{ } − { }
1
ϕ(τ) = e itτ e isτ . (26)
F − − − iτ + 1 ξ 2
2| |
(cid:0) (cid:1)
7
By using again Parseval’s relation (9) in dimension d = 1 and (26), we get
dτ
E (U(t) U(s))2 = (2π) dα µ(dξ) ϕ(τ) 2
− − H ZRd ZR |τ|2H−1 F
(cid:2) (cid:3) dτ(cid:12) 2[1 (cid:12) cos((t s)τ)]
= (2π)−dαH ZRdµ(dξ)ZR |τ|2H−(cid:12)1 −(cid:12)τ2+ |ξ4|−4
dτ µ(dξ)
= 2(2π) dα 1 cos((t s)τ) ,
− H ZR − − |τ|2H−1 ZRd τ2+ |ξ4|4
(cid:2) (cid:3)
where the last step follows from Fubini’s theorem and the convergence of the last integral
in µ(dξ) is guaranteed by relation (16). It follows that the Gaussian process U(t),t 0
{ ≥ }
has stationary increments and its spectral density is given by (24).
Under condition (19), we have
1 dξ 1
f (τ) .
U ≍ |τ|2H−1 ZRd |ξ|α τ2+ |ξ2|4 ≍ |τ|2H−d−2α+1
(cid:16) (cid:17)
This finishes the proof of (25) and thus the conclusion of Theorem 1 is obtained.
When the spatial covariance of noise WH is given by the Riesz kernel (10), we have
α 22H 1Γ(H 1) dξ
f (τ) = H − − 2
U (2π)d+12Γ(1−H)|τ|2H−1 ZRd |ξ|α τ2+ |ξ4|4
α 22H 1Γ(H 1) (cid:16) dη(cid:17)
= H − − 2 .
(2π)d+12Γ(1−H)|τ|2H−d−2α+1 ZRd |η|α 1+ |η4|4
(cid:16) (cid:17)
Therefore, in the Riesz kernel case, U(t),t 0 is, up to a constant, a fractional Brownian
{ ≥ }
motion Bγ with Hurst parameter γ = H d α.
− −4
Recall that the spectral density of the fBm Bγ with Hurst index γ (0,1) is given
∈
by f (λ) = c λ (1+2γ) with c = sin(πγ)Γ(1+2γ).) Hence, we have the following corollary.
γ γ| |− γ 2π
Corollary 1 Let U = U(t),t 0 be the Gaussian process defined by (23) such that the
{ ≥ }
spatial covariance of WH is given by the Riesz kernel (10).Then U coincides in distribution
with C Bγ with γ = H d α and
0 − −4
C2 = (2π)−d+12αH22H−1Γ(H − 21) dη .
0 sin π(d− H−4α) Γ(1+2H − d−2α)Γ(1−H) ZRd |η|α 1+ |η4|4
(cid:0) (cid:1) (cid:16) (cid:17)
Now for every t 0 we have the following decomposition
≥
u(t,x) = U(t) Y(t),
−
8
where
0
Y(t)= (G(t u,x y) G( u,x y))WH(du,dy).
Rd − − − − −
Z−∞Z
The following theorem shows that the sample function of Y(t),t 0 is smooth,
{ ≥ }
which is useful for studying the regularity properties of the solution process u(t,x),t 0
{ ≥ }
in the time variable.
Theorem 2 Let x Rd be fixed and let [a,b] (0, ). Then for any k 1 there is a
∈ ⊂ ∞ ≥
modification of Y(t),t 0 such that its sample function is almost surely continuously
{ ≥ }
differentiable on [a,b] of order k.
Proof : The method of proof is similar to those of [12, Proposition 3.1] and [41, Theorem
4.8], but is more complicated in our fractional-colored noise case.
The mean square derivative of Y at t (0, ) can be expressed as
∈ ∞
0
Y (t)= G(t u,x y)WH(du,dy),
′ ′
Rd − −
Z Z
−∞
where G := ∂G/∂t. This can be verified by checking the covariance functions. For every
′
s,t (0, ) with s t, similarly to the proof of Theorem 1, we derive
∈ ∞ ≤
0 2
E Y (t) Y (s)2 = E G(t u,x y) G(s u,x y) WH(du,dy)
′ ′ ′ ′
| − | Rd − − − − −
(cid:18)Z−∞Z (cid:19)
(cid:0) (cid:1) α (cid:0) dudv (cid:1)
= H ξ 4 αdξ ∞ ∞
−
4(2π)d ZRd | | Z0 Z0 |u−v|2−2H
e−12(t+u)|ξ|2 e−12(s+u)|ξ|2 e−12(t+v)|ξ|2 e−12(s+v)|ξ|2 .
