Stochastic Heat Equation Driven by Fractional Noise and Local Time
Stochastic Heat Equation Driven by Fractional
Noise and Local Time
Yaozhong Hu∗ and Nualart†
Department of Mathematics , University of Kansas
405 Snow Hall , Lawrence, Kansas 66045-2142
hu@math.ku.edu and nualart@math.ku.edu
Abstract
The aim of this paper is to study the d-dimensional stochastic heat
equation with a multiplicative Gaussian noise which is white in space
and it has the covariance of a fractional Brownian motion with Hurst
parameter H ∈ (0, 1) in time. Two types of equations are considered.
First we consider the equation in the Itô-Skorohod sense, and later in
the Stratonovich sense. An explicit chaos development for the solution is
obtained. On the other hand, the moments of the solution are expressed
in terms of the exponential moments of some weighted intersection local
time of the Brownian motion.
1 Introduction
This paper deals with the d-dimensional stochastic heat equation
∆u+ u ⋄ ∂
(1.1)
driven by a Gaussian noise WH which is a white noise in the spatial variable
and a fractional Brownian motion with Hurst parameter H ∈ (0, 1) in the time
variable (see (2.1) in the next section for a precise definition of this noise). The
initial condition u0 is a bounded continuous function on R
d, and the solution will
be a random field {ut,x, t ≥ 0, x ∈ Rd}. The symbol ⋄ in Equation (1.1) denotes
the Wick product. For H = 1
is a space-time white noise, and in this
case, Equation (1.1) coincides with the stochastic heat equation considered by
Walsh (see [17]). We know that in this case the solution exists only in dimension
one (d = 1).
There has been some recent interest in studying stochastic partial differential
equations driven by a fractional noise. Linear stochastic evolution equations in
∗Y. Hu is supported by the National Science Foundation under DMS0504783
†D. Nualart is supported by the National Science Foundation under DMS0604207
http://arxiv.org/abs/0704.1824v1
a Hilbert space driven by an additive cylindrical fBm with Hurst parameter H
were studied by Duncan et al. in [3] in the case H ∈ (1
, 1) and by Tindel et al.
in [15] in the general case, where they provide necessary and sufficient conditions
for the existence and uniqueness of an evolution solution. In particular, the
heat equation
on Rd has a unique solution if and only if H > d
. The same result holds when
one adds to the above equation a nonlinearity of the form b(t, x, u), where b
satisfies the usual linear growth and Lipschitz conditions in the variable u, uni-
formly with respect to (t, x) (see Maslowski and Nualart in [9]). The stochastic
heat equation on [0,∞) × Rd with a multiplicative fractional white noise of
Hurst parameter H = (H0, H1, . . . , Hd) has been studied by Hu in [6] under the
conditions 1
< Hi < 1 for i = 0, . . . , d and
i=0Hi < d−
2H0−1
The main purpose of this paper is to find conditions on H and d for the
solution to Equation (1.1) to exist as a real-valued stochastic process, and to
relate the moments of the solution to the exponential moments of weighted in-
tersection local times. This relation is based on Feynman-Kac’s formula applied
to a regularization of Equation (1.1). In order to illustrate this fact, consider
the particular case d = 1 and H = 1
. It is known that there is no Feynman-
Kac’s formula for the solution of the one-dimensional stochastic heat equation
driven by a space-time white noise. Nevertheless, using an approximation of the
solution by regularizing the noise we can establish the following formula for the
moments:
ukt,x
u0(x+B
t ) exp
i,j=1,i<j
s −Bjs)ds
 , (1.2)
for all k ≥ 2, where Bt is a k-dimensional Brownian motion independent of the
spaced-time white noise W
2 . In the case H > 1
and d ≥ 1, a similar formula
holds but
s − Bjs)ds has to be replaced by the weighted intersection
local time
Lt = H(2H − 1)
|s− r|2H−2 δ0(Bis −Bjr)dsdt, (1.3)
where
Bj , j ≥ 1
are independent d-dimensional Brownian motions (see The-
orem 5.3).
The solution of Equation (1.1) has a formal Wiener chaos expansion ut,x =∑∞
n=0 In(fn(·, t, x)). Then, for the existence of a real-valued square integrable
solution we need
n! ‖fn(·, t, x)‖2H⊗n
<∞, (1.4)
whereHd is the Hilbert space associated with the covariance of the noiseWH . It
turns out that, if H > 1
, the asymptotic behavior of the norms ‖fn(·, t, x)‖H⊗n
is similar to the behavior of the nth moment of the random variable Lt defined
in (1.3). More precisely, if u0 is a constant K, for all n ≥ 1 we have
2 ‖fn(·, t, x)‖2H⊗n
= K2E(Lnt ).
These facts leads to the following results:
i) If d = 1 and H > 1
, the series (1.4) converges, and there exists a solution
to Equation (1.1) which has moments of all orders that can be expressed in
terms of the exponential moments of the weighted intersection local times
Lt. In the case H =
we just need the local time of a one-dimensional
standard Brownian motion (see (1.2)).
ii) If H > 1
and d < 4H , the norms ‖fn(·, t, x)‖H⊗n
are finite and E(Lnt ) <
∞ for all n. In the particular case d = 2, the series (1.4) converges if t is
small enough, and the solution exists in a small time interval. Similarly,
if d = 2 the random variable Lt satisfies E(expλLt) < ∞ if λ and t are
small enough.
iii) If d = 1 and 3
< H < 1
, the norms ‖fn(·, t, x)‖H⊗n
are finite and E(Lnt ) <
∞ for all n.
A natural problem is to investigate what happens if we replace the Wick
product by the ordinary product in Equation (1.1), that is, we consider the
equation
∆u + u
. (1.5)
In terms of the mild formulation, the Wick product leads to the use of Itô-
Skorohod stochastic integrals, whereas the ordinary product requires the use
of Stratonovich integrals. For this reason, if we use the ordinary product we
must assume d = 1 and H > 1
. In this case we show that the solution exists
and its moments can be computed in terms of exponential moments of weighted
intersection local times and weighted self-intersection local times in the case
H > 3
The paper is organized as follows. Section 2 contains some preliminaries on
the fractional noiseWH and the Skorohod integral with respect to it. In Section
3 we present the results on the moments of the weighted intersection local times
assuming H ≥ 1
. Section 4 is devoted to study the Wiener chaos expansion
of the solution to Equation (1.1). The case H < 1
is more involved because
it requires the use of fractional derivatives. We show here that if 3
< H < 1
the norms ‖fn(·, t, x)‖H⊗n
are finite and they are related to the moments of a
fractional derivative of the intersection local time. We derive the formulas for
the moments of the solution in the case H ≥ 1
in Section 5. Finally, Section 6
deals with equations defined using ordinary product and Stratonovich integrals.
2 Preliminaries
Suppose that WH = {WH(t, A), t ≥ 0, A ∈ B(Rd), |A| < ∞}, where B(Rd) is
the Borel σ-algebra of Rd, is a zero mean Gaussian family of random variables
with the covariance function
E(WH(t, A)WH(s,B)) =
(t2H + s2H − |t− s|2H)|A ∩B|, (2.1)
defined in a complete probability space (Ω,F , P ), where H ∈ (0, 1), and |A|
denotes the Lebesgue measure of A. Thus, for each Borel set A with finite
Lebesgue measure, {WH(t, A), t ≥ 0} is a fractional Brownian motion (fBm)
with Hurst parameter H and variance t2H |A|, and the fractional Brownian
motions corresponding to disjoint sets are independent.
Then, the multiplicative noise ∂
appearing in Equation (1.1) is the for-
mal derivative of the random measure WH(t, A):
WH(t, A) =
dsdx.
