# Fractional Cauchy problem on random snowflakes

Raffaela Capitanelli  Department of Basic and Applied Sciences for Engineering
Sapienza University of Rome
via A. Scarpa 10, Rome, Italy<EMAIL_ADDRESS>and  Mirko
D’Ovidio  Department of Basic and Applied Sciences for Engineering
Sapienza University of Rome
via A. Scarpa 10, Rome, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We consider time-changed Brownian motions on random Koch (pre-fractal and
fractal) domains where the time change is given by the inverse to a
subordinator. In particular, we study the fractional Cauchy problem with Robin
condition on the pre-fractal boundary obtaining asymptotic results for the
corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary
conditions on the fractal domain.

Keywords: Time changes; fractional operators; time-fractional equations;
asymptotics.

AMS-MSC: 26A33, 35R11, 60J50, 58J37 .

##  1\. Introduction

Many physical and biological phenomena take place across irregular and wild
structures in which boundaries are “large”while bulk is “small”. In this
framework, domains with fractal boundaries provide a suitable setting to model
phenomena in which the surface effects are enhanced like, for example,
pulmonary system, root infiltration, tree foliage, etc..

In this paper, we consider random Koch domains which are domains whose
boundary are constructed by mixtures of Koch curves with random scales. These
domains are obtained as limit of domains with Lipschitz boundary whereas for
the limit object, the fractal given by the random Koch domain, the boundary
has Hausdorff dimension between $1$ and $2$.

Our attention will be focused on fractional Cauchy problems on the random Koch
domains with boundary conditions.

Literature on fractional Cauchy problems is extensive both from the
probability and the analysis point of view. Here, our aim is not providing a
large list of references. We mention here only few works investigating basic
and fondamental aspects: [1], [3], [14], [17], [18], [20], [22], [28], [32].

The non-local time-operator we deal with is very general and covers a huge
class of non-local (convolution type) operators. Such operators have been
recently considered in the papers [13; 31]. From the probabilistic point of
view, we consider time-changed Brownian motions where the time change is given
by an inverse to a subordinator characterized by a symbol which is a Bernstein
function. Thus, with this time-fractional operator at hand, we study the
fractional Cauchy problem with Robin condition on the pre-fractal boundary and
we obtain asymptotic results for the corresponding fractional diffusions with
Robin, Neumann and Dirichlet boundary conditions on the fractal domain.

The asymptotic problem we deal with can be illustrated, in the simple case, by
the following parabolic Dirichlet-Robin problem on the interval $(0,a)$,
$a>0$. More precisely, we consider the heat equation

$\displaystyle\partial_{t}u_{n}=\partial_{xx}u_{n},\quad t>0,\,x\in(0,a)$
$\displaystyle u_{n}(t,0)=0,\quad t>0$ (1.1) $\displaystyle
u_{n}(0,x)=f(x),\quad x\in(0,a)$

with Robin boundary condition

$\displaystyle\partial_{x}u_{n}(t,a)+c_{n}u_{n}(t,a)=0,\quad t>0$ (1.2)

where $c_{n}>0$, $n\in\mathbb{N}$. The solution can be written as follows
$u_{n}(t,x)=\sum_{k\geq 1}e^{-t\lambda_{k}^{(n)}}\phi_{k}^{(n)}(x)\,f_{k}$
where $f_{k}^{(n)}=\int f(x)\phi_{k}^{(n)}(x)\,dx$, $k\in\mathbb{N}$. Notice
that $\phi_{k}^{(n)}(x)=\sin(x\sqrt{\lambda_{k}^{(n)}})$ and
$\sqrt{\lambda_{k}^{(n)}}=z_{k}^{(n)}$ are the eigenvalues associated to
$\phi_{k}^{(n)}$ where $z_{k}^{(n)}$ are solutions to
$\tan(az_{k}^{(n)})=-z_{k}^{(n)}/c_{n}$, $k\in\mathbb{N}$.

Our aim here is to point out the asymptotic behaviour of the solution $u_{n}$
as $n\to\infty$. We obtain three different limit problems. If $c_{n}\to 0$,
then $z_{k}^{(n)}\to z_{k}^{N}=\left(\frac{\pi}{2}+\pi k\right)\frac{1}{a}$
and therefore $u_{n}(t,x)\to u(t,x)=\sum_{k\geq
1}e^{-t(z_{k}^{N})^{2}}\sin(xz_{k}^{N})\,f_{k}$ where $f_{k}=\int
f(x)\sin(xz_{k}^{N})dx$ and $u$ is the solution to (1.1) with Neumann
condition

$\displaystyle\partial_{x}u(t,a)=0,\quad t>0.$

If $c_{n}\to\infty$, then $z_{k}^{(n)}\to z_{k}^{D}=\frac{\pi k}{a}$ and
therefore $u_{n}\to u$ where the solution $u(t,x)=\sum_{k\geq
1}e^{-t(z_{k}^{D})^{2}}\sin(xz_{k}^{D})\,f_{k}$ with $f_{k}=\int
f(x)\sin(xz_{k}^{D})dx$ solves (1.1) with Dirichlet condition

$\displaystyle u(t,a)=0,\quad t>0.$

If $c_{n}\to c\in(0,\infty)$, then $z_{k}^{(n)}\to
z_{k}^{R}>0\,:\,\tan(az_{k}^{R})=-\frac{z_{k}^{R}}{c}$ and therefore $u_{n}\to
u$ where the solution $u(t,x)=\sum_{k\geq
1}e^{-t(z_{k}^{R})^{2}}\sin(xz_{k}^{R})\,f_{k}$ with $f_{k}=\int
f(x)\sin(xz_{k}^{R})dx$ solves (1.1) with Robin boundary condition

$\displaystyle\partial_{x}u(t,a)+cu(t,a)=0,\quad t>0.$

Now we wonder if a similar asymptotic behaviour holds for the analogue time-
fractional problem. A simple example is given by the problem

$\displaystyle\partial^{\beta}_{t}u_{n}=\partial_{xx}u_{n},\quad
t>0,\,x\in(0,a)$ $\displaystyle\partial_{x}u_{n}(t,a)+c_{n}u_{n}(t,a)=0,\quad
t>0$ $\displaystyle u_{n}(t,0)=0,\quad t>0$ $\displaystyle
u_{n}(0,x)=f(x),\quad x\in(0,a)$

with $c_{n}>0$, $n\in\mathbb{N}$ where $\partial_{t}^{\beta}u$ is the Caputo
fractional derivative of $u$ (see formula (4.5) below). The solution can be
written as follows

$\displaystyle u_{n}(t,x)=\sum_{k\geq
1}E_{\beta}(-\lambda_{k}^{(n)}t^{\beta})\phi_{k}^{(n)}(x)\,f_{k}^{(n)}$ (1.3)

where

$\displaystyle E_{\beta}(w)=\sum_{k\geq 0}\frac{w^{\beta}}{\Gamma(\beta
k+1)},\quad w\geq 0$

is the Mittag-Leffler function and the system
$\\{\phi_{k}^{(n)},\lambda_{k}^{(n)}:k\in\mathbb{N}\\}$ has been introduced
before. By simple arguments, we get that the solution (1.3) uniformly
converges to a function $u$ which turns out to be analogously related to the
boundary problems above (Neumann, Dirichlet, Robin) with the Caputo time-
fractional derivative $\partial_{t}^{\beta}$ in place of the ordinary
derivative $\partial_{t}$. This is due to the fact that we have explicit
representation of the system
$\\{\phi_{k}^{(n)},\lambda_{k}^{(n)}:k\in\mathbb{N}\\}$.