× − −
(cid:16) (cid:17)(cid:16) (cid:17)
Intheabove,wehaveusedthefactthattheFouriertransformofthefunctiony G(t+u,y)
′
7→
is
∂ 1
G(t+u, )(ξ) = ξ 2e−21(t+u)|ξ|2.
∂t F · −2| |
(cid:16) (cid:17)
Denote by the restricted Fourier transform of f L1(0, ) defined by ( f)(τ) =
0, 0,
0∞e−ixτf(Fx)d∞x, τ ∈ R. By applying the Parseval rela∈tion (9)∞for the restricteFd t∞ransform
(see Lemma A1 in [3]), we see that for all s,t [a,b] with s < t,
R ∈
dτ 2
E |Y′(t)−Y′(s)|2 = CZRd |ξ|4−αdξZR |τ|2H−1 (cid:12)F0,∞(cid:16)e−12(t+·)|ξ|2 −e−12(s+·)|ξ|2(cid:17)(τ)(cid:12)
(cid:0) (cid:1) = C ξ 4−αdξ dτ (cid:12)(cid:12)e−21t|ξ|2 e−12s|ξ|2 2 1 (cid:12)(cid:12)
ZRd | | ZR |τ|2H−1 (cid:12) − (cid:12) τ2+ |ξ4|4
(cid:12) 2 (cid:12) dτ
= CZRd|ξ|4−α−4He−s|ξ|2 (cid:12)1−(cid:12)e−12(t−s)|ξ|2(cid:12) dξZR(cid:12) (|τ|2+ 14)|τ|2H−1
(cid:12) (cid:12)
C t s 2 ξ 8 α 4H(cid:12)e aξ2dξ. (cid:12)
− − − | |
≤ | − | Rd| |
Z
9
Hence, as in [12, 41], by using Kolmogorov’s continuity theorem, we can find a modification
of Y such that Y(t) is continuously differentiable on [a,b]. Iterating this argument yields
the conclusion of Theorem 2.
Remark 2 In [25], the authors obtained a similar decomposition for the solution to the
linear heat equation with white noise in time and Riesz covariance in space in dimension
d = 1. Our Theorem 1 shows that the time behavior of the process u is very similar to the
behavior of the bifractional Brownian motion (see the main result in [16]).
ByapplyingTheorems1,2andtheresultsonuniformandlocalmoduliofcontinuity
for Gaussian processes (see e.g. [20, Chapter 7] or [22]), we derive the following regularity
results on the solution process u(t,x),t 0 , when x Rd is fixed. For simplicity, we
{ ≥ } ∈
avoid the point t = 0.
Proposition 1 Let x Rd be fixed. Then for any 0 <a < b < , we have
∈ ∞
u(t,x) u(s,x)
lim sup | − | = c a.s.,
3
ε→0s,t∈[a,b],|s−t|≤ε|s−t|γ log(1/(t−s)
where 0 < c < is a constant that may deppend on a,b,γ and x. Or
3
∞
u(t,x) u(s,x)
lim sup | − | =c a.s.
3
ε→0s,t∈[a,b],|s−t|≤ε εγ log(1/ε)
Here and in Propositions 2 and 3, γ = H pd α.
− −4
Proposition 2 Let x Rd be fixed. Then for any t > 0 we have
0
∈
sup u(t,x) u(t ,x)
lim |t−t0|≤ε| − 0 | = c4 a.s.,
ε 0 εγ loglog(1/ε)
→
where 0 < c < is a constant. p
4
∞
By applying Theorems 1, 2 and the Chung-type law of iterated logarithm in [23],
we derive immediately
Proposition 3 Let x Rd be fixed. Then for any t > 0 we have
0
∈
sup u(t,x) u(t ,x)
lim |t−(tε0|/≤lεo|glog(1/−ε))γ 0 | = c5 a.s.,
ε 0
→
where 0 < c < is a constant depending on the small ball probability of the Gaussian
5
∞
process U in Theorem 1.
Further properties on the local times and fractal behavior of the solution process
u(t,x),t 0 , when x Rd is fixed, can be derived from [38, 39, 40].
{ ≥ } ∈
10