We know that there is an integral representation of the form
WH(t, A) =
KH(t, s)W (ds, dx),
where W is a space-time white noise, and the square integrable kernel KH is
given by
KH(t, s) = cHs
(u− s)H−
2 du,
for some constant cH . We will set KH(t, s) = 0 if s > t.
Denote by E the space of step functions on R+. Let H be the closure of E
with respect to the inner product induced by
1[0,t],1[0,s]
= KH(t, s).
The operator K∗H : E → L2(R+) defined by K∗H(1[0,t])(s) = KH(t, s) provides
a linear isometry between H and L2(R+).
The mapping 1[0,t]×A →WH(t, A) extends to a linear isometry between the
tensor product H ⊗ L2(Rd), denoted by Hd, and the Gaussian space spanned
by WH . We will denote this isometry by WH . Then, for each ϕ ∈ Hd we have
WH(ϕ) =
(K∗H ⊗ I)ϕ(t, x)W (dt, dx).
We will make use of the notation WH(ϕ) =
ϕdWH .
If H = 1
, then H = L2(R+), and the operator K∗H is the identity. In this
case, we have Hd = L2(R+ × Rd).
Suppose now that H > 1
. The operator K∗H can be expressed as a fractional
integral operator composed with power functions (see [11]). More precisely, for
any function ϕ ∈ E with support included in the time interval [0, T ] we have
(K∗Hϕ) (t) = c
ϕ(s)sH−
where I
T− is the right-sided fractional integral operator defined by
T− f(t) =
Γ(H− 1
(s− t)H−
2 f(s)ds.
In this case the space H is not a space of functions (see [14]) because it contains
distributions. Denote by |H| the space of measurable functions on [0, T ] such
that ∫ ∞
|r − u|2H−2|ϕr||ϕu|drdu <∞.
Then, |H| ⊂ H and the inner product in the space H can be expressed in the
following form for ϕ, ψ ∈ |H|
〈ϕ, ψ〉H =
φ(r, u)ϕrϕudrdu, (2.2)
where φ(s, t) = H(2H − 1)|t− s|2H−2.
Using Hölder and Hardy-Littlewood inequalities, one can show (see [10])
‖ϕ‖Hd ≤ βH ‖ϕ‖L 1H (R+;L2(Rd)) , (2.3)
and this easily implies that
‖ϕ‖H⊗n
≤ βnH ‖ϕ‖L 1H (Rn
;L2(Rnd))
. (2.4)
If H < 1
, the operator K∗H can be expressed as a fractional derivative
operator composed with power functions (see [11]). More precisely, for any
function ϕ ∈ E with support included in the time interval [0, T ] we have
(K∗Hϕ) (t) = c
ϕ(s)sH−
where D
T− is the right-sided fractional derivative operator defined by
T− f(t) =
Γ(H+ 1
(T − t) 12−H
f(s)− f(t)
(s− t)H− 32
Moreover, for any γ > 1
− H and any T > 0 we have Cγ([0, T ]) ⊂ H =
T− (L
2([0, T ]).
If ϕ is a function with support on [0, T ], we can express the operator K∗H in
the following form
K∗Hϕ(t) = KH(T, t)ϕ(t) +
[ϕ(s) − ϕ(t)]∂KH
(s, t)ds. (2.5)
We are going to use the following notation for the operator K∗H :
K∗Hϕ =
[0,T ]
ϕ(t)K∗H(dt, r). (2.6)
Notice that if H > 1
, the kernel KH vanishes at the diagonal and we have
K∗H(dt, r) =
(t, r)1[r,T ](t)dt.
Let us now present some preliminaries on the Skorohod integral and the Wick
product. The nth Wiener chaos, denoted by Hn, is defined as the closed linear
span of the random variables of the form Hn(W
H(ϕ)), where ϕ is an element of
Hd with norm one and Hn is the nth Hermite polynomial. We denote by In the
linear isometry between H⊗nd (equipped with the modified norm
n! ‖·‖H⊗n
and the nth Wiener chaos Hn, given by In(ϕ
⊗n) = n!Hn(W
H(ϕ)), for any
ϕ ∈ Hd with ‖ϕ‖Hd = 1. Any square integrable random variable, which is
measurable with respect to the σ-field generated by WH , has an orthogonal
Wiener chaos expansion of the form
F = E(F ) +
In(fn),
where fn are symmetric elements of H⊗nd , uniquely determined by F .
Consider a random field u = {ut,x, t ≥ 0, x ∈ Rd} such that E
u2t,x
for all t, x. Then, u has a Wiener chaos expansion of the form
ut,x = E(ut,x) +
In(fn(·, t, x)), (2.7)
where the series converges in L2(Ω).
Definition 2.1 We say the random field u satisfying (2.7) is Skorohod inte-
grable if E(u) ∈ Hd, for all n ≥ 1, fn ∈ H⊗(n+1)d , and the series
WH(E(u)) +
In+1(f̃n)
converges in L2(Ω), where f̃n denotes the symmetrization of fn.We will denote
the sum of this series by δ(u) =
uδWH .
The Skorohod integral coincides with the adjoint of the derivative operator.
That is, if we define the space D1,2 as the closure of the set of smooth and
cylindrical random variables of the form
F = f(WH(h1), . . . ,W
H(hn)),
hi ∈ Hd, f ∈ C∞p (Rn) (f and all its partial derivatives have polynomial growth)
under the norm
‖DF‖1,2 =
E(F 2) + E(‖DF‖2Hd),
where
(WH(h1), . . . ,W
H(hn))hj ,
then, the following duality formula holds
E(δ(u)F ) = E
〈DF, u〉Hd
, (2.8)
for any F ∈ D1,2 and any Skorohod integrable process u.
If F ∈ D1,2 and h is a function which belongs to Hd, then Fh is Skorohod
integrable and, by definition, the Wick product equals to the Skorohod integral
of Fh:
δ(Fh) = F ⋄WH(h). (2.9)
This formula justifies the use of the Wick product in the formulation of Equation
(1.1).
Finally, let us remark that in the case H = 1
, if ut,x is an adapted stochastic
process such that E
u2t,xdxdt
<∞, then u is Skorohod integrable and
δ(u) coincides with the Itô stochastic integral:
δ(u) =
ut,xW (dt, dx).
3 Weighted intersection local times for standard
Brownian motions
In this section we will introduce different kinds of weighted intersection local
times which are relevant in computing the moments of the solutions of stochastic
heat equations with multiplicative fractional noise.
Suppose first that B1 and B2 are independent d-dimensional standard Brow-
nian motions. Consider a nonnegative measurable function η(s, t) on R2+. We
are interested in the weighted intersection local time formally defined by
η(s, t)δ0(B
s −B2t )dsdt. (3.1)
We will make use of the following conditions on the weight η:
C1) For all T > 0
‖η‖1,T := max
0≤t≤T
η(s, t)ds, sup
0≤s≤T
η(s, t)dt
C2) For all T > 0 there exist constants γT > 0 and H ∈ (0, 1) such that
η(s, t) ≤ γT |s− t|2H−2 ,
for all s, t ≤ T .
