Abstract
We consider timechanged Brownian motions on random Koch (prefractal 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 prefractal boundary obtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundary conditions on the fractal domain.
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 is 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.
The 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 fundamental aspects: [1, 3, 14, 17, 18, 23, 26, 28, 32].
The nonlocal time operator we deal with is very general and covers a huge class of nonlocal (convolution type) operators. Such operators have been recently considered in the papers [13, 31]. From the probabilistic point of view, we consider timechanged 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 timefractional operator at hand, we study the fractional Cauchy problem with Robin condition on the prefractal 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
with Robin boundary condition
where \(c_n >0\), \(n \in {\mathbb {N}}\). The solution can be written as follows \(u_n(t,x)\)\( =\sum _{k \ge 1} e^{ t \lambda _k^{(n)}} \phi _k^{(n)}(x)\, f_k\) 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 with \(\phi _k^{(n)}\) where \(z_k^{(n)}\) are solutions to \(\tan (a z_k^{(n)}) =  z_k^{(n)}/ c_n\), \(k \in {\mathbb {N}}\).
Our aim here is to point out the asymptotic behavior of the solution \(u_n\) as \(n \rightarrow \infty \). We obtain three different limit problems. If \(c_n \rightarrow 0\), then \(z_k^{(n)} \rightarrow z_k^N= \left( \frac{\pi }{2}+ \pi k\right) \frac{1}{a}\) and therefore \(u_n(t,x) \rightarrow u(t,x) = \sum _{k \ge 1} e^{ t (z_k^N)^2} \sin (x z_k^N)\, f_k\) where \(f_k = \int f(x) \sin (x z_k^N) dx \) and u is the solution to (1.1) with Neumann condition
If \(c_n \rightarrow \infty \), then \(z_k^{(n)} \rightarrow z_k^D = \frac{\pi k}{a}\) and therefore \(u_n \rightarrow u\) where the solution \(u(t,x) = \sum _{k \ge 1} e^{ t (z_k^D)^2} \sin (x z_k^D)\, f_k \) with \(f_k = \int f(x) \sin (x z_k^D) dx\) solves (1.1) with Dirichlet condition
If \(c_n \rightarrow c \in (0,\infty )\), then \(z_k^{(n)} \rightarrow z_k^R >0\, :\, \tan (a z_k^R) =  \frac{z_k^R}{c} \) and therefore \(u_n \rightarrow u\) where the solution \(u(t,x) = \sum _{k \ge 1} e^{ t (z_k^R)^2} \sin (x z_k^R)\, f_k\) with \(f_k = \int f(x) \sin (x z_k^R) dx\) solves (1.1) with Robin boundary condition
Now, we wonder if a similar asymptotic behavior holds for the analogue timefractional problem. A simple example is given by the problem
with \(c_n >0\), \(n \in {\mathbb {N}}\) where \(\partial _t^\beta u\) is the Caputo fractional derivative of u (see formula (4.5)). The solution can be written as follows
where
is the MittagLeffler 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 timefractional 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 timefractional 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 timechanged processes.
We remark that the peculiarity in studying the asymptotic behavior 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 Sect. 2, we introduce the random Koch domains; in Sect. 3, we recall the definition of Dirichlet forms with associated base processes; in Sect. 4, we introduce timefractional equations and time changes; and in the last section, we prove our main results. More precisely, in Theorem 5.1 we solve the asymptotic problem for the timechanged processes and in Theorem 5.2, we point out some peculiar aspects arising by passing from the ordinary to the fractional Cauchy problem.
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 \(\psi ^{(a)}_i : {\mathbb {C}} \rightarrow {\mathbb {C}}\) given by
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
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
and
The fractal \(K^{( \xi )}\) associated with the environment sequence \(\xi \) is defined by
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 selfsimilarity; that is, there is no scaling factor which leaves the set invariant: However, the family \(\{K^{( \xi )}, \xi \in \Xi \}\) satisfies the following relation
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). Before describing this measure, we introduce some notations. For \(\xi \in \Xi ,\) we define the word space
and, for \(w\in W,\) we set \(wn=(w_1,...,w_n)\) 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
for all \(w\in W,\) (see Sect. 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 \(n \in {\mathbb {N}}\),
and \({K^{(\xi )}_n}\) is the socalled nth prefractal curve.
Let us consider the random vector \(\varvec{\xi }= (\varvec{\xi }_1, \varvec{\xi }_2, \ldots )\) whose components \(\varvec{\xi }_i\) take values on I with probability mass function \(\mathbf{P} : \Xi \rightarrow [0,1]\). Thus, the construction of the random nth prefractal curve
depends on the realization of \(\varvec{\xi }\) with probability \(P(\varvec{\xi }_i=\xi _i)\) for its ith component. We assume that \(\{\varvec{\xi }_i\}_{i=1, \ldots , n}\) are identically distributed and \(\varvec{\xi }_i \perp \varvec{\xi }_j\) for \(i\ne j\); that is, we obtain the curve \(K_n^{(\xi )}\) with probability
