$\displaystyle\hat{\zeta}(\rho)=\frac{c_{1}}{\rho(1-c_{2}\beta^{2}\hat{\mu}(\rho))}\,.$
(3.69)

In order to ensure a proper definition for the right hand-side of (3.69), we
just want to make sure that $c_{2}\beta^{2}\hat{\mu}(\rho)<1$. This is
achieved by monotonicity arguments whenever $\rho>\rho_{c}$, where $\rho_{c}$
is the unique positive number such that

$\displaystyle\hat{\mu}(\rho_{c})=\frac{1}{c_{2}\beta^{2}}.$ (3.70)

In addition, due to the fact that $\hat{\mu}$ is a differentiable function on
$\\{\rho>0\\}$, there exists a constant $c>0$ such that

$\displaystyle\lim_{\rho\downarrow\rho_{c}}(\rho-\rho_{c})\hat{\zeta}(\rho)=c.$
(3.71)

As explained e.g. in [26], the Wiener-Ikahara theorem for Laplace transforms
with poles states that (3.70) implies

$\displaystyle\liminf_{t\to+\infty}\frac{\ln\zeta(t)}{t}=\rho_{c}.$ (3.72)

Step 3: Estimating $\rho_{c}$. In order to estimate $\rho_{c}$ in (3.72) we go
back to its definition (3.70). Then we estimate $\hat{\mu}(\rho)$ as given in
(3.68) in the following two regimes.

_(1) Case of a large $\rho$._ In this situation we use Laplace’s method.
Namely denote $f_{\rho}(t):=\rho t+{c_{3}}t^{-\frac{1}{d_{w}-1}}$. Clearly it
has unique global minimum at a point $t^{*}$ of the form
$t^{*}=(\frac{c_{3}}{(d_{w}-1)\rho})^{(d_{w}-1)/d_{w}}$. Moreover we have

$\displaystyle
f_{\rho}(t^{*})=C_{1}\rho^{\frac{1}{d_{w}}},\quad\text{and}\quad
f_{\rho}^{\prime\prime}(t^{*})=C_{2}\rho^{2-\frac{1}{d_{w}}}\,,$ (3.73)

where $C_{1},C_{2}$ are positive constants that depend on $c_{3}$ and $d_{w}$.
By Laplace’s method we know that
$\hat{\mu}(\rho)=\int_{0}^{\infty}e^{-f_{\rho}(t)}dt$ is comparable to
$e^{-f_{\rho}(t^{*})}\sqrt{\frac{2\pi}{|f_{\rho}^{\prime\prime}(t^{*})|}}.$
Plugging (3.73) into the above expression we find that for $\rho$ away from 0
we have some constants $c_{4},c_{5},c_{6},c_{7}>0$ such that

$\frac{c_{4}}{\rho^{1-\frac{1}{2d_{w}}}}e^{-c_{5}\rho^{1/d_{w}}}\leq\hat{\mu}(\rho)\leq\frac{c_{6}}{\rho^{1-\frac{1}{2d_{w}}}}e^{-c_{7}\rho^{1/d_{w}}}.$

_(2) Case $\rho$ close to $0$._ For $\rho$ close to $0$, note that

$\displaystyle\hat{\mu}(\rho)\geq\int_{1}^{\infty}e^{-\rho
t}e^{-c_{3}t^{-\frac{1}{d_{w}-1}}}dt\geq
e^{-c_{3}}\frac{1}{\rho}e^{-\rho},\quad\text{and}\quad\hat{\mu}(\rho)\leq\int_{0}^{\infty}e^{-\rho
t}dt=\frac{1}{\rho}.$ (3.74)

Therefore as $\rho\to 0$, we end up with $\hat{\mu}(\rho)\simeq C\rho^{-1}$.

Summarizing the estimates for small and large $\rho$, we have thus obtained
that there exist some positive constants $C_{1},C_{2},C_{3},C_{4}$ such that
for all $\rho>0$ we have

$\frac{C_{1}}{\rho}e^{-C_{2}\rho^{1/d_{w}}}\leq\hat{\mu}(\rho)\leq\frac{C_{3}}{\rho}e^{-C_{4}\rho^{1/d_{w}}}.$

Reporting this bound in (3.70) yields the existence of a constant $C_{5}$ such
that

$\beta^{2}\leq
C_{5}\rho_{c}e^{C_{2}\rho_{c}^{{1}/{d_{w}}}}\,\quad\Longleftrightarrow\quad\rho_{c}\geq\Phi(\beta^{2})\,,$

where $\Phi$ denotes the inverse function of $x\to
C_{5}xe^{C_{2}x^{1/d_{w}}}$. Inserting this inequality in (3.72) we have

$\displaystyle\liminf_{t\to+\infty}\frac{\ln\zeta(t)}{t}=\rho_{c}\geq\Phi(\beta^{2}).$
(3.75)

Step 4: Conclusion. We now conclude in the following way: recall that
$\psi(t)\geq\zeta(t)$ and we have seen that $\zeta(t)$ verifies (3.75).
Therefore $\ln(\psi(t))/t$ verifies (3.75) as well. In addition, owing to
(3.66), we have

$\liminf_{t\to\infty}\frac{\phi(t)}{t}=\liminf_{t\to\infty}\frac{\psi(t)}{t}-2\lambda_{1}\geq\Phi(\beta^{2})-2\lambda_{1}.$

Recalling that $\phi(t)=\inf_{y\in B(x,r/2)}\mathbf{E}(u(t,y)^{2})$, this
proves our claim (3.62). ∎

When the boundary of a domain $A$ satisfies a cone-type condition, one can
substantially improve the lower bound (3.62). This is summarized in the
theorem below.