Clearly, C2) is stronger than C1). We will denote by pt(x) the d-dimensional
heat kernel pt(x) = (2πt)
2t . Consider the approximation of the inter-
section local time (3.1) defined by
η(s, t)pε(B
s −B2t )dsdt. (3.2)
Let us compute the kth moment of Iε, where k ≥ 1 is an integer. We can write
[0,T ]2k
η(si, ti)ψε(s, t)dsdt, (3.3)
where s = (s1, . . . , sk), t = (t1, . . . , tk) and
ψε(s, t) = E
−B2t1) · · · pε(B
−B2tk)
. (3.4)
Using the Fourier transform of the heat kernel we can write
ψε(s, t) =
ξj , b
− b2tj
|ξj |2
j,l=1
ξjCov
, (3.5)
where ξ = (ξ1, . . . , ξk) and b
t, i = 1, 2, are independent one-dimensional Brow-
nian motions. Then ψε(s, t) ≤ ψ(s, t), where
ψ(s, t) = (2π)−
2 [det (sj ∧ sl + tj ∧ tl)]−
2 . (3.6)
[0,T ]2k
η(si, ti)ψ(s, t)dsdt. (3.7)
Then, if αk <∞ for all k ≥ 1, the family Iε converges in Lp, for all p ≥ 2, to a
limit I and E(Ik) = αk. In fact,
ε,δ↓0
E(IεIδ) = α2,
so Iε converges in L
2, and the convergence in Lp follows from the boundedness
in Lq for q > p. Then the following result holds.
Proposition 3.1 Suppose that C1) holds and d = 1. Then, for all λ > 0 the
random variable defined in (3.2) satisfies
E (exp (λIε)) ≤ 1 + Φ
‖η‖1,T λ
, (3.8)
where Φ(x) =
Γ(k+1
. Also, Iε converges in L
p for all p ≥ 2, and the
limit, denoted by I, satisfies the estimate (3.8).
Proof The term ψ(s, t) defined in (3.6) can be estimated using Cauchy-
Schwarz inequality:
ψ(s, t) ≤ (2π)−k [det (sj ∧ sl)]
4 [det (tj ∧ tl)]
= 2−kπ−
2 [β(s)β(t)]
4 , (3.9)
where for any element (s1, . . . , sk) ∈ (0,∞)k with si 6= sj if i 6= j, we denote
by σ the permutation of its coordinates such that sσ(1) < · · · < sσ(n) and
β(s) = sσ(1)(sσ(2) − sσ(1)) · · · (sσ(k) − sσ(k−1)). Therefore, from (3.9) and (3.7)
we obtain
αk ≤ 2−kπ−
[0,T ]2k
η(si, ti) [β(s)β(t)]
4 dsdt. (3.10)
Applying again Cauchy-Schwarz inequality yields
αk ≤ 2−kπ−
[0,T ]2k
η(si, ti) [β(s)]
2 dsdt
[0,T ]2k
η(si, ti) [β(t)]
2 dsdt
2−1π−
2 ‖η‖1,T
[β(s)]
k!2−kT
2 ‖η‖k1,T
Γ(k+1
, (3.11)
where Tk = {s = (s1, . . . , sk) : 0 < s1 < · · · < sk < T }, which implies the
estimate (3.8).
This result can be extended to the case of a d-dimensional Brownian motion
under the stronger condition C2):
Proposition 3.2 Suppose that C2) holds and d < 4H. Then, limε↓0 Iε = I,
exists in Lp, for all p ≥ 2. Moreover, if d = 2 and λ < λ0(T ), where
λ0(T ) =
H(2H − 1)4π
, (3.12)
and βH is the constant appearing in the inequality (2.3), then
E (exp (λIε)) <∞, (3.13)
and I satisfies E (exp (λI)) <∞.
Proof As in the proof of Proposition 3.1, using condition C2) and inequality
(2.4) we obtain the estimates
αk ≤ γkT 2−dkπ−
[0,T ]2k
|ti − si|2H−2 [β(s)β(t)]−
4 dsdt
≤ γkT 2−dkπ−
2 αkH
[0,T ]k
[β(s)]
4H ds
γTαH2
−2π−1
2H Γ(1− d4H )
k2HT k(1−
Γ(k(1− d
) + 1)2H
γTαH2
−2π−1Γ(1− d
)2HT 2H−
where αH =
H(2H−1)
. This allows us to conclude the proof.
If d = 2 and η(s, t) = 1 it is known that the intersection local time
B2t )dsdt exists and it has finite exponential moments up to a critical exponent
λ0 (see Le Gall [7] and Bass and Chen [1]).
Consider now a one-dimensional standard Brownian motion B, and the
weighted self-intersection local time
η(s, t)δ0(Bs −Bt)dsdt.
As before, set
η(s, t)pε(Bs −Bt)dsdt.
Proposition 3.3 Suppose that C2) holds. If H > 1
, then we have
E (exp (λ [Iε − E (Iε)])) <∞, (3.14)
for all λ > 0. Moreover, the normalized local time I −E (I) exists as a limit in
Lp of Iε − E (Iε), for all p ≥ 2, and it has exponential moments of all orders.
If H > 3
, then we have for all λ > 0
E (exp (λIε)) <∞, (3.15)
for all λ > 0, and the local time I exists as a limit in Lp of Iε, for all p ≥ 2,
and it is exponentially integrable.
Proof We will follow the ideas of Le Gall in [7]. Suppose first that H > 1
let us show (3.14). To simplify the proof we assume T = 1. It suffices to show
these results for
Jε :=
η(s, t)pε(Bs −Bt)dsdt.
Denote, for n ≥ 1, and 1 ≤ k ≤ 2n−1
An,k =
2k − 2
2k − 1
2k − 1
αεn,k =
η(s, t)pε(Bs −Bt)dsdt
ᾱεn,k = α
n,k − E
αεn,k
Notice that the random variables αεn,k, 1 ≤ k ≤ 2n−1, are independent. We
2n−1∑
αεn,k,
Jε − E (Jε) =
2n−1∑
ᾱεn,k.
We can write
αεn,k = 2
2k − 1
2k − 1
×pε(B 2k−1
−B 2k−1
)dsdt
≤ γ12−2n−(2H−2)n
|t+ s|2H−2pε(B 2k−1
−B 2k−1
)dsdt,
which has the same distribution as
βεn,k = γ12
n−(2H−2)n
|t+ s|2H−2pε2n(B1s −B2t )dsdt,
where B1 and B2 are independent one-dimensional Brownian motions. Hence,
using the estimate (3.11), we obtain
ᾱεn,k
= 1 +
ᾱεn,k
≤ 1 +
βεn,k
≤ 1 +
n−(2H−2)nλ
Γ( j+2
for some constant CT . Hence,
ᾱεn,k
≤ 1 + cλ2−3n−2(2H−2)nλ2, (3.16)
for some function cλ.
Fix a > 0 such that a < 2(2H − 1)̇. For any N ≥ 2 define
(1− 2−a(j−1)),
and notice that limN→∞ bN = b∞ > 0. Then, by Hölder’s inequality, for all
N ≥ 2 we have
2n−1∑
ᾱεn,k
 λbN
1− 2−a(N−1)
2n−1∑
ᾱεn,k
1−2−a(N−1)
λbN2a(N−1)
2N−1∑
ᾱεN,k
2−a(N−1)
λbN−1
2n−1∑
ᾱεn,k
a(N−1)ᾱεN,k
)]}2(1−a)(N−1)
Using (3.16), the second factor in the above expression can be dominated by
a(N−1)ᾱN,k
)]}2(1−a)(N−1)
1 + cλλ
2b222
2a(N−1)2−3N−2(2H−2)N
)2(1−a)(N−1)
≤ exp
22(a−2−2(2H−2))N
where κ = b222
−a−1. Thus by induction we have
2n−1∑
ᾱn,k
 ≤ exp
22(a−2−2(2H−2))n
×E (exp ᾱ1,1)
≤ exp(κcλλ2(1 − 2a+2−4H)−1)
×E (exp(ᾱ1,1)) <∞,
because a < 2(2H − 1). By Fatou lemma we see that
E (exp (λb∞ (Jε − E (Jε)))) <∞,
and (3.14) follows.