Following the same spirit, in the present paper, we move on to general domains
like the random snowflakes we have introduced before and we address the same
asymptotic problem with a general time-fractional operator. In this case we do
not have the same informations about the associated system and the compact
representation of the solution. We overcome this difficulty by using the
theory of Dirichlet forms and Markov processes. An essential tool will be
given by the convergence of forms associated with time-changed processes.

We remark that the peculiarity in studying the asymptotic behaviour of these
approximating problems is that one has to deal with an increasing sequence of
Lipschitzian domains which converges in the limit to the domain whose boundary
is a fractal.

The plan of the paper is the following: in Section 2 we introduce the random
Koch domains; in Section 3 we recall the definition of Dirichlet forms with
associated base processes; in Section 4 we introduce time-fractional equations
and time changes; in the last section we prove our main results. More
precisely, in Theorem 5.1 we solve the asymptotic problem for the the time-
changed processes and in Theorem 5.2 we point out some peculiar aspects
arising by passing from the ordinary to the fractional Cauchy problem.

##  2\. Random Koch domains (RKD)

We first introduce the Koch (snowflake) domain and then we construct the
random Koch domains. Let $\ell_{a}\in(2,4)$ with $a\in I\subset\mathbb{N}$ be
the reciprocal of the contraction factor for the family $\Psi^{(a)}$ of
contractive similitudes $\psi^{(a)}_{i}:\mathbb{C}\to\mathbb{C}$ given by

$\displaystyle\psi^{(a)}_{1}(z)=\frac{z}{\ell_{a}},\quad\psi^{(a)}_{2}(z)=\frac{z}{\ell_{a}}e^{\imath\theta(\ell_{a})}+\frac{1}{\ell_{a}},$
$\displaystyle\psi^{(a)}_{3}(z)=\frac{z}{\ell_{a}}e^{\imath\theta(\ell_{a})}+\frac{1}{2}+\imath\sqrt{\frac{1}{\ell_{a}}-\frac{1}{4}},\quad\psi^{(a)}_{4}(z)=\frac{z-1}{\ell_{a}}+1$

where $\theta(\ell_{a})=\arcsin(\sqrt{\ell_{a}(4-\ell_{a})}/2)$. Let
$\Xi=I^{\mathbb{N}}$ with $I\subset\mathbb{N}$, $|I|=N$, and let
$\xi=(\xi_{1},\xi_{2},\ldots)\in\Xi$. We call $\xi$ an environment sequence
where $\xi_{n}$ says which family of contractive similitudes we are using at
level $n$. Set $\ell^{(\xi)}(0)=1$ and

$\displaystyle\ell^{(\xi)}(n)=\prod_{i=1}^{n}\ell_{\xi_{i}}.$ (2.1)

We define a left shift $S$ on $\Xi$ such that if
$\xi=\left(\xi_{1},\xi_{2},\xi_{3},\ldots\right),$ then
$S\xi=\left(\xi_{2},\xi_{3},\ldots\right).$ For $B\subset\mathbb{R}^{2}$ set

$\Upsilon^{(a)}(B)=\bigcup_{i=1}^{4}\psi^{(a)}_{i}\left(B\right)$

and

$\Upsilon^{(\xi)}_{n}(B)=\Upsilon^{(\xi_{1})}\circ\dots\circ\Upsilon^{(\xi_{n})}\left(B\right).$

The fractal $K^{(\xi)}$ associated with the environment sequence $\xi$ is
defined by

$K^{(\xi)}=\overline{\bigcup_{n=1}^{+\infty}\Upsilon_{n}^{(\xi)}(\Gamma)}$

where $\Gamma=\\{P_{1},P_{2}\\}$ with $P_{1}=(0,0)$ and $P_{2}=(1,0).$ We
remark that these fractals do not have any exact self-similarity, that is,
there is no scaling factor which leaves the set invariant: however, the family
$\\{K^{(\xi)},\xi\in\Xi\\}$ satisfies the following relation

$K^{(\xi)}=\Upsilon^{(\xi_{1})}(K^{(S\xi)}).$ (2.2)

Moreover, the spatial symmetry is preserved and the set $K^{(\xi)}$ is locally
spatially homogeneous, that is, the volume measure $\mu^{(\xi)}$ on
$K^{(\xi)}$ satisfies the locally spatially homogeneous condition (2.3) below.
Before describing this measure, we introduce some notations. For $\xi\in\Xi,$
we define the word space

$W=W^{(\xi)}=\\{(w_{1},w_{2},...):1\leqslant w_{i}\leqslant 4\\}$

and, for $w\in W,$ we set $w|n=(w_{1},...,w_{n})$ and
$\psi_{w|n}=\psi^{(\xi_{1})}_{w_{1}}\circ\dots\circ\psi^{(\xi_{n})}_{w_{n}}.$
The volume measure $\mu^{(\xi)}$ is the unique Radon measure on $K^{(\xi)}$
such that

$\mu^{(\xi)}(\psi_{w|n}(K^{(S^{n}\xi)}))=\frac{1}{4^{n}}$ (2.3)

for all $w\in W,$ (see Section 2 in [2]) as, for each $a\in A,$ the family
$\Psi^{(a)}$ has $4$ contractive similitudes. Let $K_{0}$ be the line segment
of unit length with $P_{1}=(0,0)$ and $P_{2}=(1,0)$ as endpoints. We set, for
each $n\in\mathbb{N}$,

$K^{(\xi)}_{n}=\Upsilon_{n}^{(\xi)}(K_{0})$

and ${K^{(\xi)}_{n}}$ is the so-called ${n}$-th prefractal curve.

Let us consider the random vector
$\boldsymbol{\xi}=(\boldsymbol{\xi}_{1},\boldsymbol{\xi}_{2},\ldots)$ whose
components $\boldsymbol{\xi}_{i}$ take values on $I$ with probability mass
function ${\bf P}:\Xi\to[0,1]$. Thus, the construction of the random $n$-th
pre-fractal curve

$K^{(\boldsymbol{\xi})}_{n}=\Upsilon_{n}^{(\boldsymbol{\xi})}(K_{0})$

depends on the realization of $\boldsymbol{\xi}$ with probability
$P(\boldsymbol{\xi}_{i}=\xi_{i})$ for its $i$-th component. We assume that
$\\{\boldsymbol{\xi}_{i}\\}_{i=1,\ldots,n}$ are identically distributed and
$\boldsymbol{\xi}_{i}\perp\boldsymbol{\xi}_{j}$ for $i\neq j$, that is we
obtain the curve $K_{n}^{(\xi)}$ with probability

$\displaystyle{\bf P}(\boldsymbol{\xi}|n=\xi|n)=\prod_{i=1}^{n}{\bf
P}(\boldsymbol{\xi}_{i}=\xi_{i})$

where $\boldsymbol{\xi}|n=(\boldsymbol{\xi}_{1},\ldots,\boldsymbol{\xi}_{n})$
and $\xi|n=(\xi_{1},\ldots,\xi_{n})$. Further on we only use the superscript
$(\xi|n)$ or $(\boldsymbol{\xi}|n)$ in order to streamline the notation.

