Burkholder-Davis-Gundy inequality and Minkowski’s integral inequality, for (t, x) ∈ [0,∞)× [0,1]
e−αt(E[|u(t, x)|p])1p
≤e−αtK+
E
Z t 0
Z 1 0
e−αtPt−sb (x, y)f(s, u(s, y))ξ(s, y)dµ(y)ds
p1p
≤e−αtK+ 2√ p
Z t 0
Z 1 0
e−2αtPt−sb (x, y)2(E[|f(s, u(s, y))|p])2pdµ(y)ds 12
≤e−αtK+L2√ p
Z t 0
Z 1 0
e−2αtPt−sb (x, y)2 sup
z∈[0,1]
1 + (E[|u(s, z))|p])1p2
dµ(y)ds
!12
and further, by Lemma 5.6, Z t
0
Z 1 0
e−2α(t−s)Pt−sb (x, y)2 sup
z∈[0,1]
e−αs+e−αs(E[|u(s, z))|p])1p2
dµ(y)ds
≤ sup
(s,z)∈[0,T]×[0,1]
e−αs+e−αs(E[|u(s, z))|p])1p2Z t 0
Z 1 0
e−2α(t−s)Pt−sb (x, y)2dµ(y)ds
≤ 1 + sup
(s,z)∈[0,T]×[0,1]
e−αs(E[|u(s, z))|p])p1
!2
C92 Z t
0
e−2α(t−s)ds
≤ C92
2α 1 + sup
(s,z)∈[0,T]×[0,1]
e−αs(E[|u(s, z))|p])1p
!2
.
Letα:= 8C92L2p. Then, L2
√pC9
√2α = 12 and thus
(E[|u(t, x)|p])1p ≤2K+eαt= 2K +e8C92L2pt.
It remains to check the case p∈[1,2). For (t, x)∈[0,∞)×[0,1] and p∈[1,2), we have
(E[|u(t, x)|p])1p ≤ E
|u(t, x)|212
≤2K +e16C92L2t≤2K+e16C92L2pt. We immediately obtain forp≥1
¯
γ(p) = lim sup
t→∞
1 t sup
x∈[0,1]
logE[|u(t, x)|p]≤lim sup
t→∞
logC12
t +C13p2 =C13p2.
For the lower bound, we deal with u(t, x) :=Ctbu0(x) +
Z t 0
Z 1 0
Pt−sb (x, y)f(u(s, y))ξ(s, y)dµ(y)ds (84) for (t, x) ∈ [0,∞)×[0,1]. That is, we let u1 = 0 and f be not time-dependent in (71). We assume the following conditions.
Assumption 5.17: Let Condition 5.2(i) hold. Furthermore, let f : R→R satisfy the following Lipschitz and linear growth conditions: There exists L > 0 such that for all (x, y)∈R2
|f(x)−f(y)| ≤L|x−y|,
|f(x)| ≤L(1 +|x|).
Proposition 5.18: Assume Lf := inf
x∈R\{0}
f(x) x
>0.
(i) Let b = N and infx∈[0,1]u0(x) > 0. Then, there exists a constant κ > 0 such that γ(2, x)¯ ≥κ for all x∈[0,1].
(ii) Let b = D, ε > 0 and infx∈[ε,1−ε]u0(x) > 0. Then, there exists a constant κε>0 such that ¯γ(2, x)≥κε for all x∈[ε,1−ε].
Proof. Let ε ≥0, infx∈[ε,1−ε]u0(x)> 0, x ∈ [ε,1−ε]. It suffices to find a constant β >0 such that
Z ∞ 0
e−βtE
u(t, x)2
dt=∞ for all β ≤β (85)
(compare the proof of [13, Theorem 3.2]). Using Walsh’s isometry and the zero-mean property of the stochastic integral yields for t∈[0,∞)
E
u(t, x)2
dt =v3(t, x)2+ Z t
0
Z 1 0
Pt−sb (x, y)2E
f(u(s, y))2
dµ(y)ds +v3(t, x)E
Z t 0
Z 1 0
Pt−sb (x, y)f(u(s, y))ξ(s, y)dµ(y)ds
=v3(t, x)2+ Z t
0
Z 1 0
Pt−sb (x, y)2E
f(u(s, y))2
dµ(y)ds
and thus, by Laplace transformation for β >0, Z ∞
0
e−βtE
u(t, x)2 dt =
Z ∞ 0
e−βtv3(t, x)2dt +
Z ∞ 0
e−βt Z t
0
Z 1 0
Pt−sb (x, y)2E
f(u(s, y))2
dµ(y)dsdt.