On the other hand, one can easily show that
ε,δ↓0
E((Jε − E (Jε)) (Jδ − E (Jδ))) =
s<t<1,s′<t′<1
η(s, t)η(s′, t′)
t− s |[s, t] ∩ [s′, t′]|
|[s, t] ∩ [s′, t′]| t′ − s′
])− 1
− ((t− s)(t′ − s′))−
dsdtds′dt′ <∞,
which implies the convergence of Iε in L
2. The convergence in Lp for p ≥ 2 and
the estimate (3.14) follow immediately.
The proof of the inequality (3.15) is similar. The estimate (3.16) is replaced
αεn,k
≤ 1 + dλ2−3n−2(2H−2)nλ, (3.17)
for a suitable function dλ, and we obtain
2n−1∑
≤ exp
κdλλ2
(− 52−2H)n
E (exp (α1,1))
≤ exp(
2(1− 2(−
−2H)n)−1)E (exp (α1,1)) <∞,
because H > 3
. By Fatou lemma we see that
E (exp (λb∞ (Jε − E (Jε)))) <∞,
which implies (3.15). The convergence in Lp of Iε is proved as usual.
Notice that condition H > 3
cannot be improved because
|t− s|−
2 δ0(Bs −Bt)dsdt
|t− s|−1dsdt = ∞.
4 Stochastic heat equation in the Itô-Skorohod
sense
In this section we study the stochastic partial differential equation (1.1) on
Rd, where WH is a zero mean Gaussian family of random variables with the
covariance function (2.1), defined on a complete probability space (Ω,F , P ),
and the initial condition u0 belongs to Cb(R
d). First we give the definition of a
solution using the Skorhohod integral, which corresponds formally to the Wick
product appearing in Equation (1.1).
For any t ≥ 0, we denote by Ft the σ-field generated by the random variables
{W (s, A), 0 ≤ s ≤ t, A ∈ B(Rd), |A| < ∞} and the P -null sets. A random field
u = {ut,x, t ≥ 0, x ∈ R} is adapted if for any (t, x), ut,x is Ft-measurable.
For any bounded Borel function ϕ on R we write ptϕ(x) =
pt(x −
y)ϕ(y)dy.
Definition 4.1 An adapted random field u = {ut,x, t ≥ 0, x ∈ Rd} such that
E(u2t,x) < ∞ for all (t, x) is a solution to Equation (1.1) if for any (t, x) ∈
[0,∞) × Rd, the process {pt−s(x − y)us,y1[0,t](s), s ≥ 0, y ∈ Rd} is Skorohod
integrable, and the following equation holds
ut,x = ptu0(x) +
pt−s(x − y)us,yδWHs,y . (4.1)
The fact that Equation (1.1) contains a multiplicative Gaussian noise allows
us to find recursively an explicit expression for the Wiener chaos expansion of
the solution. This approach has extensively used in the literature. For instance,
we refer to the papers by Hu [6], Buckdahn and Nualart [2], Nualart and Zakai
[13], Nualart and Rozovskii [12], and Tudor [16], among others.
Suppose that u = {ut,x, t ≥ 0, x ∈ Rd} is a solution to Equation (1.1). Then,
for any fixed (t, x), the random variable ut,x admits the following Wiener chaos
expansion
ut,x =
In(fn(·, t, x)), (4.2)
where for each (t, x), fn(·, t, x) is a symmetric element in H⊗nd . To find the
explicit form of fn we substitute (4.2) in the Skorohod integral appearing in
(4.1) we obtain
pt−s(x− y)us,yδWHs,y =
In(pt−s(x− y)fn(·, s, y)) δWHs,y
In+1( ˜pt−s(x− y)fn(·, s, y)) .
Here, ( ˜pt−s(x− y)fn(·, s, y) denotes the symmetrization of the function
pt−s(x− y)fn(s1, x1; . . . ; sn, xn; s, y)
in the variables (s1, x1), . . . , (sn, xn), (s, y), that is,
˜pt−s(x− y)fn(·, s, y) =
[pt−s(x− y)fn(s1, x1, . . . , sn, xn, s, y)
pt−sj (x − yj)
×fn(s1, x1, . . . , sj−1, xj−1, s, y, sj+1, xj+1, . . . , sn, yn, sj , yj)].
Thus, Equation (4.1) is equivalent to say that f0(t, x) = ptu0(x), and
fn+1(·, t, x) = ˜pt−s(x− y)fn(·, s, y) (4.3)
for all n ≥ 0. Notice that, the adaptability property of the random field u
implies that fn(s1, x1, . . . , sn, xn, t, x) = 0 if sj > t for some j.
This leads to the following formula for the kernels fn, for n ≥ 1
fn(s1, x1, . . . , sn, xn, t, x) =
×pt−sσ(n)(x − xσ(n)) · · · psσ(2)−sσ(1)(xσ(2) − xσ(1))psσ(1)u0(xσ(1)), (4.4)
where σ denotes the permutation of {1, 2, . . . , n} such that 0 < sσ(1) < · · · <
sσ(n) < t. This implies that there is a unique solution to Equation (4.1), and the
kernels of its chaos expansion are given by (4.4). In order to show the existence
of a solution, it suffices to check that the kernels defined in (4.4) determine
an adapted random field satisfying the conditions of Definition 4.1. This is
equivalent to show that for all (t, x) we have
n! ‖fn(·, t, x)‖2H⊗n
<∞. (4.5)
It is easy to show that (4.5) holds ifH = 1
and d = 1. In fact, we have, assuming
|u0| ≤ K, and with the notation x = (x1, . . . , xn), and s = (s1, . . . , sn):
‖fn(·, t, x)‖2H⊗n1
[0,t]n
pt−sσ(n)(x− xσ(n))
2 · · · psσ(2)−sσ(1)(xσ(2) − xσ(1))
×psσ(1)u0(xσ(1))
2 dxds
≤ K2 (4π)
[0,t]n
(sσ(j+1) − sσ(j))−
K2 (4π)
(sj+1 − sj)−
2 ds,
where Tn = {(s1, . . . , sn) ∈ [0, t]n : 0 < s1 < · · · < sn < t} and by convention
sn+1 = t. Hence,
‖fn(·, t, x)‖2H⊗n1 ≤
K22−nt
n!Γ(n+1
which implies (4.5). On the other hand, if H = 1
and d ≥ 2, these norms are
infinite.
Notice that if u0 = 1, then (n!)
2 ‖fn(·, t, x)‖2H⊗n1 coincides with the moment
of order n of the local time at zero of the one-dimensional Brownian motion
with variance 2t, that is,
(n!)2 ‖fn(·, t, x)‖2H⊗n1 = E
[(∫ t
δ0(B2s)ds
To handle the case H > 1
, we need the following technical lemma.
Lemma 4.2 Set
gs(x1, . . . , xn) = pt−sσ(n)(x− xσ(n)) · · · psσ(2)−sσ(1)(xσ(2) − xσ(1))). (4.6)
Then,
〈gs, gt〉L2(Rnd) = ψ(s, t),
where ψ(s, t) is defined in (3.4).
Proof By Plancherel’s identity
〈gs, gt〉L2(Rnd) = (2π)
−dn 〈Fgs,Fgt〉L2(Rnd) ,
where F denotes the Fourier transform, given by
Fgs(ξ1, . . . , ξn) = (2π)−
(sσ(j+1) − sσ(j))−
i 〈ξj , xj〉 −
∣∣xσ(j+1) − xσ(j)
sσ(j+1) − sσ(j)
with the convention xn+1 = x and sn+1 = t. Making the change of variables
uj = xσ(j+1) − xσ(j) if 1 ≤ j ≤ n− 1, and un = x− xσ(n), we obtain
Fgs(ξ1, . . . , ξn) = (2π)−
(sσ(j+1) − sσ(j))−
ξσ(j), x− un − · · · − uj
− |uj |
sσ(j+1) − sσ(j)
ξσ(j), x−Bt −Bsσ(j)
ξj , x−Bt −Bsj
As a consequence,
〈gs, gt〉L2(Rnd) = (2π)
ξj , B
−B2tj
 dξ,
which implies the desired result.