The fractal $K^{(\boldsymbol{\xi})}$ associated with the random environment
sequence $\boldsymbol{\xi}$ is therefore defined by

$K^{(\boldsymbol{\xi})}=\overline{\bigcup_{n=1}^{+\infty}\Upsilon_{n}^{(\boldsymbol{\xi})}(\Gamma)}$

where $\Gamma=\\{P_{1},P_{2}\\}$ with $P_{1}=(0,0)$ and $P_{2}=(1,0).$

Let $\Omega^{(\xi|n)}$ be the planar domain obtained from a regular polygon by
replacing each side with a pre-fractal curve $K_{n}^{(\xi)}$ and
$\Omega^{(\xi)}$ be the planar domain obtained by replacing each side with the
corresponding fractal curve $K^{(\xi)}$. We introduce the random planar
domains $\Omega^{(\boldsymbol{\xi}|n)}$ and $\Omega^{(\boldsymbol{\xi})}$ by
considering the random curves $K_{n}^{(\boldsymbol{\xi})}$ and
$K^{(\boldsymbol{\xi})}$. Examples of (pre-fractal) random Koch domains are
given in figures 1 (outward curves), 2 (inward curves), 3 (inward curves) by
choosing as regular polygon the square.

Figure 1. Outward curves Figure 2. Inward curves Figure 3. Inward curves

Since $\boldsymbol{\xi}_{i}\stackrel{{\scriptstyle
law}}{{=}}\boldsymbol{\xi}_{1}$, $\forall\,i$, we have that the Hausdorff
dimension $d^{(\boldsymbol{\xi})}$ of the curve $K^{(\boldsymbol{\xi})}$ can
be obtained by considering the strong law of large numbers and the fact that

$\displaystyle\frac{\ln
4^{n}}{\sum_{i=1}^{n}\ell_{\boldsymbol{\xi}_{i}}}=\frac{\ln
4}{\frac{1}{n}\sum_{i=1}^{n}\ell_{\boldsymbol{\xi}_{i}}}\stackrel{{\scriptstyle
a.s.}}{{\to}}\frac{\ln 4}{\mathbf{E}[\ln\ell_{\boldsymbol{\xi}_{1}}]},\quad
n\to\infty.$

Then (see [2, Lemma 2.3]),

$\displaystyle d^{(\boldsymbol{\xi})}=\frac{\ln 4}{\ln\prod_{a\in
I}(\ell_{a})^{P(\boldsymbol{\xi}_{1}=a)}}=\frac{\ln
4}{\mathbf{E}[\ln\ell_{\boldsymbol{\xi}_{1}}]}.$ (2.4)

Moreover the measure $\mu^{(\xi)}$ in (2.3) has the property that there exist
two positive constants $C_{1},C_{2},$ such that,

$C_{1}r^{d^{(\xi)}}\leq\mu^{{(\xi)}}(\mathcal{B}(P,r)\cap K^{(\xi)})\leq
C_{2}r^{d^{(\xi)}}\ ,\quad\forall\,P\in K^{(\xi)},$ (2.5)

where $\mathcal{B}(P,r)$ denotes the Euclidean ball with center in $P$ and
radius $0<r\leq 1$ (see [2]). According to Jonsson and Wallin (see [25]), we
say that $K^{(\xi)}$ is a $d$-set with respect to the Hausdorff measure
$\mathcal{H}^{d},$ with $d=d^{(\xi)}.$ The sequence

$\displaystyle\sigma^{(\xi|n)}=\frac{\ell^{(\xi|n)}}{4^{n}},\qquad\textrm{where}\qquad\ell^{(\xi|n)}=\prod_{i=1}^{n}\ell_{\xi_{i}}$
(2.6)

is obtained from the realization of $\boldsymbol{\xi}|n$ and therefore, from
the realization of the random variable $\ell^{(\boldsymbol{\xi}|n)}$ with mean
value given by

$\displaystyle\mathbf{E}[\ell^{(\boldsymbol{\xi}|n)}]=\prod_{i=1}^{n}\mathbf{E}[\ell_{\boldsymbol{\xi}_{i}}]=\left(\mathbf{E}[\ell_{\boldsymbol{\xi}_{1}}]\right)^{n}.$

Thus, for $\alpha=\mathbf{E}\ell_{\boldsymbol{\xi}_{1}}\in(2,4)$ we find the
mean value $\mathbf{E}[\sigma^{(\boldsymbol{\xi}|n)}]=\alpha^{n}/4^{n}$.

The realization $\xi|n$ can be regarded as the vector
$\mathbf{a}|n=(a_{1},\ldots,a_{n})$ which is a $n$-dimensional vector with $N$
different values of $I$, that is $\mathbf{a}|n\in I^{n}$. We introduce the
multinomial distribution

$\displaystyle
p_{\mathbf{a}|n}=n!\prod_{i=1}^{N}\frac{p_{i}^{\sharp(a_{i})}}{\sharp(a_{i})!},\qquad\sum_{i=1}^{N}\sharp(a_{i})=n,\qquad\sum_{i=1}^{N}p_{i}=1$

where $p_{i}=P(\boldsymbol{\xi}_{1}=a_{i})$ and write
$p=\\{p_{i}\\}_{i=1,\ldots,N}$. Thus, for the realization of the vector
$\boldsymbol{\xi}|n$ we have that

$\displaystyle\mathbf{E}[\mathbf{1}_{\overline{\Omega^{(\boldsymbol{\xi}|n)}}}]=\sum_{\xi|n}p_{\xi|n}\mathbf{1}_{\overline{\Omega^{(\xi|n)}}}\quad\textrm{with}\quad
p_{\xi|n}=P(\boldsymbol{\xi}|n=\xi|n).$

or equivalently

$\displaystyle\mathbf{E}[\mathbf{1}_{\overline{\Omega^{(\boldsymbol{\xi}|n)}}}]=\sum_{\mathbf{a}|n}p_{\mathbf{a}|n}\mathbf{1}_{\overline{\Omega^{(\mathbf{a}|n)}}}.$

We notice that

$\displaystyle\mathbf{E}[\mathbf{1}_{\overline{\Omega^{(\boldsymbol{\xi}|n)}}}(x)]=1=(p_{1}+\ldots+p_{N})^{N}\quad\textrm{if}\quad
x\in\bigcap_{\Xi}\bigcup_{i=1}^{n}\overline{\Omega^{(\xi|i)}}.$ (2.7)

##  3\. Dirichlet forms and Base processes

Let $E$ be a locally compact, separable metric space and
$E_{\partial}=E\cup\\{\partial\\}$ be the one-point compactification of $E$.
Denote by $\mathcal{B}(E)$ the $\sigma$-field of the Borel sets in $E$
($\mathcal{B}_{\partial}$ is the $\sigma$-field in $E_{\partial}$). Let
$X=\\{X_{t},t\geq 0\\}$ with infinitesimal generator $(A,D(A))$ be the
symmetric Markov process on $(E,\mathcal{B}(E))$ with transition function
$p(t,x,B)$ on $[0,\infty)\times E\times\mathcal{B}(E)$. The point $\partial$
is the cemetery point for $X$ and a function $f$ on $E$ can be extended to
$E_{\partial}$ by setting $f(\partial)=0$. The associated semigroup is
uniquely defined by

$\displaystyle P_{t}f(x):=\int_{E}f(y)p(t,x,dy)=\mathbf{E}_{x}[f(X_{t})],\quad
f\in C_{\infty}(E)$

with $X_{0}=x\in E$ where $\mathbf{E}_{x}$ denote the mean value with respect
to the probability measure

$\displaystyle\mathbf{P}_{x}(X_{t}\in dy)=p(t,x,dy)$

and $C_{\infty}$ is the set of continuous function $C(E)$ on $E$ such that
$f(x)\to 0$ as $x\to\partial$. Let $\mathcal{E}(u,v)=(\sqrt{-A}u,\sqrt{-A}v)$
with domain $D(\mathcal{E})=D(\sqrt{-A})$ be the Dirichlet form associated
with (the non-positive definite, self-adjoint operator) $A$. Then $X$ is
equivalent to an $m$-symmetric Hunt process whose Dirichlet form
$(\mathcal{E},D(\mathcal{E}))$ is on $L^{2}(E)$ (see the books [15; 23]).
Without restrictions we assume that the form is regular ([23, page 143]).