In order to bound the first term on the right-hand side from below, note that v3(0, x) = u0(x) ≥ infy∈[ε,1−ε]u0(y) > 0. By Proposition 5.8, v3 is Hölder con-tinuous int uniformly for all x∈[0,1]. We thus obtain the existence of a constant t0 >0 such that v3(t, x)> u20, (t, x) ∈ [0, t0]×[ε,1−ε]. Let Kβ := 16βu20 . Then, for x∈[ε,1−ε]
Z ∞ 0
e−βtE
u(t, x)2 dt
≥Kβ+L2f Z ∞
0
e−βt Z t
0
Z 1 0
Pt−sb (x, y)2E
u(s, y)2
dµ(y)dsdt and for (t, y)∈[0,∞)×[ε,1−ε]with Pb(x, y) : [0,∞)→R, t7→Ptb(x, y),
Z t 0
Pt−sb (x, y)2E
(u(s, y)2
ds= Pb(x, y)∗E
u(·, y)2 (t),
where ∗ denotes the time convolution. It holds Lβ(f ∗g) = Lβf · Lβg, where L denotes the Laplace transformation. Hence,
Z ∞ 0
e−βt Z t
0
Z 1 0
Pt−sb (x, y)2E
u(s, y)2
dµ(y)dsdt
= Z 1
0
Z ∞ 0
e−βt Z t
0
Pt−sb (x, y)2E
u(s, y)2
dsdtdµ(y)
= Z 1
0
Z ∞ 0
e−βtPtb(x, y)2dt Z ∞
0
e−βsE
u(s, y)2
dsdµ(y).
LetMβ(x) :=R∞
0 e−βsE[u(s, x)2]ds. Then, Mβ(x)≥Kβ+L2f
Z 1 0
Z ∞ 0
e−βtPtb(x, y)2Mβ(y)dtdµ(y). (86)
Ifb =N, we set ε:= 0 and have for all (t, x)∈[0,∞)×[0,1]
PtN(x,·)
2
µ=t2+X
k≥2
sin2 p
λNkt
λNk ϕNk2
(x)
≥t2 and thus
Z ∞ 0
Z 1 0
e−βtPtN(x, y)2Kβdµ(y)dt =Kβ Z ∞
0
e−βt
PtN(x,·)
2 µdt
≥Kβ Z ∞
0
e−βtt2dt
= 2Kββ−3. By iterating this in (86), we obtain for all x∈[0,1]
Mβ(x)≥Kβ
∞
X
n=0
2L2fβ−3n
.
This sum diverges if and only ifβ ≤q3
2L2f, which verifies (85).
Now, let b = D, ε > 0 and c0 := infx∈[ε,1−ε]ϕD1(x)2. As ϕD1 (x) 6= 0, x ∈ (0,1) (see [29, Proposition 2.5]), we have c0 >0. Then,
Z ∞ 0
Z 1 0
e−βtPtD(x, y)2Kβdµ(y)dt≥Kβ Z ∞
0
e−βt
∞
X
k=1
sin2p λDkt λDk ϕDk2
(x)dt
≥Kβ Z ∞
0
e−βt
sin2p λD1t λD1 ϕD12
(x)dt
≥Kβ Z ∞
0
e−βt
sin2p λD1t λD1 c0dt
= Kβc0 (λD1)32
Z ∞ 0
e
−√β
λD1
t
sin2(t)dt
= Kβc0 (λD1)32
2
√β λD1
3
+ 4
√β λD1
>0.
By iterating this in (86) we obtain
Mβ(x)≥Kβ,ε
∞
X
n=0
2c0 λD1−3
2 L2f
√β λD1
3
+ 4
√β λD1
n
.