######  Theorem 3.18.

Under the same conditions as for Theorem 3.17 , let $A$ be an open subset of
$U$ such that $\overline{A}\subset U$ and such that $A$ satisfies the
hypothesis (2.18). We consider an initial condition $u_{0}\in L^{\infty}(U)$
such that $u_{0}$ is non negative and $u_{0}\geq\varepsilon>0$ on $A$. Then
the solution $u$ to (3.31) is such that for all $p\geq 2$ and for all $x\in A$
we have

$\displaystyle\liminf_{t\to+\infty}\frac{\ln\mathbf{E}(u(t,x)^{2})}{t}\geq
c\beta^{\frac{2d_{w}}{d_{w}-d_{h}}}-2\lambda_{1},$ (3.76)

where $c>0$ does not depend on $\beta$.

###### Proof.

The expression (3.63) for the second moment of $u(t,y)$ is still valid.
Furthermore, we have $u_{0}(z)\geq\varepsilon 1_{A}(z)$. Hence we get

$\displaystyle\mathbf{E}\left(u(t,y)^{2}\right)\geq\varepsilon^{2}\left(P_{t}^{D}1_{A}(y)\right)^{2}+\beta^{2}\int_{0}^{t}\int_{A}p_{t-s}^{D}(y,z)^{2}\mathbf{E}\left(u(s,z)^{2}\right)d\mu(z)ds.$

Similarly to what we did in the proof of Theorem 3.17, set

$\phi(t)=e^{2\lambda_{1}t}\inf_{x\in A}\mathbf{E}\left(u(t,x)^{2}\right).$

Along the same lines as for (3.64)-(3.65)-(3.66) and resorting to relation
(2.20), which is valid whenever (2.18) is fulfilled, we get that
$\phi(t)\geq\zeta(t)$ where $\zeta$ solves the renewal equation

$\displaystyle\zeta(t)=c_{1}+c_{2}\beta^{2}\int_{0}^{t}\frac{1}{(t-s)^{d_{h}/d_{w}}}\zeta(s)ds.$
(3.77)

Next we still follow closely the argument of Theorem 3.17, the main difference
being that now we can compute explicitly the Laplace transform of $t\mapsto
t^{-d_{h}/d_{w}}$. Namely applying Laplace transform on both sides of (3.77)
we get

$\hat{\zeta}(\rho)=\frac{c_{1}}{\rho}+c_{2}\beta^{2}\hat{\mu}(\rho)\hat{\zeta}(\rho),\quad\mathrm{where}\
\ \hat{\mu}(\rho)=\int_{0}^{+\infty}\frac{e^{-\rho
t}}{t^{d_{h}/d_{w}}}dt=c_{3}\rho^{-1+d_{h}/d_{w}}.$

This equation can be solved algebraically and we get

$\hat{\zeta}(\rho)=\frac{c_{1}}{\rho^{d_{h}/d_{w}}\left(\rho^{1-d_{h}/d_{w}}-\beta^{2}c_{2}c_{3}\right)}.$

In particular, the positive pole for $\hat{\zeta}$ is of the form

$\rho_{c}=c_{4}\beta^{\frac{2d_{w}}{d_{w}-d_{h}}}.$

The same steps as for Theorem 3.17 yield our claim (3.76). ∎

###  3.7  Some properties of the solutions, Intermittency

As mentioned in the introduction, when considering a Dirichlet type boundary
condition in an inner uniform domain, the parabolic Anderson model exhibits an
interesting phase transition for the behavior of moments. This is spelled out
below.

######  Corollary 3.19.

Let $U$ be an inner uniform domain and $u_{0}\in L^{\infty}(U)$ a non negative
initial condition. We consider a noise $W_{\alpha}$ with $\alpha\geq 0$ and
the corresponding solution $u$ to the equation (3.31). Let also $p\geq 2$ and
$x\in U$. Then there exists $\beta_{c,1}>0$ such that for all
$\beta\leq\beta_{c,1}$, we have

$\displaystyle\lim_{t\to+\infty}\mathbf{E}(u(t,x)^{p})=0.$ (3.78)

If we assume moreover that $\alpha>0$, and that $u_{0}$ satisfies
$u_{0}>\varepsilon>0$ in a neighborhood of $x$, then there exists
$\beta_{c,2}>0$ such that for every $\beta\geq\beta_{c,2}$