In the case H > 1
, and assuming that u0 = 1, the next proposition shows
that the norm (n!)2 ‖fn(·, t, x)‖2H⊗n
coincides with the nth moment of the in-
tersection local time of two independent d-dimensional Brownian motions with
weight φ(t, s).
Proposition 4.3 Suppose that H > 1
and d < 4H. Then, for all n ≥ 1
(n!)2 ‖fn(·, t, x)‖2H⊗n
≤ ‖u0‖2∞E
[(∫ t
φ(s, r)δ0(B
s −B2r )dsdr
(4.7)
with equality if u0 is constant. Moreover, we have:
1. If d = 1, there exists a unique solution to Equation (4.1).
2. If d = 2 , then there exists a unique solution in an interval [0, T ] provided
T < T0, where
βHΓ(1−
)−1/(2H−1)
. (4.8)
Proof We have
(n!)2 ‖fn(·, t, x)‖2H⊗n
≤ ‖u0‖2∞
[0,t]n
φ(sj , tj) 〈gs, gt〉L2(Rnd) dsdt, (4.9)
where gs is defined in (4.6). Then the results follow easily from from Lemma
4.2 and Proposition 3.2.
In the two-dimensional case and assuming H > 1
, the solution would exists
in any interval [0, T ] as a distribution in the Watanabe space Dα,2 for any α > 0
(see [18]).
4.1 Case H < 1
and d = 1
We know that in this case, the norm in the space H is defined in terms of
fractional derivatives. The aim of this section is to show that ‖fn(·, t, x)‖2H⊗n1
is related to the nth moment of a fractional derivative of the self-intersection
local time of two independent one-dimensional Brownian motions, and these
moments are finite for all n ≥ 1, provided 3
< H < 1
Consider the operator (K∗H)
on functions of two variables defined as the
action of the operator K∗H on each coordinate. That is, using the notation (2.5)
we have
(K∗H)
f(r1, r2) = KH(T, r1)KH(T, r2)f(r1, r2)
+KH(T, r1)
(s, r2) (f(r1, s)− f(r1, r2)) ds
+KH(T, r2)
(v, r1) (f(v, r2)− f(r1, r2)) dv
(s, r2)
(v, r1) [f(v, s)− f(r1, s)− f(v, r2)− f(r1, r2)] dsdv.
Suppose that f(s, t) is a continuous function on [0, T ]2. Define the Hölder norms
‖f‖1,γ = sup
|f(s1, t)− f(s2, t)|
|s1 − s2|γ
, s1, s2, t ∈ T, s1 6= s2
‖f‖2,γ = sup
|f(s, t1)− f(s, t2)|
|t1 − t2|γ
, t1, t2, s ∈ T, t1 6= t2
‖f‖1,2,γ = sup
|f(s1, t1)− f(s1, t2)− f(s2, t1) + f(s2, t2)|
|s1 − s2|γ |t1 − t2|γ
where the supremum is taken in the set {t1, t2, s2, s2 ∈ T, s1 6= s2, t1 6= t2}. Set
‖f‖0,γ = ‖f‖1,γ + ‖f‖2,γ + ‖f‖1,2,γ
Then, (K∗H)
f is well defined if ‖f‖0,γ < ∞ for some γ >
− H . As
a consequence, if B1 and B2 are two independent one-dimensional Brownian
motions, the following random variable is well defined for all ε > 0
(K∗H)
· −B2· )(r, r)dr. (4.10)
The next theorem asserts that Jε converges in L
p for all p ≥ 2 to a fractional
derivative of the intersection local time of B1 and B2.
Proposition 4.4 Suppose that 3
< H < 1
.Then, for any integer k ≥ 1 and,
T > 0 we have E
≥ 0 and
Moreover, for all p ≥ 2, Jε converges in Lp as ε tends to zero to a random
variable denoted by
(K∗H)
· −B2· )(r, r)dr.
Proof Fix k ≥ 1. Let us compute the moment of order k of Jε. We can write
[0,T ]k
(K∗H)
p2ε(B
1 −B2)(ri, ri)
dr. (4.11)
Using the expression (2.6) for the operator K∗H , and the notation (3.4) yields
[0,T ]3k
ψε(s, t)
K∗H(dsi, ri)K
H(dti, ri)dr. (4.12)
As a consequence, using (3.5) we obtain
= (2π)−k
[0,T ]k
[0,T ]2k
j,l=1 ξjξlCov
−B2tj
,B1sl
−B2tl
K∗H(dsi, ri)K
H(dti, ri)e
j=1 ξ
j dξdr
≤ (2π)−k
[0,T ]k
[0,T ]k
K∗H(dti, ri)
dξdr.
Then, it suffices to show that for each k the following quantity is finite
[sj − sj−1 + tj − tj−1]−
K∗H(dsi, ri)
K∗H(dti, ri)dr, (4.13)
where Tk = {0 < t1 < · · · < tk < T }. Fix a constant a > 0. We are going to
compute
[tj − tj−1 + a]−2
K∗H(dti, ri).
To do this we need some notation. Let ∆j and Ij be the operators defined on
a function f(t1, . . . , tk) by
∆jf = f − f |tj=rj ,
Ijf = f |tj=rj .
The operatorK∗H(dti, ri) is the sum of two components (see (2.5)), and it suffices
to consider only the second one because the first one is easy to control. In this
way we need to estimate the following term
[0,T ]k
∆1 · · ·∆k
j [tj − tj−1 + a]
2 1{tj−1<tj}
(tj − rj)H−
j 1{rj<tj}
Because t
j ≤ 1, we can disregard the factors r
j and t
j . Using
the rule
∆j(FG) = F (tj)G(tj)− F (rj)G(rj)
= [F (tj)− F (rj)]G(tj) + F (rj) [G(tj)−G(rj)]
= ∆jFG+ IjF∆jG,
we obtain
∆1 · · ·∆k
[tj − tj−1 + a]−
2 1{tj−1<tj}
[tj − tj−1 + a]−
2 1{tj−1<tj}
where Sj is an operator of the form:
IIj , I∆j ,∆j−1Ij ,∆j−1∆j ,
and for each j, ∆j must appear only once in the product
j=1 Sj . Let us
estimate each one of the possible four terms. Fix ε > 0 such that H − 3
> 2ε.
1. Term IIj :
[tj − tj−1 + a]−
2 1{tj−1<tj}
= [rj − tj−1 + a]−
2 1{tj−1<rj},
2. Term I∆j :
∣∣∣I∆j
[tj − tj−1 + a]−
2 1{tj−1<tj}
∣∣∣[tj − tj−1 + a]−
2 1{tj−1<tj} − [rj − tj−1 + a]
2 1{tj−1<rj}
≤ C [tj − rj ]
[rj − tj−1 + a]H−1−ε 1{tj−1<rj}
+C [tj − tj−1 + a]−
2 1{rj<tj−1}.
3. Term ∆j−1I:
∣∣∣∆j−1I
[tj − tj−1 + a]−
2 1{tj−1<tj}
∣∣∣[tj − tj−1 + a]−
2 1{tj−1<tj} − [tj − rj−1 + a]
2 1{rj−1<tj}
≤ C [tj−1 − rj−1]
[tj − tj−1 + a]H−1−ε 1{rj−1<tj−1<tj}.