We say that $X$ is the base process. Our aim is to consider time changes of
the base process $X$. Such random times will be introduced in the next
section.

##  4\. Time fractional equations and Time changes

We first introduce the subordinator $H=\\{H_{t},t\geq 0\\}$ for which

$\mathbf{E}_{0}[\exp(-\lambda H_{t})]=\exp(-t\Phi(\lambda))$

where $\Phi$ is the symbol of $H$. The symbol $\Phi$ may be associated also to
the inverse $L$ of $H$, that is $L=\\{L_{t},t\geq 0\\}$ defined as

$\displaystyle L_{t}=\inf\\{s\geq 0\,:\,H_{s}>t\\},\quad t\geq 0.$

We assume that $H_{0}=0$, $L_{0}=0$. By definition, we also have that

$\displaystyle\mathbf{P}_{0}(H_{t}<s)=\mathbf{P}_{0}(L_{s}>t),\quad s,t>0.$
(4.1)

The symbol $\Phi$ we consider hereafter is a Bernstein function with
representation

$\Phi(\lambda)=\int_{0}^{\infty}\left(1-e^{-\lambda
z}\right)\Pi(dz),\quad\lambda\geq 0$ (4.2)

where $\Pi$ on $(0,\infty)$ with $\int_{0}^{\infty}(1\wedge z)\Pi(dz)<\infty$
is the associated _Lévy measure_. We also recall that

$\displaystyle\frac{\Phi(\lambda)}{\lambda}=\int_{0}^{\infty}e^{-\lambda
z}\overline{\Pi}(z)dz,\qquad\overline{\Pi}(z)=\Pi((z,\infty))$ (4.3)

and $\overline{\Pi}$ is the so called _tail of the Lévy measure_. Both random
times $H,L$ are non-decreasing. We do not consider step-processes with
$\Pi((0,\infty))<\infty$ and therefore we focus only on strictly increasing
subordinators with infinite measures. Thus, the inverse process $L$ turns out
to be a continuous process. For details, see the books [5; 30].

We now introduce the fractional operators and the fractional equations
governing the time-changed process $X^{L}=\\{X\circ L_{t},t\geq 0\\}$, that is
the base process $X=\\{X_{t},t\geq 0\\}$ with the time change $L$
characterized by the symbol $\Phi$.

Let $M>0$ and $w\geq 0$. Let $\mathcal{M}_{w}$ be the set of (piecewise)
continuous function on $[0,\infty)$ of exponential order $w$ such that
$|u(t)|\leq Me^{wt}$. Denote by $\widetilde{u}$ the Laplace transform of $u$.
Then, we define the operator
$\mathfrak{D}^{\Phi}_{t}:\mathcal{M}_{w}\mapsto\mathcal{M}_{w}$ such that

$\displaystyle\int_{0}^{\infty}e^{-\lambda
t}\mathfrak{D}^{\Phi}_{t}u(t)\,dt=\Phi(\lambda)\widetilde{u}(\lambda)-\frac{\Phi(\lambda)}{\lambda}u(0),\quad\lambda>w$

where $\Phi$ is given in (4.2). Since $u$ is exponentially bounded, the
integral $\widetilde{u}$ is absolutely convergent for $\lambda>w$. By Lerch’s
theorem the inverse Laplace transforms $u$ and $\mathfrak{D}^{\Phi}_{t}u$ are
uniquely defined. Notice that

$\displaystyle\Phi(\lambda)\widetilde{u}(\lambda)-\frac{\Phi(\lambda)}{\lambda}u(0)=$
$\displaystyle\left(\lambda\widetilde{u}(\lambda)-u(0)\right)\frac{\Phi(\lambda)}{\lambda}.$
(4.4)

Simple arguments say that $\mathfrak{D}^{\Phi}_{t}$ can be written as a
convolution involving the ordinary derivative and the inverse transform of
(4.3) iff $u\in\mathcal{M}_{w}\cap C([0,\infty),\mathbb{R}_{+})$ and
$u^{\prime}\in\mathcal{M}_{w}$, that is,

$\displaystyle\mathfrak{D}^{\Phi}_{t}u(t)=\int_{0}^{t}u^{\prime}(s)\overline{\Pi}(t-s)ds.$

We notice that when $\Phi(\lambda)=\lambda$ (that is, the ordinary derivative)
we have that a.s. $H_{t}=t$ and $L_{t}=t$.

We also notice that for $\Phi(\lambda)=\lambda^{\beta}$, the symbol of a
stable subordinator, the operator $\mathfrak{D}^{\Phi}_{t}$ becomes the Caputo
fractional derivative

$\displaystyle\mathfrak{D}^{\Phi}_{t}u(t)=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}\frac{u^{\prime}(s)}{(t-s)^{\beta}}ds$
(4.5)

with $u^{\prime}(s)=du/ds$.

For $\Phi(\lambda)=(\lambda+\eta)^{\beta}-\eta^{\beta}$, with $\eta\geq 0$ and
$\beta\in(0,1)$, the operator $\mathfrak{D}^{\Phi}_{t}$ becomes the Caputo
tempered fractional derivative

$\displaystyle\mathfrak{D}^{\Phi}_{t}u(t)=\frac{1}{\Gamma(1-\beta)}\int_{0}^{t}\frac{u^{\prime}(s)}{(t-s)^{\beta}}e^{-\eta(t-s)}\,ds$

with $u^{\prime}(s)=du/ds$.

For explicit representation of the operator $\mathfrak{D}^{\Phi}_{t}$ see also
the recent works [13; 31].

Let $X$ be the process with generator $(A,D(A))$ introduced above. In the
present work we consider the time fractional equation

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=Au,\qquad u_{0}=f\in D(A).$ (4.6)

The probabilistic representation of the solution to (4.6) is written in terms
of the time-changed process $X^{L}$, that is

$\displaystyle
u(t,x)=\mathbf{E}_{x}[f(X^{L}_{t})]=\int_{0}^{\infty}P_{s}f(x)\,\mathbf{P}_{0}(L_{t}\in
ds),\quad x\in E,\,t>0.$ (4.7)

We notice that $\eqref{sol-time-frac-problem}$ is not a semigroup, indeed the
random time $L$ is not Markovian and therefore, the composition $X^{L}$ is not
a Markov process.

The fractional Cauchy problem has been investigated by many authors by
considering Caputo derivative and only recently, by taking into account more
general operators. The following theorem has been obtained in [11] for Feller
processes (not necessarily Feller diffusions, see [11]) and we mention here
such a result for the reader’s convenience.

######  Theorem 4.1.

The function (4.7) is the unique strong solution in $L^{2}(E)$ to (4.6) in the
sense that:

  1. (1)

$\varphi:t\mapsto u(t,\cdot)$ is such that $\varphi\in
C([0,\infty),\mathbb{R}_{+})$ and $\varphi^{\prime}\in\mathcal{M}_{0}$,

  2. (2)

$\vartheta:x\mapsto u(\cdot,x)$ is such that $\vartheta,A\vartheta\in D(A)$,

  3. (3)

$\forall\,t>0$, $\mathfrak{D}^{\Phi}_{t}u(t,x)=Au(t,x)$ holds a.e in $E$

  4. (4)

$\forall\,x\in E$, $u(t,x)\to f(x)$ as $t\downarrow 0$.