$\displaystyle\lim_{t\to+\infty}\mathbf{E}(u(t,x)^{p})=+\infty.$ (3.79)

###### Proof.

The first assertion (3.78) is derived from the upper bound (3.38) in Theorem
3.11 for $p\geq 2$ (see (3.32) for the particular case $p=2$). Specifically,
identity (3.78) is satisfied for a fixed $p$ as long as the right hand side

$h_{\alpha,\beta}:=\frac{p}{2}\Theta_{\alpha}(\beta\sqrt{p-1})-p\lambda_{1}$

verifies $h_{\alpha,\beta}<0$ for $\beta<\beta_{c,1}$. Now it is trivially
checked from Theorem 3.10 that for all $\alpha\geq 0$

$\lim_{\beta\to 0}\Theta_{\alpha}(\beta\sqrt{p-1})=0.$

We thus get our desired conclusion $h_{\alpha,\beta}<0$ for $\beta$ small
enough. This ends the proof of (3.78).

We now turn to the claim (3.79). Since we assume that $u_{0}>\varepsilon$ in a
neighborhood of $x$ and $U$ is an open set, one can work under the conditions
of Theorem 3.15 with $A=B(x,r)$ for $r$ small enough. Also recall that we are
considering $\alpha>0$. Hence the lower bound (3.51) holds true. Since the
right hand side

$\displaystyle G(\beta):=C\beta^{2}p(p-1)-p\lambda_{1}(A)$ (3.80)

of (3.51) verifies $\lim_{\beta\to\infty}G(\beta)=\infty$, we clearly have
$G(\beta)>0$ for $\beta>\beta_{c,2}$ and a large enough $\beta_{c,2}$. This
yields relation (3.79). ∎

We now get closer to the intermittency phenomenon, by proving that large
moments of $u(t,x)$ are divergent as $t\to\infty$.

######  Corollary 3.20.

Let the conditions of Corollary 3.19 prevail. Consider $\alpha>0$, $\beta>0$
and $x\in U$ such that $u_{0}>\varepsilon>0$ in a neighborhood of $x$. Then
there exists $p_{0}$ such that for all $p\geq p_{0}$ we have

$\lim_{t\to+\infty}\mathbf{E}(u(t,x)^{p})=+\infty.$

###### Proof.

The proof is very similar to the proof of (3.79) above. It is based on Theorem
3.15 and the asymptotic behavior of the function $G$ in (3.80), now seen as a
function of $p$. Details are left to the reader. ∎

One way to quantify intermittency for the parabolic Anderson model is to show
that in large time the $L^{p}(\Omega)$-norm of $u(t,x)$ is much larger than
the $L^{q}(\Omega)$-norm whenever $p>q$. We are now able to prove this for
inner uniform domains.

######  Corollary 3.21.

We still work under the conditions of Corollary 3.19, with $\alpha,\beta>0$,
an inner uniform domain $U$, and $x\in U$ such that $u_{0}>\varepsilon$ in a
neighborhood of $x$. Let $q\geq 2$. Then, there exists $p_{0}>q$ such that for
$p\geq p_{0}$,

$\lim_{t\to+\infty}\frac{\mathbf{E}(u(t,x)^{p})^{1/p}}{\mathbf{E}(u(t,x)^{q})^{1/q}}=+\infty.$
(3.81)

###### Proof.

Denote by $Q_{p,q}(t,x)$ the ratio
$\|u(t,x)\|_{L^{p}(\Omega)}/\|u(t,x)\|_{L^{q}(\Omega)}$. Then

$\frac{1}{t}\ln(Q_{p,q}(t,x))=\frac{1}{t}\left(\frac{1}{p}\ln(\mathbf{E}[u(t,x)^{p}])-\frac{1}{q}\ln(\mathbf{E}[u(t,x)^{q}])\right).$

According to the lower bound in Theorem 3.15 (for a domain $A$ of the form
$B(x,r)$ like in Corollary 3.21) and the upper bound in Theorem 3.11, we
obtain

$\liminf_{t\to\infty}\frac{1}{t}\ln(Q_{p,q}(t,x))\geq
C\beta^{2}(p-1)-\lambda_{1}(A)-\left(\frac{1}{2}\Theta_{\alpha}(\beta\sqrt{q-1})-\lambda_{1}\right).$
(3.82)

As a function of $p$, the dominant term in (3.82) is $C\beta^{2}p$. Therefore
if $p$ is large enough we get

$\liminf_{t\to\infty}\frac{1}{t}\ln(Q_{p,q}(t,x))>0.$

This is enough to prove (3.81). ∎

###  3.8 Scaling invariance in the Sierpiński gasket

Up to now we have stated results and bounds which are valid for a wide range
of underlying spaces $X$. When particularizing our study to sets with
convenient invariances, the stochastic heat equation also exhibits nice
scaling properties. In this section we examine the case of the Sierpiński
gasket with zero boundary condition at the vertices.