4. Term ∆j−1∆j :
∣∣∣∆j−1∆j
[tj − tj−1 + a]−
2 1{tj−1<tj}
∣∣∣[tj − tj−1 + a]−
2 1{tj−1<tj} − [rj − tj−1 + a]
2 1{tj−1<rj}
− [tj − rj−1 + a]−
2 1{rj−1<tj} + [rj − rj−1 + a]
2 1{rj−1<rj}
≤ C [tj − rj ]
[tj−1 − rj−1]
[rj − tj−1 + a]2H−
1{tj−1<rj<tj}
+C [tj−1 − rj−1]
[tj − tj−1 + a]H−1−ε 1{rj<tj−1<tj}
+C [rj − rj−1 + a]−
2 1{rj−1<rj<tj−1<tj}.
If we replace the constant a by sj−sj−1 and we treat the the term sj−sj−1
in the same way, using the inequality
(a+ b)−α ≤ a−
we obtain the same estimates as if we had started with
[tj − tj−1]−
K∗H(dtj , rj)
instead of (4.13). As a consequence, it suffices to control the following integral
j (t, r)dt
dr, (4.14)
where a, b ∈ {0, 1}, and Aj has one of the following forms
j = [rj − tj−1]
4 1{tj−1<rj},
j,1 = [tj − rj ]
[rj − tj−1]H−
1{tj−1<rj}
j,2 = [tj − tj−1]
4 [tj − rj ]H−
2 1{rj<tj−1},
j = [tj−1 − rj−1]
[tj − tj−1]H−
1{rj−1<tj−1<tj},
j,1 = [tj − rj ]
[tj−1 − rj−1]−1+ε [rj − tj−1]2H−
1{tj−1<rj<tj},
j,2 = [tj−1 − rj−1]
[tj − tj−1]H−
[tj − rj ]H−
2 1{rj<tj−1<tj},
j,3 = [rj − rj−1]
4 [tj − rj ]H−
2 [tj−1 − rj−1]H−
2 1{rj−1<rj<tj−1<tj},
and with the convention that any term of the form A
j or A
j must be followed
j or A
j and any term of the form A
j or A
j must be followed by
j or A
j . It is not difficult to check that the integral (4.14) is finite. For
instance, for a product of the form A
j,1 we get
{rj−1<tj−1<rj<tj}
[rj−1 − tj−2]−
4 [tj−1 − rj−1]−1+ε [rj − tj−1]2H−
× [tj − rj ]−1+ε dtj−1
= [rj−1 − tj−2]−
4 [rj − rj−1]2H−
−ε [tj − rj ]−1+ε ,
and the integral in the variable rj of the square of this expression will be finite
because 4H − 5
− 2ε > −1.
So, we have proved that supε E(J
ε ) < ∞ for all k. Notice that all these
moments are positive. It holds that limε,δ↓0 E(JεJδ) exists, and this implies the
convergence in L2, and also in Lp, for all p ≥ 2.
On the other hand, if the initial condition of Equation (1.1) is a constant
K, then for all n ≥ 1 we have
(n!)2 ‖fn(·, t, x)‖2H⊗n1 = K
[(∫ T
(K∗H)
· −B2· )(r, r)dr
provided H ∈
. In fact, by Lemma 4.2 we have
(n!)2 ‖fn(·, t, x)‖2H⊗n1 = K
[0,t]2n
〈gs, gt〉L2(Rn)
K∗H(dti, ri)
K∗H(dsi, ri)dsdt
[0,t]2n
ψ(s, t)
K∗H(dti, ri)
K∗H(dsi, ri)dsdt,
and it suffices to apply the above proposition.
However, we do not know the rate of convergence of the sequence ‖fn(·, t, x)‖2H⊗n1
as n tends to infinity, and for this reason we are not able to show the existence
of a solution to Equation (1.1) in this case.
5 Moments of the solution
In this section we introduce an approximation of the Gaussian noise WH by
means of an approximation of the identity. In the space variable we choose the
heat kernel to define this approximation and in the time variable we choose a
rectangular kernel. In this way, for any ε > 0 and δ > 0 we set
t,x =
ϕδ(t− s)pε(x− y)dWHs,y, (5.1)
where
ϕδ(t) =
1[0,δ](t).
Now we consider the approximation of Equation (1.1) defined by
t,x + u
t,x ⋄ Ẇ
t,x . (5.2)
We recall that the Wick product u
t,x⋄Ẇ
t,x is well defined as a square integrable
random variable provided the random variable u
t,x belongs to the space D
(see (2.9)), and in this case we have
uε,δs,y ⋄ Ẇ ε,δs,y =
ϕδ(s− r)pε(y − z)uε,δs,yδWHr,z . (5.3)
The mild or evolution version of Equation (5.2) will be
t,x = ptu0(y) +
pt−s(x− y)uε,δs,y ⋄ Ẇ ε,δs,y dsdy. (5.4)
Substituting (5.3) into (5.4), and formally applying Fubini’s theorem yields
t,x = ptu0(y)+
pt−s(x− y)ϕδ(s− r)pε(y − z)uε,δs,ydsdy
δWHr,z .
(5.5)
This leads to the following definition.
Definition 5.1 An adapted random field uε,δ = {uε,δt,x, t ≥ 0, x ∈ Rd} is a mild
solution to Equation (5.2) if for each (r, z) ∈ R+ × Rd the integral
Y t,xr,z =
pt−s(x − y)ϕδ(s− r)pε(y − z)uε,δs,ydsdy
exists and Y t,x is a Skorohod integrable process such that (5.5) holds for each
(t, x).
The above definition is equivalent to saying that u
t,x ∈ L2(Ω), and for any
random variable F ∈ D1,2 , we have
t,x) = E(F )ptu0(y)
〈(∫ t
pt−s(x− y)ϕδ(s− ·)pε(y − ·)uε,δs,ydsdy
Our aim is to construct a solution of Equation (5.2) using a suitable version
of Feynman-Kac’s formula. Suppose that B = {Bt, t ≥ 0} is a d-dimensional
Brownian motion starting at 0, independent of W . Set
t−s,x+Bs
ϕδ(t− s− r)pε(Bs + x− y)dWHr,yds
Aε,δr,ydW
r,y ,
where
Aε,δr,y =
ϕδ(t− s− r)pε(Bs + x− y)ds. (5.6)
Define
t,x = E
u0(x+Bt) exp
Aε,δr,ydW
r,y −
, (5.7)
where αε,δ =
∥∥Aε,δ
Proposition 5.2 The random field u
t,x given by (5.7) is a solution to Equation
(5.2).
Proof The proof is based on the notion of S transform from white noise
analysis (see [5]). For any element ϕ ∈ H1 we define
St,x(ϕ) = E
t,xFϕ
where
Fϕ = exp
WH(ϕ)− 1
‖ϕ‖2Hd
From (5.7) we have
St,x(ϕ) = E
u0(x+Bt) exp
WH(Aε,δ + ϕ)−
αε,δ −
‖ϕ‖2Hd
u0(x+Bt) exp
Aε,δ, ϕ
u0(x+Bt) exp
〈ϕδ(t− s− ·)pε(Bs + x− ·), ϕ〉Hd ds
By the classical Feynman-Kac’s formula, St,x(ϕ) satisfies the heat equation with
potential V (t, x) = 〈ϕδ(t− ·)pε(x− ·), ϕ〉Hd , that is,
∂St,x(ϕ)
∆St,x(ϕ) + St,x(ϕ) 〈ϕδ(t− ·)pε(x − ·), ϕ〉Hd .