In [13] the author proves existence and uniqueness of strong solutions to
general time fractional equations with initial datum $f\in D(A)$. In [16] the
authors establish existence and uniqueness for weak solutions and initial
datum $f\in L^{2}$. The result in Theorem 4.1 has been proved in a general
setting, that is by considering a generator of a Feller process as in [13] but
following a very different approach. We notice that the condition on the
initial datum $f$ must be better specified for the compact representation of
the solution, this is the case investigated in [17] for instance (the domain
has no boundary) or the case investigated in [14] (with Dirichlet condition on
the boundary).

In the next section we will study continuous base processes time changed by
continuous random times, thus we do not stress the fact that the previous
result holds for Feller process (right-continuous with no discontinuity other
than jumps).

##  5\. Main results

We consider the prefractal RKD $\Omega^{(\xi|n)}$ defined in Section 2 and we
construct the set $\Omega^{(\xi|n)}\setminus B$ where
$B\subset\Omega^{(\xi|1)}$ is a ball.

Then, we consider Brownian diffusions on the Random Koch Domain
$\Omega^{(\xi|n)}\setminus B$. Let $X^{n}=\\{X_{t}^{n},t\geq 0\\}$ with
$X_{0}^{n}=x\in\Omega^{(\xi|n)}\setminus B$ be a sequence of planar Brownian
motions for a given $\xi\in\Xi$. Let $(A^{n},D(A^{n}))$ be the generator of
$X^{n}$, in particular $A^{n}=\Delta$ and

$D(A^{n})=$ $\displaystyle\\{u\in H^{1}(\Omega^{(\xi|n)}\setminus B):\,\Delta
u\in L^{2}(\Omega^{(\xi|n)}\setminus B),\,u|_{\partial B}=0,\,(\partial_{\bf
n}u+c_{n}\sigma^{(\xi|n)}u)|_{\partial\Omega^{(\xi|n)}}=0\\}$

where $c_{n}\geq 0$, $\mathbf{n}(x)$ denote the inward normal vector at
$x\in\partial\Omega^{(\xi|n)}$ and $\sigma^{(\xi|n)}$ is defined in (2.6). It
is well-known that there is one to one correspondence between the
infinitesimal generator of $X^{n}$ and the closed symmetric form
$(\mathcal{E}^{n},D(\mathcal{E}^{n})$ (see [23, Theorem 1.3.1]).

We recall that a form $(\mathcal{E}^{n},D(\mathcal{E}^{n})$ can be defined in
the whole of $L^{2}(F,m)$ by setting
$\mathcal{E}^{n}(u,u)=+\infty\quad\forall\,u\in L^{2}(F,m)\setminus
D(\mathcal{E}^{n}).$ Similarly a forms $\mathcal{E}$, $\mathcal{E}$ can be
defined in the whole of $L^{2}(F,m)$ by setting
$\mathcal{E}(u,u)=+\infty\quad\forall\,u\in L^{2}(F,m)\setminus
D(\mathcal{E}).$

For the convenience of the readers we recall the definition of convergence of
forms introduced by Mosco in [27], denoted by $M$-convergence.

######  Definition 1.

A sequence of forms $\\{\mathcal{E}^{n}(\cdot,\cdot)\\}$ $M$-converges to a
form $\mathcal{E}(\cdot,\cdot)$ in $L^{2}(F)$ if

  * (a)

For every $v_{n}$ converging weakly to $u$ in $L^{2}(F)$

$\underline{\lim}\ \mathcal{E}^{n}(v_{n},v_{n})\geq\mathcal{E}(u,u)\
,\quad\hbox{\rm as }n\to\infty.$ (5.1)

  * (b)

For every $u\in L^{2}(F)$ there exists $v_{n}$ converging strongly in
$L^{2}(F)$ such that

$\overline{\lim}\ \mathcal{E}^{n}(v_{n},v_{n})\leq\mathcal{E}(u,u)\
,\quad\hbox{\rm as }n\to\infty\ .$ (5.2)

In our framework, we consider the pre-fractal form
$\mathcal{E}^{n}(\cdot,\cdot)$ on $L^{2}(\Omega^{(\xi)}\setminus B)$ by
defining

$\mathcal{E}^{n}(u,u)=$
$\begin{cases}\displaystyle\int_{\Omega^{(\xi|n)}\setminus B}|\nabla
u|^{2}\,dxdy+c_{n}\sigma^{(\xi|n)}\int_{\partial\Omega^{(\xi|n)}}|u|^{2}ds&\hbox{\text{f}or}\,u|_{\Omega^{(\xi|n)}\setminus
B}\in H^{1}(\Omega^{(\xi|n)}\setminus B),\,u|_{\partial B}=0\\\
+\infty\quad&\hbox{\text{o}therwise }.\end{cases}$

We now introduce the time-changed process $X^{L,n}=X^{n}\circ L$ and we study
the asymptotic behaviour of $X^{L,n}$ depending on the asymptotics for
$c_{n}$. The process $X^{L,n}$ can be considered in order to study the
corresponding time-fractional Cauchy problem on $\Omega^{(\xi|n)}\setminus B$

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=A^{n}u,\quad u_{0}=f\in D(A^{n}).$

Let $\mathbb{D}$ be the set of continuous functions from $[0,\infty)$ to
$E_{\partial}=\Omega^{(\xi)}\cup\partial$ which are right continuous on
$[0,\infty)$ with left limits on $(0,\infty)$. We denote by $\partial$ the
cemetery point, that is $E^{n}_{\partial}$ is the one-point compactification
of $E^{n}=\Omega^{(\xi|n)}$, $n\in\mathbb{N}$. Let $\mathbb{D}_{0}$ the set of
non-decreasing continuous function from $[0,\infty)$ to $[0,\infty)$.

######  Proposition 5.1.

(Kurtz, [21]. Random time change theorem). Suppose that $X^{n}$, $X$ are in
$\mathbb{D}$ and $L^{n}$, $L$ are in $\mathbb{D}_{0}$. If $(X^{n},L^{n})$
converges to $(X,L)$ in distribution as $n\to\infty$, then $X^{n}\circ L^{n}$
converges to $X\circ L$ in distribution as $n\to\infty$.