Let us thus go back to the setting of Section 2.4.3, with the functions
$\mathfrak{f}_{i}$ given by (2.67) and the Sierpiński gasket $K$ defined by
(2.68). The inner uniform domain we consider here is $U=K\setminus V_{0}$. For
a word $w=(i_{1},\dots,i_{n})$ in $\\{1,2,3\\}^{n}$ we define

$\mathfrak{f}_{w}=\mathfrak{f}_{i_{1}}\circ\cdots\circ\mathfrak{f}_{i_{n}},\quad\text{and}\quad
U_{w}:=\mathfrak{f}_{w}(K)\setminus\mathfrak{f}_{w}(V_{0})=\mathfrak{f}_{w}(U).$
(3.83)

Since uniform curves in $U$ are transformed by $\mathfrak{f}_{w}$ into uniform
curves in $U_{w}$, it follows that every $U_{w}$ is a uniform and hence also a
inner uniform domain. With this additional notation in mind, we can now state
our invariance result.

######  Theorem 3.22.

Let $\alpha,\beta$ be two positive parameters. On the inner uniform domain
$U=K\setminus V_{0}$ we define the Dirichlet Laplace operator $\Delta_{D,U}$
as in Section 2.2.1 and the noise $W^{D,U}_{\alpha}$ through Definition 3.2.
We denote by $u$ the unique solution of

$\partial_{t}u(t,x)=\Delta_{D,U}u(t,x)+\beta
u(t,x)\,\dot{W}_{\alpha}^{D,U}(t,x),\quad\text{for}\quad t\geq 0,\,x\in U,$
(3.84)

with initial condition $u(0,\cdot)=1$. For a word
$w=(i_{1},\dots,i_{n})\in\\{1,2,3\\}^{n}$ and $U_{w}$ defined by (3.83), we
similarly introduce $\Delta_{D,U_{w}}$ and $W^{D,U_{w}}_{\alpha}$. Furthermore
we define $v$ as the unique solution to

$\partial_{t}v(t,x)=\Delta_{D,U_{w}}v(t,x)+\beta\left(\frac{5^{\alpha+1/2}}{3}\right)^{n}v(t,x)\,\dot{W}_{\alpha}^{D,U_{w}}(t,x),\quad\text{for}\quad
t\geq 0,\,x\in U,$ (3.85)

with initial condition $v(0,\cdot)=1$. Then the following identity in the
sense of finite dimensional distributions holds true:

$\displaystyle(v(t,x))_{t\geq 0,x\in
U_{w}}\stackrel{{\scriptstyle\mathrm{(fdd)}}}{{=}}(u(5^{n}t,\mathfrak{f}^{-1}_{w}(x)))_{t\geq
0,x\in U_{w}}.$ (3.86)

###### Proof.

We will prove the identity for one single value of $(t,x)$. The corresponding
property for finite dimensional distributions is done in the same way and left
to the reader. We have seen in Theorem 2.39 that $\Delta_{D,U}$ could be
obtained as the extension of the Kigami Laplacian defined through the limiting
procedure (2.73)-(2.74). A similar construction can be performed for
$\Delta_{D,U_{w}}$. Hence it is easily seen from (2.73) that for a generic
Schwartz function $g$ (see Definition 2.18 for the notion of Schwartz
function) we have

$\displaystyle(\Delta_{D,U_{w}}g)\circ\mathfrak{f}_{w}=5^{n}\Delta_{D,U}(g\circ\mathfrak{f}_{w}).$
(3.87)

Moreover, our representation (2.26) implies the corresponding relations for
heat semigroups and heat kernels, valid for $t\geq 0$ and $x,y\in U$:

$\displaystyle(P_{t}^{D,U_{w}}g)\circ\mathfrak{f}_{w}=P_{5^{n}t}^{D,U}(g\circ\mathfrak{f}_{w}),\quad\text{and}\quad
p_{t}^{D,U_{w}}(\mathfrak{f}_{w}(x),\mathfrak{f}_{w}(y))=3^{n}p^{D,U}_{5^{n}t}(x,y).$
(3.88)

In the same way, one can invoke either (2.28) or (2.29) to see that

$\displaystyle((-\Delta_{D,U_{w}})^{-\alpha}g)\circ\mathfrak{f}_{w}=5^{-n\alpha}(-\Delta_{D,U})^{-\alpha}(g\circ\mathfrak{f}_{w}).$
(3.89)

Let us now specify the invariance for the noise $W_{\alpha}$. To begin with,
from (3.89) one deduces that for $h\in L^{2}(U,\mu)$ and $\alpha\geq 0$

$\displaystyle\int_{U_{w}}\left[(-\Delta_{D,U_{w}})^{-\alpha}(h\circ\mathfrak{f}^{-1}_{w})\right]^{2}d\mu$
$\displaystyle=5^{-2n\alpha}\int_{U_{w}}\left[((-\Delta_{D,U})^{-\alpha}h)\circ\mathfrak{f}^{-1}_{w}\right]^{2}d\mu.$