As a consequence,
St,x(ϕ) = ptu0(x) +
pt−s(x− y)Ss,y(ϕ) 〈ϕδ(s− ·)pε(y − ·), ϕ〉Hd dsdy.
Notice that DFϕ = ϕFϕ. Hence, for any exponential random variable of this
form we have
t,xFϕ) = ptu0(x) +
pt−s(x− y)E
t,x 〈ϕδ(s− ·)pε(y − ·), DFϕ〉Hd
and we conclude by the duality relationship between the Skorohod integral and
the derivative operator.
The next theorem says that the random variables u
t,x have moments of all
orders, uniformly bounded in ε and δ, and converge to the solution to Equation
(1.1) as δ and ǫ tend to zero. Moreover, it provides an expression for the
moments of the solution to Equation (1.1).
Theorem 5.3 Suppose that H ≥ 1
and d = 1. Then, for any integer k ≥ 1 we
[∣∣∣uε,δt,x
<∞, (5.8)
and the limit limε↓0 limδ↓0 u
t,x exists in L
p, for all p ≥ 1, and it coincides with
the solution ut,x of Equation (1.1). Furthermore, if U
0 (t, x) =
j=1 u0(x+B
where Bj are independent d-dimensional Brownian motions, we have for any
k ≥ 2
ukt,x
UB0 (t, x) exp
s −Bjs)ds
 . (5.9)
if H = 1
, and
ukt,x
UB0 (t, x) exp
φ(s, r)δ0(B
s −Bjr)dsdr
 . (5.10)
if H > 1
In the case d = 2, for any integer k ≥ 2 there exists t0(k) > 0 such that
for all t < t0(k) (5.8) holds. If t < t0(M) for some M ≥ 3 then the limit
limε↓0 limδ↓0 u
t,x exists in L
p for all 2 ≤ p < M , and it coincides with the
solution ut,x of Equation (1.1). Moreover, (5.10) holds for all 1 ≤ k ≤M − 1.
Proof Fix an integer k ≥ 2. Suppose that Bi =
Bit , t ≥ 0
, i = 1, . . . , k are
independent d-dimensional standard Brownian motions starting at 0, indepen-
dent of WH . Then, using (5.7) we have
u0(x +B
t ) exp
Aε,δ,B
r,y dW
r,y −
αε,δ,B
where Aε,δ,B
r,y and α
ε,δ,Bj are computed using the Brownian motion Bj . There-
fore,
= E B
 exp
∥∥∥∥∥∥
Aε,δ,B
∥∥∥∥∥∥
αε,δ,B
u0(x+B
 exp
Aε,δ,B
, Aε,δ,B
u0(x+B
That is, the correction term 1
αε,δ in (5.7) due to the Wick product produces a
cancellation of the diagonal elements in the square norm of
j=1 A
ε,δ,Bj . The
next step is to compute the scalar product
Aε,δ,B
, Aε,δ,B
for i 6= j. We
consider two cases.
Case 1. Suppose first that H = 1
and d = 1. In this case we have
Aε,δ,B
, Aε,δ,B
ϕδ(t− s1 − r)pε(Bis1 + x− y)
×ϕδ(t− s2 − r)pε(Bjs2 + x− y)ds1ds2drdy
ϕδ(t− s1 − r)ϕδ(t− s2 − r)
×p2ε(Bis1 −B
)ds1ds2dr.
We have
ϕδ(t− s1 − r)ϕδ(t− s2 − r)dr
= δ−2
(t− s1) ∧ (t− s2)− (t− s1 − δ)+ ∨ (t− s2 − δ)+
= ηδ(s1, s2).
It it easy to check that ηδ is a a symmetric function on [0, t]
2 such that for any
continuous function g on [0, t]2,
ηδ(s1, s2)g(s1, s2)ds1ds2 =
g(s, s)ds.
As a consequence the following limit holds almost surely
Aε,δ,B
, Aε,δ,B
p2ε(B
s −Bjs)ds,
and by the properties of the local time of the one-dimensional Brownian motion
we obtain that, almost surely.
Aε,δ,B
, Aε,δ,B
s −Bjs)ds.
The function ηδ satisfies
0≤r≤t
ηδ(s, r)ds ≤ 1,
and, as a consequence, the estimate (3.8) implies that for all λ > 0
λ exp
Aε,δ,B
, Aε,δ,B
Hence (5.8) holds and limε↓0 limδ↓0 u
t,x := vt,x exists in L
p, for all p ≥ 1.
Moreover, E(vkt,x) equals to the right-hand side of Equation (5.9). Finally,
Equation (5.5) and the duality relationship (2.8) imply that for any random
variable F ∈ D1,2 with zero mean we have
pt−s(x− y)ϕδ(s− ·)pε(y − ·)uε,δs,ydsdy
and letting δ and ε tend to zero we get
Fvt,x
pt−s(x − y)ϕδ(s− ·)pε(y − ·)vs,ydsdy
which implies that the process v is the solution of Equation (1.1), and by the
uniqueness vt,x = ut,x.
Case 2. Consider now the case H > 1
and d = 2. We have
Aε,δ,B
, Aε,δ,B
ϕδ(t− s1 − r1)pε(Bis1 + x− y)
×ϕδ(t− s2 − r2)pε(Bjs2 + x− y)ds1ds2 φ(r1, r2)dr1dr2dy
ϕδ(t− s1 − r1)ϕδ(t− s2 − r2)
×p2ε(Bis1 −B
)ds1ds2φ(r1, r2)dr1dr2.
This scalar product can be written in the following form
Aε,δ,B
, Aε,δ,B
ηδ(s1 − s2)p2ε(Bis1 −B
)ds1ds2,
where
ηδ(s1, s2) =
ϕδ(t− s1 − r1)ϕδ(t− s2 − r2) φ(r1, r2)dr1dr2. (5.11)
We claim that there exists a constant γ such that
ηδ(s1, s2) ≤ γ|s1 − s2|2H−2. (5.12)
In fact, if |s2 − s1| = s we have
ηδ(s1, s2) ≤ H(2H − 1)δ−2
∫ s+δ
|u− v|2H−2dudv
(s+ δ)2H − (s− δ)2H − 2s2H
∫ s+δ
y2H−1 − (y − δ)2H−1
dy ≤ Hδ2H−2 ≤ H22−2Hs2H−2,
if s ≤ 2δ. On the other hand, if s ≥ 2δ, we have
(s+ δ)2H − (s− δ)2H − 2s2H
s2H−1 − (s− δ)2H−1
≤ H(2H − 1)(s− δ)2H−2
≤ H(2H − 1)22−2Hs2H−2.
It it easy to check that for any continuous function g on [0, t]2,
ηδ(s1, s2)g(s1, s2)ds1ds2 =
φ(s1, s2)g(s1, s2)ds1ds2.
As a consequence the following limit holds almost surely
Aε,δ,B
, Aε,δ,B
φ(s1, s2)δ0(B
−Bjs2)ds1ds2.
From (5.12) and the estimate (3.13) we get
Aε,δ,B
, Aε,δ,B
<∞, (5.13)
if λ < λ0(t), where λ0(t) is defined in (3.12) with gammaT replaced by γ.
Hence, for any integer k ≥ 2, if t < t0(k), where k(k−1)2 = λ0(t0(k)), then
(5.8) holds because
≤ ‖u0‖k
k(k − 1)
Aε,δ,B
, Aε,δ,B
)]) 2
k(k−1)
Finally, if t < t0(M) and M ≥ 3, the limit limε↓0 limδ↓0 uε,δt,x := vt,x exists in Lp,
for all 2 ≤ p < M and it is equal to the right-hand side of Equation (5.10). As
in the case H = 1
we show that vt,x = ut,x.