where \(\varvec{\xi } n = (\varvec{\xi }_1, \ldots , \varvec{\xi }_n)\) and \(\xi n = (\xi _1, \ldots , \xi _n)\). Further on, we only use the superscript \((\xi n)\) or \((\varvec{\xi }n)\) in order to streamline the notation.
The fractal \(K^{(\varvec{\xi })}\) associated with the random environment sequence \(\varvec{\xi }\) is therefore defined by
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 prefractal 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 \(\Omega ^{(\varvec{\xi }n)}\) and \(\Omega ^{(\varvec{\xi })}\) by considering the random curves \(K_n^{(\varvec{\xi })}\) and \(K^{(\varvec{\xi })}\). Examples of (prefractal) random Koch domains are given in Figs. 1 (outward curves), 2 (inward curves), 3 (inward curves) by choosing as regular polygon the square.
Since \(\varvec{\xi }_i {\mathop {=}\limits ^{law}} \varvec{\xi }_1\), \(\forall \, i\), we have that the Hausdorff dimension \(d^{(\varvec{\xi })}\) of the curve \(K^{(\varvec{\xi })}\) can be obtained by considering the strong law of large numbers and the fact that
Then (see [2, Lemma 2.3]),
Moreover, the measure \(\mu ^{(\xi )}\) in (2.3) has the property that there exist two positive constants \(C_1, C_2,\) such that
where \({\mathcal {B}}(P,r)\) denotes the Euclidean ball with center in P and radius \(0<r\le 1\) (see [2]). According to Jonsson and Wallin (see [21]), we say that \(K^{(\xi )} \) is a dset with respect to the Hausdorff measure \({\mathcal {H}}^d,\) with \(d=d^{(\xi )}.\) The sequence
is obtained from the realization of \(\varvec{\xi }n\) and therefore, from the realization of the random variable \(\ell ^{(\varvec{\xi }n)}\) with mean value given by
Thus, for \(\alpha ={\mathbf {E}} \ell _{\varvec{\xi }_1} \in (2,4)\) we find the mean value \({\mathbf {E}}[\sigma ^{(\varvec{\xi }  n)}] = \alpha ^n / 4^n\).
The realization \(\xi n\) can be regarded as the vector \({\mathbf {a}}n=(a_1, \ldots , a_n)\) which is a ndimensional vector with N different values of I, that is \({\mathbf {a}}n \in I^n\). We introduce the multinomial distribution
where \(p_i=P(\varvec{\xi }_1= a_i)\) and write \(p=\{p_i\}_{i=1, \ldots , N}\). Thus, for the realization of the vector \(\varvec{\xi }n\) we have that
or equivalently
We notice that
Dirichlet forms and base processes
Let E be a locally compact, separable metric space and \(E_\partial = E \cup \{\partial \}\) be the onepoint 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 \(X=\{X_t, t\ge 0 \}\) 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
with \(X_0=x \in E\) where \({\mathbf {E}}_x\) denotes the mean value with respect to the probability measure
and \(C_\infty \) is the set of continuous function C(E) on E such that \(f(x)\rightarrow 0\) as \(x\rightarrow \partial \). 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 nonpositive definite, selfadjoint operator) A. Then, X is equivalent to an msymmetric Hunt process whose Dirichlet form \(({\mathcal {E}}, D({\mathcal {E}}))\) is on \(L^2(E)\) (see the books [15, 19]). Without restrictions, we assume that the form is regular ([19, 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.
Timefractional equations and time changes
We first introduce the subordinator \(H=\{H_t, t\ge 0 \}\) for which
where \(\Phi \) is the symbol of H. The symbol \(\Phi \) may be associated also with the inverse L of H, that is \(L=\{L_t , t\ge 0\}\) defined as
We assume that \(H_0=0\), \(L_0=0\). By definition, we also have that
The symbol \(\Phi \) we consider hereafter is a Bernstein function with representation
where \(\Pi \) on \((0, \infty )\) with \(\int _0^\infty (1 \wedge z) \Pi (dz) < \infty \) is the associated Lévy measure. We also recall that
and \({\overline{\Pi }}\) is the socalled tail of the Lévy measure. Both random times H, L are nondecreasing. 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 timechanged process \(X^L = \{ X \circ L_t, t\ge 0\}\), that is the base process \(X=\{ X_t, t\ge 0\}\) with the time change L characterized by the symbol \(\Phi \).
Let \(M>0\) and \(w\ge 0\). Let \({\mathcal {M}}_w\) be the set of (piecewise) continuous function on \([0, \infty )\) of exponential order w such that \(u(t) \le M e^{wt}\). 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
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 \({\mathfrak {D}}^\Phi _tu\) are uniquely defined. Notice that
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,
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
with \(u^\prime (s)=du/ds\).
For \(\Phi (\lambda ) = (\lambda +\eta )^\beta  \eta ^\beta \), with \(\eta \ge 0\) and \(\beta \in (0,1)\), the operator \({\mathfrak {D}}^\Phi _t\) becomes the Caputo tempered fractional derivative
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 timefractional equation
The probabilistic representation of the solution to (4.6) is written in terms of the timechanged process \(X^{L}\), that is
We notice that (4.7) 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)
\(\varphi : t \mapsto u(t, \cdot )\) is such that \(\varphi \in C([0, \infty ), {\mathbb {R}}_+)\) and \(\varphi ^\prime \in {\mathcal {M}}_0\),