Let β¯ := √β
λD1 . The above sum is equal to ∞ for all β such that β¯3 + 4 ¯β ≤ 2c0 λD1−32
L2f. This verifies (85).
We directly obtain the main result of this section.
Corollary 5.19: Let Lf >0 and let u be given by (84).
(i) Let b=N and infx∈[0,1]u0(x)>0. Then, u is weakly intermittent on [0,1].
(ii) Let b = D, ε > 0, infx∈[ε,1−ε]u0(x) > 0. Then, u is weakly intermittent on [ε,1−ε].
A Some Technical Details
Letµbe an atomless Borel probability measure on [0,1] such that 0,1∈supp(µ) and letK := supp(µ)6= [0,1]. Recall that[0,1]\K =S∞
i=1(ai, bi) (see (18)), D1 =n
f : [0,1]→R: there exists f0 ∈L2 [0,1], λ1 : f(x) =f(0) +
Z x 0
f0(y)dy, x∈[0,1]o and thatF is defined to be the space of allL2([0,1], µ)-equivalence classes possessing aD1-representative.
Lemma A.1: Let f ∈ F. Then, there exists a unique continuous representative g in the equivalence class of f such that g is for i≥1 linear on [ai, bi] and g ∈ D1. Proof. The uniqueness is clear. For the proof of existence, letf¯be aD1-representative of f. We define a function h: [0,1]→R by
h(x) :=
f¯0(x) if x∈K,
f(b¯ i)−f¯(ai)
bi−ai if x∈(ai, bi), i≥1 and g : [0,1]→Rby
g(x) := ¯f(0) + Z x
0
h(y)dy, x∈[0,1].
Fori≥1, g is obviously linear on [ai, bi]. Further, let x∈K andJx :={i≥1 :bi ≤ x}. Then, for x∈K,
Z x 0
h(y)dy = Z
K∩[0,x]
h(y)dy+X
i∈Jx
Z bi
ai
h(y)dy
= Z
K∩[0,x]
f¯0(y)dy+X
i∈Jx
f¯(bi)−f¯(ai)
= Z
K∩[0,x]
f¯0(y)dy+X
i∈Jx
Z bi
ai
f¯0(y)dy
= Z x
0
f¯0(y)dy.
Forx∈K, it follows that g(x) = ¯f(x). It remains to show that h∈L2([0,1], λ1).
By the Cauchy-Schwarz inequality, we have fora, b∈[0,1]
|f(b)¯ −f¯(a)|2 ≤ |b−a|
Z b a
f¯0)(x)2
dx and thus, fori≥1,
Z bi
ai
h2(x)dx= ( ¯f(bi)−f¯(ai))2 (bi−ai) ≤
Z bi
ai
f¯0(x)2
dx.
We conclude
Z 1 0
h2(x)dx=X
i∈N
Z bi
ai
h2(x)dx+ Z
K
h2(x)dx
≤X
i∈N
Z bi
ai
f¯0(x)2
dx+ Z
K
f¯0(x)2
dx
= Z 1
0
f¯0(x)2
dx, which is finite due to f¯∈ D1.
Lemma A.2: Let b ∈ {N, D}, ψ :L2([0,1], µ)→L2(K, µ), u7→ u|K and
∆ebµ(u) := ψ∆bµψ−1u, D
∆ebµ
:=ψ D ∆bµ . Then,
(i)
∆ebµ,D
∆ebµ
is self-adjoint and dissipative. In particular, ∆ebµ is the gen-erator of a unique strongly continuous semigroup
Tetb
t≥0. Further, u is an eigenfunction of ∆ebµfor the eigenvalueλif and only ifψ−1uis an eigenfunction of ∆bµ for the eigenvalue λ.
(ii) Ee(u,e ev) := E(ψ−1eu, ψ−1ev), u,e ev ∈ Fe := ψ(F) defines a Dirichlet form, which is associated to ∆eNµ and Ee(eu,ev),eu,ve∈Ff0 := ψ(F0) defines a Dirichlet form associated to ∆eDµ.