Next we apply the scaling invariance of the Hausdorff measure on $K$, which
stipulates that $\int_{U_{w}}f\circ f^{-1}_{w}d\mu=3^{-n}\int_{U}fd\mu$. We
end up with

$\displaystyle\int_{U_{w}}\left[(-\Delta_{D,U_{w}})^{-\alpha}(h\circ\mathfrak{f}^{-1}_{w})\right]^{2}d\mu=3^{-n}5^{-2n\alpha}\int_{U}((-\Delta_{D,U})^{-\alpha}h)^{2}d\mu,$
(3.90)

and note that one can recast (3.90) as

$\displaystyle\|h\circ\mathfrak{f}^{-1}_{w}\|^{2}_{\mathcal{W}_{D}^{-\alpha}(U_{w})}=3^{-n}5^{-2n\alpha}\|h\|^{2}_{\mathcal{W}_{D}^{-\alpha}(U)}.$
(3.91)

Taking into account the covariance function (2.29) of $W_{\alpha}^{D,U}$ and
$W_{\alpha}^{D,U_{w}}$, we get the following equality in distribution for
processes:

$\\{W_{\alpha}^{D,U_{w}}(g\circ\mathfrak{f}^{-1}_{w});\,g\in
W_{D}^{-\alpha}(U)\\}\stackrel{{\scriptstyle(d)}}{{=}}\\{3^{-n/2}5^{-n\alpha}W_{\alpha}^{D,U}(g);\,g\in\mathcal{W}_{D}^{-\alpha}(U)\\}.$
(3.92)

We are ready to translate the previous observations into invariances for the
stochastic heat equation. That is start from the mild formulation of (3.84):

$u(t,x)=(P_{t}^{D,U}1)(x)+\beta\int_{0}^{t}\int_{U}p^{D,U}_{t-s}(x,y)u(s,y)W^{D,U}_{\alpha}(ds,dy).$

Composing the above relation with $\mathfrak{f}^{-1}_{w}$ and applying a
Brownian scaling in time we get

$\displaystyle u(5^{n}t,\mathfrak{f}_{w}^{-1}(x))$
$\displaystyle=(P_{t}^{D,U_{w}}1)(x)+\beta\int_{0}^{5^{n}t}\int_{U}p^{D,U}_{5^{n}t-s}(\mathfrak{f}_{w}^{-1}(x),y)u(s,y)W^{D,U}_{\alpha}(ds,dy)$
$\displaystyle=(P_{t}^{D,U_{w}}1)(x)+\beta
5^{n/2}\int_{0}^{t}\int_{U}p^{D,U}_{5^{n}t-5^{n}s}(\mathfrak{f}_{w}^{-1}(x),y)u(5^{n}s,y)\tilde{W}^{D,U}_{\alpha}(ds,dy),$

where $\tilde{W}^{D,U}$ is a field such that $\tilde{W}^{D,U}=W^{D,U}$
(specifically $\tilde{W}^{D,U}(t,\cdot)$ is obtained as
$5^{n/2}W^{D,U}(t/5^{n},\cdot)$). We now resort to (3.88) in order to write

$\displaystyle u(5^{n}t,\mathfrak{f}_{w}^{-1}(x))$
$\displaystyle=(P_{t}^{D,U_{w}}1)(x)+\beta
5^{n/2}3^{-n}\int_{0}^{t}\int_{U}p^{D,U_{w}}_{t-s}(x,\mathfrak{f}_{w}(y))u(5^{n}s,y)\tilde{W}^{D,U}_{\alpha}(ds,dy).$

Eventually, thanks to (3.92) we get the existence of a field
$\tilde{W}^{D,U_{w}}$ such that
$\tilde{W}^{D,U_{w}}\stackrel{{\scriptstyle(d)}}{{=}}{W}^{D,U_{w}}$ and

$\displaystyle u(5^{n}t,\mathfrak{f}_{w}^{-1}(x))=$
$\displaystyle\,(P_{t}^{D,U_{w}}1)(x)$
$\displaystyle+\beta\left(\frac{5^{\alpha+1/2}}{3}\right)^{n}\int_{0}^{t}\int_{U_{w}}p^{D,U_{w}}_{t-s}(x,y)u(5^{n}s,\mathfrak{f}_{w}^{-1}(y))\tilde{W}^{D,U_{w}}_{\alpha}(ds,dy).$

Otherwise stated, $u(5^{n}t,\mathfrak{f}_{w}^{-1}(x))$ solves equation (3.85).
This shows identity (3.86) and finishes our proof. ∎

##  4 Neumann parabolic Anderson model in uniform domains

In this section we investigate the behavior of a parabolic Anderson model
subject to Neumann boundary conditions. Since many of the computations are
similar to the Dirichlet case, we sill just sketch the main differences. To
begin with, let us label the hypothesis on the domain $U$ considered below.

######  Assumption 4.1.

The domain $U$ is a uniform domain as specified in Definition 2.24. On this
domain we consider the Neumann Laplace operator from Proposition 2.27.