(2)
\(\vartheta : x \mapsto u(\cdot , x)\) is such that \(\vartheta , A\vartheta \in D(A)\),

(3)
\(\forall \, t > 0\), \({\mathfrak {D}}^\Phi _t u(t,x) = Au(t,x)\) holds a.e in E

(4)
\(\forall \, x \in E\), \(u(t,x) \rightarrow f(x)\) as \(t \downarrow 0\).
In [13], the author proves existence and uniqueness of strong solutions to general timefractional 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, and 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, and thus, we do not stress the fact that the previous result holds for Feller process (right continuous with no discontinuity other than jumps).
Main results
We consider the prefractal RKD \(\Omega ^{(\xi  n)}\) defined in Sect. 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 \(X^n=\{X_t^n, t\ge 0\}\) 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
where \(c_n \ge 0\), \({\mathbf {n}}(x)\) 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 onetoone correspondence between the infinitesimal generator of \(X^n\) and the closed symmetric form \(({\mathcal {E}}^n, D({\mathcal {E}}^n)\) (see [19, 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, 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 Mconvergence.
Definition 1
A sequence of forms \(\{{\mathcal {E}}^n(\cdot ,\cdot )\}\) Mconverges to a form \({\mathcal {E}}(\cdot ,\cdot )\) in \(L^2(F)\) if
 (a):

For every \(v_n\) converging weakly to u in \(L^2(F)\)
$$\begin{aligned} {\underline{\lim }}\ {\mathcal {E}}^n(v_n,v_n)\ge {\mathcal {E}}(u,u)\ ,\quad {\mathrm{as }} n\rightarrow \infty . \end{aligned}$$(5.1)  (b):

For every \(u\in L^2(F)\), there exists \(v_n\) converging strongly in \(L^2(F)\) such that
$$\begin{aligned} {\overline{\lim }}\ {\mathcal {E}}^n(v_n,v_n )\le {\mathcal {E}}(u,u)\ ,\quad \hbox { as }n\rightarrow \infty \ . \end{aligned}$$(5.2)
In our framework, we consider the prefractal form \({\mathcal {E}}^n(\cdot ,\cdot )\) on \(L^2(\Omega ^{(\xi )}\setminus B)\) by defining
We now introduce the timechanged process \(X^{L,n} = X^n \circ L\), and we study the asymptotic behavior 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 timefractional Cauchy problem on \(\Omega ^{(\xi n)} \setminus B\)
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 onepoint compactification of \(E^n=\Omega ^{(\xi  n)}\), \(n \in {\mathbb {N}}\). Let \({\mathbb {D}}_0\) be the set of nondecreasing continuous function from \([0, \infty ) \) to \([0, \infty )\).
Proposition 5.1
(Kurtz, [24]. 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 \(n\rightarrow \infty \), then \(X^n \circ L^n\) converges to \(X \circ L\) in distribution as \(n \rightarrow \infty \).
Proof
The proof follows from part b) of Theorem 1.1 and part a) of Lemma 2.3 in [24]. 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 nondecreasing and continuous. Then, part b) holds for the random time changes \(L^n\).
\(\square \)
Theorem 5.1
As \(n\rightarrow \infty \),
In particular, as \(c_n \rightarrow c \ge 0\),