6 Pathwise heat equation
In this section we consider the one-dimensional stochastic partial differential
equation
∆u+ uẆHt,x, (6.1)
where the product between the solution u and the noise ẆHt,x is now an ordinary
product. We first introduce a notion of solution using the Stratonovich integral
and a weak formulation of the mild solution. Given a random field v = {vt,x, t ≥
0, x ∈ R} such that
|vt,x| dxdt < ∞ a.s. for all T > 0, the Stratonovich
integral ∫ T
vt,xdW
is defined as the following limit in probability if it exists
vt,xẆ
t,x dxdt,
where W
t,x is the approximation of the noise W
H introduced in (5.1).
Definition 6.1 A random field u = {ut,x, t ≥ 0, x ∈ R} is a weak solution to
Equation (6.1) if for any C∞ function ϕ with compact support on R, we have
ut,xϕ(x)dx =
u0(x)ϕ(x)dx+
us,xϕ
′′(x)dxds+
us,xϕ(x)dW
Consider the approximating stochastic heat equation
∂uε,δ
∆uε,δ + uε,δẆ
t,x . (6.2)
Theorem 6.2 Suppose that H > 3
. For any p ≥ 2, the limit
t,x = ut,x
exists in Lp, and defines a weak solution to Equation (6.2) in the sense of
Definition 6.1. Furthermore, for any positive integer k
ukt,x
UB0 (t, x) exp
i,j=1
φ(s1, s2)δ(B
−Bjs2)ds1ds2
where UB0 (t, x) has been defined in Theorem (5.3).
Proof By Feynman-Kac’s formula we can write
t,x = E
u0(x+Bt) exp
Aε,δr,ydW
, (6.3)
where Aε,δr,y has been defined in (5.6). We will first show that for all k ≥ 1
[∣∣∣uε,δt,x
<∞. (6.4)
Suppose that Bi =
Bit , t ≥ 0
, i = 1, . . . , k are independent standard Brownian
motions starting at 0, independent of WH .Then, we have, as in the proof of
Theorem 5.3
= E B
 exp
i,j=1
Aε,δ,B
, Aε,δ,B
UB0 (t, x)
Notice that
Aε,δB
, Aε,δB
ηδ(s1, s2)p2ε(B
−Bjs2)ds1ds2,
where ηδ(s1, s2) satisfies (5.12). As a consequence, the inequalities (3.8) and
(3.15) imply that for all λ > 0, and all i,j we have
Aε,δB
, Aε,δB
Thus, (6.4) holds, and
= EB exp
UB0 (t, x) exp
i,j=1
φ(s1, s2)δ0(B
−Bjs2)ds1ds2
In a similar way we can show that the limit limε,ε′↓0 limδ,δ′↓0E
ε′,δ′
ists. Therefore, the iterated limit limε↓0 limδ↓0E
exists in L2.
Finally we need to show that
us,xϕ(x)dW
s,x −
uε,δs,xϕ(x)Ẇ
s,xdsdx
in probability. We know that
uε,δs,xϕ(x)Ẇ
s,xdsdx converges in L
2 to some
random variable G. Hence, if
Bε,δ =
uε,δs,x − us,x
ϕ(x)ẆHs,xdsdx (6.5)
converges in L2 to zero, us,xϕ(x) will be Stratonovich integrable and
us,xϕ(x)Ẇ
s,xdsdx = G.
The convergence to zero of (6.5) is done as follows. First we remark that Bε,δ =
δ(φε,δ), where
φε,δr,z =
uε,δs,x − us,x
ϕ(x)ϕδ(s− r)pε(x− z)dsdx.
Then, from the properties of the divergence operator, it suffices to show that
(∥∥Dφε,δ
H1⊗H1
= 0. (6.6)
It is clear that limε↓0 limδ↓0E
(∥∥φε,δ
= 0. On the other hand,
φε,δr,z
uε,δs,x
−D (us,x)
ϕ(x)ϕδ(s− r)pε(x − z)dsdx,
uε,δs,x
= E B
u0(x+Bt) exp
Aε,δs,ydW
Then, as before we can show that
ε,ε′↓0
δ.δ′↓0
uε,δs,x
= E B
u0(x+B1t )u0(x+B
t ) exp
i,j=1
φ(s1, s2)δ0(B
−Bjs2)ds1ds2
φ(s1, s2)δ0(B
−B2s2)ds1ds2
This implies that uε,δs,x converges in the space D
1,2 to us,x as δ ↓ 0 and ε ↓ 0.
Actually, the limit is in the norm of the space D1,2(H1). Then, (6.6) follows
easily.
Since the solution is square integrable it admits a Wiener-Itô chaos expan-
sion. The explicit form of the Wiener chaos coefficients are given below.
Theorem 6.3 The solution to (6.1) is given by
ut,x =
In(fn(·, t, x)) (6.7)
where
fn(t1, x1, . . . , tn, xn, t, x)
u0(x+Bt) exp
φ(s1, s2)δ0(Bs1 −Bs2)ds1ds2
×δ0(Bt1 + x− x1) · · · δ0(Btn + x− xn)] . (6.8)
Proof From the Feynman-Kac formula it follows that
t,x = E
u0(x +Bt) exp
Aε,δr,ydW
u0(x +Bt) exp
‖Aε,‖2H1
Aε,δr,ydW
r,y −
‖Aε,δ‖2H1
n (t, x)),
where
f ε,δn (t1, x1, . . . , tn, xn, t, x) = E
u0(x+Bt) exp
‖Aε,δ‖2H1
t1,x1
· · ·Aε,δtn,xn
Letting δ and ε go to 0, we obtain the chaos expansion of ut,x.
Consider the stochastic partial differential equation (6.1) and its approxima-
tion (6.2). The initial condition is u0(x). We shall study the strict positivity of
the solution. In particular we shall show that E [|ut(x)|−p] <∞.
Theorem 6.4 Let H > 3/4. If E (|u0(Bt)|) > 0, then for any 0 < p < ∞, we
have that
|ut,x|−p
<∞ (6.9)
and moreover,
|ut(x)|−p
≤ (E|u0(x +Bt)|)−p−1E B
|u0(x+Bt)|
× exp
δ(Bs1 −Bs2)φ(s1, s2)ds1ds2
. (6.10)
Proof Denote κp =
E B (|u0(x+Bt)|)
)−p−1
. Then, Jensen’s inequality ap-
plied to the equality u
t,x = E
u0(x +Bt) exp
Aε,δr,ydW
implies
|uε,δt,x|−p ≤ κpE B
|u0(x +Bt)| exp
Aε,δr,ydW
Therefore
|uε,δt,x|−p
≤ κpE B
|u0(x+Bt)|E
Aε,δr,ydW
= κpE
|u0(x+Bt)|E
∥∥Aε,δ
and we can conclude as in the proof of Theorem 6.2.
Using the theory of rough path analysis (see [8]) and p-variation estimates,
Gubinelli, Lejay and Tindel [4] have proved that for H > 3
, the equation
∆u+ σ(u)ẆHt,x
had a unique mild solution up to a random explosion time T > 0, provided
σ ∈ C2b (R). In this sense, the restriction H > 34 , that we found in the case
σ(x) = x is natural, and in this particular case, using chaos expansion and
Feynman-Kac’s formula we have been able to show the existence of a solution
for all times.
Acknowledgment. We thank Carl Mueller for discussions.
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	Introduction
	Preliminaries
	Weighted intersection local times for standard Brownian motions
	Stochastic heat equation in the Itô-Skorohod sense
	Case H<12 and d=1
	Moments of the solution
	Pathwise heat equation