We now proceed to a description of the noise in Section 4.1, and we state the
main results about the stochastic heat equation in Sections 4.2 to 4.4.

###  4.1 Fractional noise

The family of noises we consider for the Neumann case are still described as
white noises in time, with a spatial covariance based on the Neumann
Laplacian.

######  Definition 4.2.

For $\alpha>0$ the noise $W^{N}_{\alpha}$ is introduced similarly to
Definition 3.2, with a space $\mathcal{H}_{\alpha}$ given as
$\mathcal{H}_{\alpha}=L^{2}(\mathbb{R}_{+},\mathcal{W}_{N}^{-\alpha}(U))$.
Notice that the space $\mathcal{W}_{N}^{-\alpha}(U)$ is introduced in
Definition 2.35, and we suppose that the constant $C_{U}$ in (2.65) is such
that $G^{N,U}_{2\alpha}(x,y)+C_{U}$ is lower bounded by a positive constant
(see Remark 2.36). The Neumann fractional field
$\\{W_{\alpha}^{N}(\phi);\phi\in\mathcal{H}_{\alpha}\\}$ is then the centered
isonormal Gaussian family with covariance

$\displaystyle\mathbf{E}\left[W_{\alpha}^{N}(\varphi)W_{\alpha}^{N}(\psi)\right]=\int_{\mathbb{R}_{+}}\
\left\langle\varphi(t,\cdot),\psi(t,\cdot)\right\rangle_{\mathcal{W}_{N}^{-\alpha}(U)}dt,$
(4.1)

It is assumed to be defined on a complete probability space
$(\Omega,\mathcal{G},\mathbf{P})$.

######  Remark 4.3.

For $\alpha=0$, the space-time white noise $W^{N}_{0}$ is in fact defined
exactly as in Definition 3.3. Otherwise stated, $W^{N}_{0}=W^{D}_{0}$.

######  Remark 4.4.

As we did in Remark 3.5 for the Dirichlet case, let us mention that Neumann
noises on fractals have been introduced in [6]. For
$\alpha\leq{d_{h}}/{2d_{w}}$ it had been obtained that $W^{N}_{\alpha}$ is a
space-time distribution, while for $\alpha>{d_{h}}/{2d_{w}}$ a density
property similar to (3.4) holds true.

###  4.2 Existence, uniqueness and $L^{p}$ upper bounds

We are now in a position to state a result about the $L^{p}$ moment of the
parabolic Anderson model with Neumann Laplacian.

######  Theorem 4.5.

Let $\alpha,\beta\geq 0$ and let $W^{N}_{\alpha}$ be the noise from Definition
4.2. We consider the equation

$\partial_{t}u(t,x)=\Delta_{N}u(t,x)+\beta u(t,x)\,\dot{W}_{\alpha}^{N}(t,x)$
(4.2)

interpreted in the mild sense as in (3.6). The initial condition $u_{0}$ lies
in $L^{\infty}(U,\mu)$. Then the equation admits a unique solution $u$
according to Definition 3.6. Moreover, for any integer $p\geq 2$, the solution
$u$ satisfies the following $L^{p}$-upper bound estimates:

$\displaystyle\limsup_{t\to+\infty}\frac{\ln\mathbf{E}(u(t,x)^{p})}{t}\leq\frac{p}{2}\widehat{\Theta}_{\alpha}\left(\beta\sqrt{p-1}\right),$
(4.3)

where $\widehat{\Theta}_{\alpha}$ is a continuous increasing function whose
asymptotic behavior is exactly the same as $\Theta_{\alpha}$ in Theorem 3.10.

###### Proof.

We proceed using chaos expansions like in Theorem 3.10. We thus have to check
a bound like (3.33), with $p_{t}^{D}$ replaced by $p_{t}^{N}$. In order to
check (3.33) we then adopt the same strategy as in Lemma 3.9. This leads to
estimate a quantity like $M_{k}(t,x)$ in (3.19). Let us just highlight a small
difference between the definition of the current $M_{k}(t,x)$ and the one we
had in the Dirichlet case: relation (3.19) involved the positive kernel
$G_{2\alpha}^{D}$, which should be replaced by $G^{N}_{2\alpha}$. Now the
problem is that $G_{2\alpha}^{N}$ is not positive anymore. Nevertheless as
mentioned in Definition 4.2, Remark 2.36 entails the existence of a constant
$C_{U}$ such that $G_{2\alpha}+C_{U}\geq 0$. With the definition (4.1) of
inner product in $\mathcal{H}$ and similarly to (3.19) one can express
$M_{k}(t,x)$ as

$\begin{split}M_{k}(t,x)=&\int_{[0,t]^{k}_{<}}\int_{U^{2k}}g^{N}(s,y,t,x)\\\
&\times\prod_{i=1}^{k}(G_{2\alpha}^{N}(y_{i},y_{i}^{\prime})+C_{U})g^{N}(s,y^{\prime},t,x)d\mu(y)d\mu(y^{\prime})e^{-2\lambda_{1}s_{1}}ds,\end{split}$
(4.4)

where the function $g^{N}$ is defined by

$g^{N}(s,y,t,x)=p_{t-s_{k}}^{N}(x,y_{k})\cdots
p_{s_{2}-s_{1}}^{N}(y_{2},y_{1}).$