###### Proof.

The proof follows from part b) of Theorem 1.1 and part a) of Lemma 2.3 in
[21]. Lemma 2.3 gives convergence for strictly increasing time changes. Since
$H$ is strictly increasing, we use part c) of Theorem 1.1 and find results for
$L$ which is non-decreasing and continuous. Then, part b) holds for the random
time changes $L^{n}$. ∎

######  Theorem 5.1.

As $n\to\infty$,

$\displaystyle X^{L,n}\to X^{L}\quad\textrm{in distribution
in}\quad\mathbb{D}\quad\boldsymbol{\xi}-a.s.\quad\textrm{on}\quad\Omega^{(\boldsymbol{\xi})}.$

In particular, as $c_{n}\to c\geq 0$,

  * i)

if $c=0$, then $X^{L}$ is reflected on $\partial\Omega^{(\xi)}$, that is the
process driven by

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=\Delta_{N}u,\quad u_{0}=f\in
D(\Delta_{N})$

where

$\displaystyle D(\Delta_{N})=\\{u\in H^{1}(\Omega^{(\xi)}\setminus B):\,\Delta
u\in L^{2}(\Omega^{(\xi)}\setminus B),\,u|_{\partial B}=0,\,(\partial_{\bf
n}u)|_{\partial\Omega^{(\xi)}}=0\\};$

  * ii)

if $c\in(0,\infty)$, then $X^{L}$ is (elastic) partially reflected on
$\partial\Omega^{(\xi)}$, that is the process driven by

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=\Delta_{R}u,\quad u_{0}=f\in
D(\Delta_{R})$

where

$\displaystyle D(\Delta_{R})=\\{u\in H^{1}(\Omega^{(\xi)}\setminus B):\,\Delta
u\in L^{2}(\Omega^{(\xi)}\setminus B),\,u|_{\partial B}=0,\,(\partial_{\bf
n}u+cu)|_{\partial\Omega^{(\xi)}}=0\\};$

  * ii)

if $c=\infty$, then $X^{L}$ is killed on $\partial\Omega^{(\xi)}$, that is the
process driven by

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=\Delta_{D}u,\quad u_{0}=f\in
D(\Delta_{D})$

where

$\displaystyle D(\Delta_{D})=\\{u\in H^{1}(\Omega^{(\xi)}\setminus B):\,\Delta
u\in L^{2}(\Omega^{(\xi)}\setminus B),\,u|_{\partial
B}=0,\,u|_{\partial\Omega^{(\xi)}}=0\\}.$

######  Remark 5.1.

We point out that the condition on the boundary $\partial\Omega^{(\xi)}$ must
be meant in the dual of certain Besov spaces (for details, see [8], [26] and
the references therein).

###### Proof.

Fix $\xi\in\Xi$. First we prove the M-convergence in
$L^{2}(\Omega^{(\xi)}\setminus B)$ of the Dirichlet forms $\mathcal{E}^{n}$.

The case of finite limit has been addressed in Theorem 5.2 in [9]: in
particular, it has been proved that if $c_{n}\to c\geq 0$, then the sequence
of forms $\mathcal{E}^{n}(\cdot,\cdot)$ $M$–converges in the space
$L^{2}({\Omega}^{(\xi)}\setminus B)$ to the form

$\mathcal{E}_{c}(u,u)=\begin{cases}\displaystyle\int_{\Omega^{(\xi)}\setminus
B}|\nabla
u|^{2}\,dxdy+c\int_{\partial\Omega^{(\xi)}}|u|^{2}d\mu^{(\xi)}&\hbox{\text{f}or}\,u|_{\Omega^{(\xi)}\setminus
B}\in H^{1}(\Omega^{(\xi)}\setminus B),\,u|_{\partial B}=0\\\
+\infty\quad&\hbox{\text{o}therwise}\end{cases}.$

The last form $\mathcal{E}_{c},$ for $c\in(0,\infty)$, is associated with the
semigroup ([6; 15])

$\displaystyle\mathbf{E}_{x}[f(X_{t})]=\mathbf{E}_{x}[f({{}^{*}X}_{t})M_{t}]$
(5.3)

where the multiplicative functional $M_{t}$ is associated to the Revuz measure
given by the perturbation of the form $\mathcal{E}_{c}$. Thus, (5.3) is the
solution to $\partial_{t}u=\Delta_{R}u$, $u_{0}=f\in D(\Delta_{R})$.

For $c=0$, the form $\mathcal{E}_{c}$ is associated with

$\displaystyle\mathbf{E}_{x}[f(X_{t})]=\mathbf{E}_{x}[f({{}^{*}X}_{t})]$

solution to $\partial_{t}u=\Delta_{N}u$, $u_{0}=f\in D(A)$.

Now we prove that if $c_{n}\to\infty$, the sequence of forms $\mathcal{E}^{n}$
M-converges on $L^{2}(\Omega^{(\xi)})$ to the form

$\mathcal{E}_{\infty}(u,u)=\begin{cases}\displaystyle\int_{\Omega^{(\xi)}\setminus
B}|\nabla u|^{2}\,dxdy&\hbox{\text{f}or}\,u|_{\Omega^{(\xi)}\setminus B}\in
H_{0}^{1}(\Omega^{(\xi)}\setminus B)\\\
+\infty\quad&\hbox{\text{o}therwise.}\end{cases}$

First we prove condition $(a)$ of Definition 1. Up to passing to a
subsequence, which we still denote by $v_{n}$, we can suppose that

$\,v_{n}|_{\Omega^{(\xi|n)}\setminus B}\in H^{1}(\Omega^{(\xi|n)}\setminus
B),$ (5.4)

and, for every $n,$

$||v_{n}||_{H^{1}(\Omega^{(\xi|n)}\setminus B)}\leqslant c^{*},$ (5.5)

with $c^{*}$ independent of $n.$ First we extend $v_{n}$ by Jones extension
operator (Theorem 1 in [24]) and after we restrict it to the domain
$\Omega^{(\xi)}\setminus B$ : more precisely, we extend $v_{n}$ to a function
$v^{*}_{n}=Ext_{J}\,v_{n}|_{\Omega^{(\xi)}\setminus B},$ such that

$||v^{*}_{n}||_{H^{1}(\Omega^{(\xi)}\setminus B)}\leqslant
C_{J}||v_{n}||_{H^{1}(\Omega^{(\xi)}_{n}\setminus B)}\leqslant C_{J}c^{*}.$
(5.6)

We point out that the constant $C_{J}$ independent of $n$ (see Theorem 3.4 in
[9]) that is the norm of extension operator is independent of the (increasing)
number of sides.

Then, there exists $v^{*}$ such that the sequence $v^{*}_{n}$ weakly converges
to $v^{*}$ in $H^{1}(\Omega^{(\xi)}\setminus B):$ for the uniqueness of the
limit in the weak topology, we obtain that $v^{*}=u$ and, in particular, $u\in
H^{1}(\Omega^{(\xi)}\setminus B).$ Since the sequence $v^{*}_{n}$ weakly
converges to $u$ in $H^{1}(\Omega^{(\xi)}\setminus B),$ we have that

$\underline{\lim}\ \int_{\Omega^{(\xi|n)}\setminus B}|\nabla
v_{n}|^{2}\,dxdy\geq\int_{\Omega^{(\xi)}\setminus B}|\nabla u|^{2}\,dxdy.$
(5.7)

From the compact embedding of $H^{1}(\Omega^{(\xi)})$ in
$H^{\alpha}(\Omega^{(\xi)})$ ($\frac{1}{2}<\alpha<1$), we have that

$||v^{*}_{n}-u||_{H^{\alpha}(\Omega^{(\xi)}\setminus B)}\to 0$ (5.8)

and by using Trace theorems (see [25] and [10]) we obtain that

$\sigma^{(\xi|n)}\int_{\partial\Omega^{(\xi|n)}}|v_{n}|^{2}ds\to
k\int_{\partial\Omega^{(\xi)}}|u|^{2}d\mu^{(\xi)}$ (5.9)

when $n\to\infty$ (see Theorem 2.1 in [9]). We stress the fact that the value
of $\sigma^{(\xi|n)}$ play a crucial role in the previous limit.