Proof. (i) First, we show that∆ebµ is self-adjoint. We denote the inner product on L2(K, µ) also by h·,·iµ. Let eu∈ D
∆ebµ
and u := ψ−1u.e D ∆bµ
is dense in L2([0,1], µ). Therefore, for any u ∈ L2([0,1], µ), there is a sequence (un)n∈N
with un ∈ D ∆bµ
, n ∈ N such that kun−ukµ → 0 for n → ∞. Because of
kun−ukµ =kψun−eukµ for all n ∈ N and ψun ∈ D
∆ebµ , D
∆ebµ
is dense in L2(K, µ).
Now, let u,e ev ∈ D
∆ebµ
and u := ψ−1eu, v := ψ−1ve. It is straight forward to check that v 7→
u,∆bµv
µ is a linear continuous mapping on D ∆bµ if and only if ev 7→ D
u,e ∆ebµevE
µ is linear and continuous on D
∆ebµ
, which yields D
∆ebµ
=D
∆ebµ∗
. Further, for all eu,ev ∈ D
∆ebµ D
∆ebµu,e evE
µ
=
ψ∆bµψ−1ψu, ψv
µ
=
ψ∆bµu, ψv
µ
=
∆bµu, v
µ
=
u,∆bµv
µ
=
ψu, ψ∆bµψ−1ψv
µ
=D
u,e ∆ebµevE
µ
.
The self-adjointness of ∆bµ follows. The dissipativity of ∆ebµ implies the dissi-pativity of ∆bµ since
D
∆ebµu,e euE
µ=
∆bµu, u
µ≤0.
The self-adjointness along with the dissipativity implies that ∆ebµ generates a strongly continuous semigroup
Tetb
t≥0 (see [48, Theorem B.2.2]). It remains to verify the statement about eigenvalues and eigenfunctions of ∆ebµ. For that, letλ <0,u∈ D ∆bµ
. The bijectivity of ψ implies that ∆bµ−λ
u= 0 if and only if ψ ∆bµ−λ
u = 0. The results about eigenvalues and eigenfunctions follow.
(ii) Again, let eu,ev ∈ D
∆eNµ
and u = ψ−1eu, v = ψ−1ev. The density of Fe in L2(K, µ) can be checked exactly like the density of D
∆eNµ
in L2([0,1], µ).
Further, it is obvious that Ee defines a positive definite, symmetric bilinear form. We verify that, with α >0 and Eeα(eu,ev) :=Ee(eu,ev) +αheu,eviµ,
F ,e Eeα is a Hilbert space. Note that Eeα(eu,ev) = Eα(u, v), which implies that Eeα defines an inner product. Now, let(uen)n∈
Nbe a Cauchy sequence inFe. Then, un = ψ−1eun, n ∈ N is a Cauchy sequence in F with limit, say u. Since
kuen−ψukµ = kun−ukµ for all n, ψu is the limit of (uen)n∈
N in F. For thee Markov property, we calculate
Ee(0∨ue∧1) =E(0∨u∧1)≤ E(u) = Ee(eu).
To verify that ∆eNµ is associated to Ee, we apply the correspondence between
∆Nµ and E to get
−D
∆eNµeu,evE
µ=−
∆Nµu, v
µ=E(u, v) = Ee(eu,ev). The case b=D works similarly.
B Directions for Further Research
Remark B.1: Consider the heat equation (9) with initial value given by the Delta distributionδy :g 7→g(y) for y∈supp(µ). Then, the heat kernel
pbt(x, y) =X
k≥1
e−λbktϕbk(x)ϕbk(y), (t, x)∈[0,∞)×[0,1]
solves the equation in the distributional sense. The heat kernel is of particular importance in the context of the associated Markov process (compare the remark below) and stochastic partial differential equations (compare Section 4). It is an open question whether weak measure convergence implies convergence of the corre-sponding heat kernels in an appropriate sense.