Starting from (4.4), the proof is completely similar to the proof of Lemma
3.9. We just mention that the bound for
$\|(-\Delta_{D})^{-\alpha}p_{t}^{D}\|_{L^{2}(U,\mu)}$ now has to be changed
into a bound for
$\left\|\left[(-\Delta_{N,U})^{-\alpha}+C_{U}\right]p_{t}^{N}\right\|_{L^{2}(U,\mu)}$.
However, it is easy to see from the proof of Lemma 2.33 that

$\sup_{x\in
U}\left\|\left[(-\Delta_{N,U})^{-\alpha}+C_{U}\right]p_{t}^{N,U}(x,\cdot)\right\|_{L^{2}(U,\mu)}\leq\begin{cases}C(t^{\alpha-\frac{d_{h}}{2d_{w}}}+1),\quad
0<\alpha<\frac{d_{h}}{2d_{w}}\\\
C|\ln(\max(t,1/2))|,\quad\alpha=\frac{d_{h}}{2d_{w}}\\\
C,\quad\alpha>\frac{d_{h}}{2d_{w}},\end{cases}$

The bound (4.3) is thus obtained with the same estimates as in Lemma 3.9 and
Theorem 3.10. Details are left to the patient reader. ∎

###  4.3 $L^{p}$ lower bounds

In this section we turn to an analog of the lower bounds in Section 3.6. We
start with a general lower bound in the case $\alpha>0$.

######  Theorem 4.6.

Let $\beta\geq 0$ and $\alpha>0$, and consider the noise $W^{N}_{\alpha}$
given in Definition 4.2. Let $u$ be the solution to equation (4.2), where the
initial condition $u_{0}$ is an element of $L^{\infty}(U,\mu)$ such that
$u_{0}\geq\varepsilon>0$ on $U$. Let $p$ be an integer with $p\geq 2$. Then
for $x\in U$ and $t\geq 0$ we have

$\displaystyle\mathbf{E}(u(t,x)^{p})\geq\varepsilon^{p}\exp\left(C\beta^{2}p(p-1)t\right),$
(4.5)

where $C$ is a positive constant. In particular, for $x\in U$

$\displaystyle\liminf_{t\to+\infty}\frac{\ln\mathbf{E}(u(t,x)^{p})}{t}\geq
C\beta^{2}p(p-1).$ (4.6)

###### Proof.

From the Feynman-Kac representation of the moments, which is established along
the same lines as in the Dirichlet boundary case, one has

$\displaystyle\mathbf{E}(u(t,x)^{p})=\mathbf{E}\left(\prod_{j=1}^{p}u_{0}(\hat{B}_{t}^{x,j})\exp\left\\{\beta^{2}\sum_{i\not=k}^{p}\int_{0}^{t}\left(G_{2\alpha}^{N}(\hat{B}^{x,i}_{s},\hat{B}^{x,k}_{s})+C_{U}\right)ds\right\\}\right),$

where $\hat{B}$ is the reflected Brownian motion on $U$. Otherwise stated,
$\hat{B}$ is the diffusion with generator $\Delta_{N}$. Replacing
$G_{2\alpha}^{D}$ by $G_{2\alpha}^{N}$ in the computations going from (3.54)
to (3.6.1), we easily get that

$\mathbf{E}\left(u(t,x)^{p}\right)\geq\varepsilon^{p}\exp\left(C\beta^{2}p(p-1)t\right)\left(P_{t}^{N,U}1(x)\right)^{p}.$

In addition, for Neumann boundary conditions we have $P^{N,U}_{t}1(x)=1$, from
which (4.5) is easily deduced. Relation (4.6) is a trivial consequence of
(4.5). ∎

######  Remark 4.7.

In the Dirichlet case, we had observed a phase transition for moments in
Corollary 3.19. This phase transition is absent from the Neumann boundary
picture. Indeed relation (4.5) implies that

$\lim_{t\to\infty}\mathbf{E}\left(u(t,x)^{p}\right)=\infty,\quad\mathrm{for\
all}\ \beta>0.$

This difference in the asymptotic behavior stems from the fact that
$\Delta_{D,U}$ has a strictly negative principal eigenvalue $\lambda_{1}$.

As in the Dirichlet case, one can improve the bounds of Theorem 4.6 when the
noise is rough enough. This is summarized in the theorem below.

######  Theorem 4.8.

Under the same conditions as in Theorem 4.6, let $u$ be the solution to
equation (4.2). The initial condition $u_{0}$ is bounded, non-negative on $U$
and bounded from below by a positive constant on the neighborhood of a given
$x\in U$. Then, for an integer $p\geq 2$, we have the following lower bounds.