Now, if $c_{n}\to\infty,$ for any $k>0$ there exists $n_{1}$ such that, for
all $n>n_{1},$ $c_{n}\geq k.$ Then

$c_{n}\sigma^{(\xi|n)}\int_{\partial\Omega^{(\xi|n)}}|v_{n}|^{2}ds\geq
k\sigma^{(\xi|n)}\int_{\partial\Omega^{(\xi|n)}}|v_{n}|^{2}ds\to
k\int_{\partial\Omega^{(\xi)}}|u|^{2}d\mu^{(\xi)}$ (5.10)

when $n\to\infty.$ Dividing for $k$ and letting $k\to\infty$ we obtain that

$\int_{\partial\Omega^{(\xi)}}|u|^{2}d\mu^{(\xi)}=0$ (5.11)

and so $u=0$ on $\partial\Omega^{(\xi)}.$ By combining (5.7), (5.9), (5.11) we
have proved condition $(a)$ of Definition 1.

In order to prove condition (b) of Definition 1, we can assume that $u\in
H_{0}^{1}(\Omega^{(\xi)}\setminus B)$ without loss of generality: then, the
choice of $v_{n}=u$ suffices to achieve the result. So we have proved the
M-convergence of the forms $\mathcal{E}^{n}(\cdot,\cdot)$ on
$L^{2}(\Omega^{(\xi)}\setminus B)$ to the form $\mathcal{E}_{\infty}$ when
$c_{n}\to\infty$.

From the M-convergence of the forms $\mathcal{E}^{n}(\cdot,\cdot)$ on
$L^{2}(\Omega^{(\xi)}\setminus B),$ by using the results in the recent paper
[11], we obtain the convergence of the time changed processes.

More precisely, from the $M$-convergence of the forms we have the strong
convergence of semigroups. From Theorem 17.25 (Trotter, Sova, Kurtz,
Mackevičius) in [19] we have that strong convergence of semigroups (Feller
semigroups) is equivalent to weak convergence of measures if $X^{n}_{0}\to
X_{0}$ in distribution. Then we obtain that $X^{n}\stackrel{{\scriptstyle
d}}{{\to}}X$ in $\mathbb{D}$.

From Proposition 5.1, we have that

$\displaystyle\forall\,\xi\in\Xi,\quad X^{n}\circ L=:X^{L,n}\to X^{L}:=X\circ
L\quad\textrm{on}\quad\Omega^{(\xi)}$

in distribution as $n\to\infty$ in $\mathbb{D}$.

From the pointwise convergence, we get that $\boldsymbol{\xi}-$a.s.

$\displaystyle X^{L,n}\to
X^{L}\quad\textrm{on}\quad\Omega^{(\boldsymbol{\xi})}$

in distribution as $n\to\infty$ in $\mathbb{D}$.

∎

Let us consider now the process $X^{L}$ on $E$. We point out some peculiar
aspects of $X^{L}$ and the corresponding lifetimes.

######  Theorem 5.2.

Let us consider the Cauchy problems

$\displaystyle\partial_{t}w=Aw,\quad w_{0}=f\in D(A)$ (5.12)

and

$\displaystyle\mathfrak{D}^{\Phi}_{t}u=Au,\quad u_{0}=f\in D(A)$ (5.13)

with $\Phi$ such that

$\displaystyle\lim_{\lambda\to 0}\frac{\Phi(\lambda)}{\lambda}\in(0,\infty).$

We have that, $\forall\,x\in E$:

  * -

if $\Phi^{\prime}(0)<1$, then

$\int_{0}^{\infty}u(t,x)dt<\int_{0}^{\infty}w(t,x)dt,$

  * -

if $\Phi^{\prime}(0)>1$, then

$\int_{0}^{\infty}u(t,x)dt>\int_{0}^{\infty}w(t,x)dt,$

  * -

if $\Phi^{\prime}(0)=1$, then

$\int_{0}^{\infty}u(t,x)dt=\int_{0}^{\infty}w(t,x)dt.$

###### Proof.

The solution to (5.12) has the following probabilistic representation

$\displaystyle
w(t,x)=\mathbf{E}_{x}[f(X_{t})]=\mathbf{E}_{x}[f({{}^{*}X}_{t})M_{t}]$

where $M_{t}=\mathbf{1}_{(t<\zeta)}$ is the multiplicative functional written
in terms of the lifetime $\zeta$ of the process $X$ on $E$. Then, we consider
the part process $X$ of ${{}^{*}X}$ where ${{}^{*}X}_{0}=x\in E$. It is well
known that $M_{t}$ characterizes uniquely the associated semigroup ([6]), that
is the solution $w$. We also have that

$\displaystyle\mathbf{E}_{x}\left[\int_{0}^{\zeta}f(X_{s})ds\right]=\int_{0}^{\infty}w(t,x)dt=:\overline{w}(x)$

is the solution to the elliptic problem on $E$

$\displaystyle-A\overline{w}=f.$

From Theorem 4.1 we have that the time-changed process $X^{L}$ can be
considered in order to solve the problem (5.13), that is

$\displaystyle
u(t,x)=\mathbf{E}_{x}[f(X_{t}^{L})]=\mathbf{E}_{x}[f({{}^{*}X}^{L}_{t})\mathbf{1}_{(t<\zeta^{L})}]$

where $\zeta^{L}$ is the lifetime of $X^{L}$. As before we introduce the

$\displaystyle\overline{u}(x):=\int_{0}^{\infty}u(t,x)dt=\mathbf{E}_{x}\left[\int_{0}^{\zeta^{L}}f(X_{s}^{L})ds\right]$

which is the solution to the elliptic problem associated with the fractional
Cauchy problem (5.13). We are able to obtain the key relation between
$\overline{w}$ and $\overline{u}$ by taking into consideration the following
plain calculations. First we recall (4.7) where $P_{s}f(x)$ here is given by
$w(s,x)$ with $w(s,x)\to f(x)$ as $s\to 0$. Moreover (see [12]),

$\displaystyle\int_{0}^{\infty}e^{-\lambda t}\mathbf{P}_{0}(L_{t}\in
ds)dt=\frac{\Phi(\lambda)}{\lambda}e^{-s\Phi(\lambda)}ds.$ (5.14)

We have that

$\displaystyle\overline{u}(x)=$ $\displaystyle\lim_{\lambda\to
0}\int_{0}^{\infty}e^{-\lambda t}u(t,x)dt$ $\displaystyle=$
$\displaystyle[\textrm{by}\;\eqref{sol-time-frac-problem}]$ $\displaystyle=$
$\displaystyle\lim_{\lambda\to 0}\int_{0}^{\infty}e^{-\lambda
t}\int_{0}^{\infty}w(s,x)\mathbf{P}_{0}(L_{t}\in ds)\,dt$ $\displaystyle=$
$\displaystyle[\textrm{by}\;\eqref{ResL}]$ $\displaystyle=$
$\displaystyle\lim_{\lambda\to
0}\int_{0}^{\infty}w(s,x)\frac{\Phi(\lambda)}{\lambda}e^{-s\Phi(\lambda)}ds$
$\displaystyle=$ $\displaystyle\left(\lim_{\lambda\to
0}\frac{\Phi(\lambda)}{\lambda}\right)\int_{0}^{\infty}w(s,x)ds.$

That is

$\displaystyle\overline{u}(x)=\left(\lim_{\lambda\to
0}\frac{\Phi(\lambda)}{\lambda}\right)\overline{w}(x)$

and this gives a connection between solutions of elliptic problems introduced
above in the proof. Since $\Phi$ is a Bernstein function with $\Phi(0)=0$ we
get the result. ∎

The characterization given in the previous result admits a probabilistic
interpretation in terms of mean lifetime of the base and time-changed
processes. The problems (5.12) and (5.13) with $f=\mathbf{1}_{E}$ are
associated with $\overline{w}(x)=\mathbf{E}_{x}[\zeta]$ and
$\overline{u}(x)=\mathbf{E}_{x}[\zeta^{L}]$ as described in the previous proof
and the mean lifetime says how much the time change $L$ modifies the base
process $X$. By following the definition given in [12] and the relation
between $\overline{w}$ and $\overline{u}$ we say that $X$ is delayed or rushed
on $E$ by $L$. An example is given by the tempered fractional derivative ([4;
29]) associated with the symbol
$\Phi(\lambda)=(\lambda+\eta)^{\beta}-\eta^{\beta}$ with $\eta>0$ and
$\beta\in(0,1)$. We get that

$\displaystyle\mathbf{E}_{x}[\zeta^{L}]=\beta\eta^{\beta-1}\mathbf{E}_{x}[\zeta]$

that is, if $\beta\eta^{\beta-1}<1$ then the process $X$ is rushed by $L$,
whereas if $\beta\eta^{\beta-1}>1$ then the process $X$ is delayed by $L$.