Remark B.2: The operator ∆bµ on L2([0,1], µ) is the infinitesimal generator of a Markov process, called a quasi-diffusion (compare e.g. [47,53–55]). Convergence of semigroups raises the question whether the associated Markov processes also converge weakly. Ifµn* µ, our results imply that for eachf ∈(C[0,1])bµ,t∈[0,∞) and each starting pointx∈[0,1]
E
f Xnb(t)
=Tt,nb f(x)→Ttbf(x) = E
f Xb(t)
, n→ ∞,
whereXb is associated to∆bµandXnb is associated to∆bµn. A direct argument extends this to all continuous functions on [0,1]. Then, a modification of the corresponding
proof in [14] gives convergence of all finite-dimensional distributions. The tightness would be required to establish that Xnb → Xb weakly in the Skorokhod space of càdlàg functions.
Remark B.3: Let µbe of full support. Consider the wave equation
∂2u
∂t2(t) = ∆bµu(t), t∈[0,∞) (87) onL2([0,1], µ). This hyperbolic equation describes the motion of a vibrating string with mass distribution µ such that, if it is deflected, a tension force drives it back towards its state of equilibrium. Ifµwere not of full support, the string would have massless parts. It is not clear how to interpret massless parts of a string. We suppose that the motion of such a string behaves approximately like the motion of a string with very little mass on these gaps, analogous to our results about the diffusion of heat.
Assume thatu(0) ∈ D ∆bµ
and, for reasons of simplicity, that the initial velocity vanishes. Then, there exists a unique solution to (87) in L2([0,1], µ)given byu(t) = Ctbu(0), t ≥ 0, where Ctb
t≥0 denotes the strongly continuous cosine family of ∆bµ (compare Section5.1). We have already shown thatµn* µimplies strong resolvent convergence of the corresponding operators restricted to continuous functions. It is well-known that this implies convergence of the corresponding cosine families
Ct,nb
t≥0, which implies convergence of the solutions to the corresponding wave equations, provided that there exist M > 0 and w ≥ 0 such that for all n ≥ 1, t≥ 0
Ct,nb
≤M ewt (see [39]). Proving that the restriction of Ctb to (C[0,1])bµ for t ≥ 0 is the cosine family of ∆¯bµ (and analogously for µn) and verifying the above estimate would be a way to establish the desired convergence of solutions to the wave equations.
Remark B.4: We have shown that under some regularity conditions, the mild solution to a stochastic heat equation given by (68) satisfies the upper moment bound
(E[|u(t, x)|p])1p ≤
2ku0kq+ 1
eC4p
1 1−γδt
.
for a constant C4 > 0. We conjecture that there are constants C13, C14 > 0 such that for all (t, x)∈[0,∞)×[0,1], p≥1 we have
(E[|u(t, x)|p])1p ≥C13eC14p
1 1−γδt,
where we probably have to assume further regularity conditions, asf(t, x) = 0, g(t, x) = x, u0(x) = 1,(t, x)∈[0,∞)×[0,1], which leads to the so-called parabolic Anderson model. Comparable results are known for the parabolic Anderson model defined by the standard Laplacian (compare e.g. [3, Theorem 2.6]). The proof relies on the fact that the moments of the mild solution can be expressed in terms of the local times of Brownian motion. We suppose that a generalization of this concept to Cantor-like sets would lead to the desired lower moment bound.
Treating the same problem for the stochastic wave equation is probably even more difficult as there is no such lower bound known for the stochastic wave equation defined by the standard one-dimensional Laplacian, according to the knowledge of the author.
Remark B.5: The investigation of stochastic heat and wave equations raises the question of further stochastic PDEs defined by ∆bµ, such as the stochastic Burgers equation
∂
∂tu(t, x) = ∆bµu(t, x) +f(t, u(t, x))−u(t, x) ∂
∂xu(t, x) +ξ(t, x). (88) for(t, x)∈[0, T]×[0,1]. It is known (see e.g. [4]) that, assuming sufficient regularity and appropriate initial data, the mild solution to (88) forµ=λ1 possesses a version that is essentially 12-Hölder continuous in space and essentially 14-Hölder continuous in time, which coincides with our results concerning the regularity of the stochastic heat equation. This suggests the assumption that one could establish the same Hölder exponents for the mild solution to (88) as for the stochastic heat equation defined by∆bµ. However, as the proof in [4] makes use of stochastic calculus, it seems that our results are not trivially generalizable to this equation.
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