  * •

For $0<\alpha<\frac{d_{h}}{2d_{w}}$,

$\liminf_{\beta\to+\infty}\liminf_{t\to+\infty}\frac{1}{\beta^{\frac{2d_{w}}{(1+2\alpha)d_{w}-d_{h}}}}\frac{\ln\mathbf{E}(u(t,x)^{p})}{t}\geq
Cp(p-1),$

  * •

For $\alpha=\frac{d_{h}}{2d_{w}}$,

$\liminf_{\beta\to+\infty}\liminf_{t\to+\infty}\frac{1}{\beta^{2}\ln\beta}\frac{\ln\mathbf{E}(u(t,x)^{p})}{t}\geq
Cp(p-1).$

###### Proof.

The proof is very similar to the proof of Theorem 3.16. In particular, notice
that the equivalent of (3.59) still involves the exit time
$T^{1}_{\varepsilon}$. We thus end up with (3.60) as before. From there, the
computations proceed as in Theorem 3.16. ∎

We now label the result for the space-time white noise case. Its proof goes
along the same lines as for Theorem 3.18.

######  Theorem 4.9.

Let the conditions of Theorem 4.6 and 4.8 prevail. We assume that we are in
the white noise case $\alpha=0$. Moreover, the initial condition $u_{0}$ is
assumed to be in $L^{\infty}(U,\mu)$ and lower bounded by $\varepsilon>0$.
Then the solution $u$ to (4.2) satisfies the following $L^{2}$-lower bound
estimate:

$\liminf_{t\to+\infty}\frac{\ln\mathbf{E}(u(t,x)^{2})}{t}\geq
c\beta^{\frac{2d_{w}}{d_{w}-d_{h}}},\quad\mathrm{for\ all}\ x\in U,$

where $c>0$ is a positive constant that does not depend on $\beta$.

###  4.4 Scaling invariance in the Sierpiński gasket

In this section we go back to the setting of Section 3.8 and we establish
scaling factors in the Sierpinski gasket $K$. Our findings are summarized
below.

######  Theorem 4.10.

Let the notation and assumptions of Theorem 3.22 prevail. On the uniform
domain $U=K\setminus V_{0}$, we consider the unique solution $u$ to

$\partial_{t}u(t,x)=\Delta_{N,U}u(t,x)+\beta
u(t,x)\,\dot{W}_{\alpha}^{N,U}(t,x),\quad\mathrm{for}\ t\geq 0,x\in U,$

with initial condition $u(0,\cdot)=1$. For the set $U_{w}$ defined by (3.83),
we also consider the solution $v$ to

$\partial_{t}v(t,x)=\Delta_{N,U_{w}}v(t,x)+\beta\left(\frac{5^{\alpha+1/2}}{3}\right)^{n}v(t,x)\,\dot{W}_{\alpha}^{N,U_{w}}(t,x),\quad\mathrm{for}\
t\geq 0,x\in U_{w},$ (4.7)

with initial condition $v(0,\cdot)=1$. Then the following identity in the
sense of finite dimensional distribution holds true:

$\displaystyle(v(t,x))_{t\geq 0,x\in
U_{w}}\stackrel{{\scriptstyle(f\\!dd)}}{{=}}(u(5^{n}t,\mathfrak{f}^{-1}_{w}(x)))_{t\geq
0,x\in U_{w}}.$

###### Proof.

The proof goes along the same lines as for Theorem 3.22. Details are spared
for sake of conciseness. Let us just highlight a subtle change in the
definition of the kernel $G_{2\alpha}^{N,U_{w}}$ related to (4.7). Indeed, in
the current situation we have

$\displaystyle\int_{U_{w}}(-\Delta_{N,U_{w}})^{-\alpha}(g\circ\mathfrak{f}^{-1}_{w})^{2}d\mu=3^{-n}5^{-2n\alpha}\int_{U}((-\Delta_{N,U})^{-\alpha}g)^{2}d\mu.$

Hence if one wishes (3.90) to hold in our context with the operator
$\left(-\Delta_{N,U_{w}}\right)^{-\alpha}$ replaced by
$\left(-\Delta_{N,U_{w}}\right)^{-\alpha}+C_{U_{w}}$, we need to choose
$C_{U_{w}}=3^{n}5^{-2n\alpha}C_{U}.$ ∎

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Fabrice Baudoin<EMAIL_ADDRESS>
Department of Mathematics, Aarhus University, Denmark

Li Chen: lichen<EMAIL_ADDRESS>
Department of Mathematics, Louisiana State University, USA & Department of
Mathematics, Aarhus University, Denmark

Che-Hung Huang<EMAIL_ADDRESS>
Department of Mathematics, Purdue University, USA

Cheng Ouyang<EMAIL_ADDRESS>
Department of Mathematics, Statistics, and Computer Science, University of
Illinois at Chicago, USA

Samy<EMAIL_ADDRESS>
Department of Mathematics, Purdue University, USA

Jing<EMAIL_ADDRESS>
Department of Mathematics, Purdue University, USA