The previous discussion on either delayed or rushed processes holds according
to specific regularity conditions on the boundary $\partial E$. We must have
that $\sup_{E}w(x)<\infty$ which is the characterization of trap domains
(written here for $X$ with generator $A$) given in [7] for the Brownian
motion. By applying the result in [7] it follows that the following
proposition holds true.

######  Proposition 5.2.

For $\xi\in\Xi$, the domains $\Omega^{(\xi|n)}$, $n\in\mathbb{N}$ are non trap
for the Brownian motion.

Since the previous statement holds pointwise for any contraction factor, we
immediately obtain the following general statement.

######  Proposition 5.3.

For the $\Xi$-valued random vector $\boldsymbol{\xi}$ the domains
$\Omega^{(\boldsymbol{\xi}|n)}$, $n\in\mathbb{N}$ are $\boldsymbol{\xi}-$a.s.
non trap for the Brownian motion.

Grant. The authors are members of GNAMPA (INdAM) and are partially supported
by Grants Ateneo “Sapienza” 2018.

## References

  * [1] M. Allen, L. Caffarelli, A. Vasseur, A parabolic problem with a fractional time derivative. Arch. Ration. Mech. Anal. 221 (2016), no. 2, 603-630.
  * [2] M. T. Barlow and B. M. Hambly, Transition density estimates for Brownian motion on scale irregular Sierpinski gasket, Ann. Inst. H. Poincaré Probab. Statist., 33 (1997), 531–557.
  * [3] E. Bazhlekova, The abstract Cauchy problem for the fractional evolution equation, Fract. Calc. Appl. Anal., 1 (1998), 255-270.
  * [4] L. Beghin, On fractional tempered stable processes and their governing differential equations, Journal of Computational Physics, 293 (2015), 29-39.
  * [5] J. Bertoin, Subordinators: Examples and Applications. In: Bernard P. (eds) Lectures on Probability Theory and Statistics. Lecture Notes in Mathematics, vol 1717. Springer, Berlin, Heidelberg, 1999.
  * [6] R. M. Blumenthal, R. K. Getoor, _Markov Processes and Potential Theory._ Academic Press, New York, 1968.
  * [7] K. Burdzy, Z.-Q. Chen, D. E. Marshall, Traps for reflected Brownian motion, Math. Z. 252, (2006) 103 - 132.
  * [8] R.Capitanelli, Transfer across scale irregular domains, Series on Advances in Mathematics for Applied Sciences 82 (2009), 165–174
  * [9] R.Capitanelli, Robin boundary condition on scale irregular fractals, Commun. Pure Appl. Anal. 9 (2010), 1221–1234.
  * [10] R. Capitanelli, M.A. Vivaldi, Trace Theorems on Scale Irregular Fractals in “Classification and Application of Fractals”, Nova Publishers, 2011.
  * [11] R. Capitanelli, M. D’Ovidio, Fractional equations via convergence of forms. Fractional Calculus and Applied Analysis, 22 (2019), 844-870.
  * [12] R. Capitanelli, M. D’Ovidio, Delayed and rushed motions through time change. ALEA, Lat. Am. J. Probab. Math. Stat. 17 (2020), 183-204.
  * [13] Z.-Q. Chen, Time fractional equations and probabilistic representation, Chaos, Solitons & Fractals, 102, (2017), 168 -174.
  * [14] Z.-Q. Chen, M. M. Meerschaert, E. Nane, Space-time fractional diffusion on bounded domains. J. Math. Anal. Appl. 393 (2012), no. 2, 479-488.
  * [15] Z.-Q. Chen, M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, London Mathematical Society Monographs, Princeton University Press, 2012.
  * [16] Z.-Q. Chen, P. Kim, T. Kumagai, J. Wang, Heat kernel estimates for time fractional equations, Forum Math. 30 (2018), no. 5, 1163-1192 .
  * [17] M. D’Ovidio, E. Nane, Fractional Cauchy problems on compact manifolds. Stoch. Anal. Appl. 34 (2016), no. 2, 232-257.
  * [18] S. D. Eidelman, A. N. Kochubei, Cauchy problem for fractional diffusion equations. J. Differ. Equ. 199 (2004), 211-255.
  * [19] O. Kallenberg, _Foundations of Modern Probability,_ Springer-Verlag New York, Inc., New York 1997.
  * [20] V. Keyantuo, C. Lizama, M. Warma, Existence, regularity and representation of solutions of time fractional diffusion equations. Adv. Differential Equations 21 (2016), no. 9-10, 837-886.
  * [21] T. G. Kurtz, Random Time Changes and Convergence in Distribution Under the Meyer-Zheng Conditions. _Ann. Probab._ 19, No 3 (1991), 1010–1034.
  * [22] M. M. Meerschaert, E. Nane, P. Vellaisamy, Fractional Cauchy problems on bounded domains, The Annals of Probability, Vol. 37 (2009), No. 3, 979-1007.
  * [23] M. Fukushima, Y. Oshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co, New York, 1994.
  * [24] P. W. Jones, Quasiconformal mapping and extendability of functions in Sobolev spaces, Acta Math., 147 (1981), 71–88.
  * [25] A. Jonsson, H. Wallin, Function spaces on subsets of $\mathbb{R}^{n}$, Math. Rep. 2 (1984), no. 1, xiv+221.
  * [26] M. R. Lancia, P. Vernole, Venttsel’ problems in fractal domains, J. Evol. Equ. 14 (2014), 681-712.
  * [27] U. Mosco, Convergence of convex sets and of solutions of variational inequalities, Adv. Math. 3, 510-585 (1969).
  * [28] E. Orsingher, L. Beghin, Fractional diffusion equations and processes with randomly varying time, Ann. Probab. 37, Number 1 (2009), 206-249.
  * [29] F. Sabzikar, M. M. Meerschaert, J. Chen, Tempered Fractional Calculus, J Comput Phys. 2015 293:14-28.
  * [30] R. L. Schilling, R. Song, Z. Vondraček, Bernstein Functions, Theory and Applications, Series:De Gruyter Studies in Mathematics 37, Berlin, 2010.
  * [31] B. Toaldo, Convolution-type derivatives, hitting-times of subordinators and time-changed $C_{\infty}$-semigroups, Potential Analysis, 42 (2015), 115 - 140.
  * [32] V.Vergara, R. Zacher, Stability, instability, and blowup for time fractional and other nonlocal in time semilinear subdiffusion equations. J. Evol. Equ. 17 (2017), no. 1, 599-626.