(i)
if \(c=0\), then \(X^L\) is reflected on \(\partial \Omega ^{(\xi )}\), that is the process driven by
$$\begin{aligned} {\mathfrak {D}}^\Phi _t u = \Delta _N u, \quad u_0=f \in D(\Delta _N) \end{aligned}$$where
$$\begin{aligned} D(\Delta _N) = \{u \in H^1(\Omega ^{(\xi )}\setminus B): \, \Delta u \in L^2(\Omega ^{(\xi )}\setminus B),\, u_{\partial B}=0,\, (\partial _{\mathbf{n}} u)_{\partial \Omega ^{(\xi )}}=0\}; \end{aligned}$$ 
(ii)
if \(c \in (0, \infty )\), then \(X^L\) is (elastic) partially reflected on \(\partial \Omega ^{(\xi )}\), that is the process driven by
$$\begin{aligned} {\mathfrak {D}}^\Phi _t u = \Delta _R u, \quad u_0=f \in D(\Delta _R) \end{aligned}$$where
$$\begin{aligned} D(\Delta _R) = \{u \in H^1(\Omega ^{(\xi )}\setminus B): \, \Delta u \in L^2(\Omega ^{(\xi )}\setminus B),\, u_{\partial B}=0,\, (\partial _{\mathbf{n}} u + c u)_{\partial \Omega ^{(\xi )}}=0\}; \end{aligned}$$ 
(ii)
if \(c=\infty \), then \(X^L\) is killed on \(\partial \Omega ^{(\xi )}\), that is the process driven by
$$\begin{aligned} {\mathfrak {D}}^\Phi _t u = \Delta _D u, \quad u_0=f \in D(\Delta _D) \end{aligned}$$where
$$\begin{aligned} D(\Delta _D) {=} \{u \in H^1(\Omega ^{(\xi )}\setminus B): \, \Delta u \in L^2(\Omega ^{(\xi )}\setminus B),\, u_{\partial B}{=}0,\, u_{\partial \Omega ^{(\xi )}}{=}0\}. \end{aligned}$$
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, 25] and the references therein).
Proof
Fix \(\xi \in \Xi \). First, we prove the Mconvergence 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 \(c_n\rightarrow c\ge 0\), then the sequence of forms \({\mathcal {E}}^n(\cdot ,\cdot )\) Mconverges in the space \(L^2({\Omega }^{(\xi )}\setminus B)\) to the form
The last form \({\mathcal {E}}_c,\) for \(c \in (0,\infty )\), is associated with the semigroup ([6, 15])
where the multiplicative functional \(M_t\) is associated with 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
solution to \(\partial _t u = \Delta _N u\), \(u_0=f \in D(A)\).
Now, we prove that if \(c_n \rightarrow \infty \), the sequence of forms \({\mathcal {E}}^n\) Mconverges on \(L^2(\Omega ^{(\xi )})\) to the form
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
and, for every n,
with \(c^*\) independent of n. First, we extend \(v_n\) by Jones extension operator (Theorem 1 in [20]) 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
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
From the compact embedding of \(H^1(\Omega ^{(\xi )})\) in \(H^\alpha (\Omega ^{(\xi )})\) (\(\frac{1}{2}<\alpha <1\)), we have that
and by using Trace theorems (see [21] and [10]), we obtain that
when \(n\rightarrow \infty \) (see Theorem 2.1 in [9]). We stress the fact that the value of \(\sigma ^{(\xi  n)}\) plays a crucial role in the previous limit.
Now, if \(c_n\rightarrow \infty ,\) for any \(k>0\) there exists \(n_1\) such that, for all \(n>n_1,\) \(c_n\ge k.\) Then,
when \(n\rightarrow \infty .\) Dividing for k and letting \(k\rightarrow \infty \), we obtain that
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 Mconvergence of the forms \({\mathcal {E}}^n(\cdot ,\cdot )\) on \(L^2(\Omega ^{(\xi )}\setminus B)\) to the form \({\mathcal {E}}_\infty \) when \(c_n \rightarrow \infty \).
From the Mconvergence 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 timechanged processes.
More precisely, from the Mconvergence of the forms we have the strong convergence of semigroups. From Theorem 17.25 (Trotter, Sova, Kurtz, Mackevičius) in [22], we have that strong convergence of semigroups (Feller semigroups) is equivalent to weak convergence of measures if \(X^n_0 \rightarrow X_0\) in distribution. Then, we obtain that \(X^n {\mathop {\rightarrow }\limits ^{d}} X\) in \({\mathbb {D}}\).
From Proposition 5.1, we have that
in distribution as \(n\rightarrow \infty \) in \({\mathbb {D}}\).
From the pointwise convergence, we get that \(\varvec{\xi }\)a.s.
in distribution as \(n\rightarrow \infty \) in \({\mathbb {D}}\). \(\square \)
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
and
with \(\Phi \) such that
We have that, \(\forall \, x\in E\):

if \(\Phi ^\prime (0) < 1\), then
$$\begin{aligned} \int _0^\infty u(t,x)dt < \int _0^\infty w(t,x) dt, \end{aligned}$$ 
if \(\Phi ^\prime (0) > 1\), then
$$\begin{aligned} \int _0^\infty u(t,x)dt > \int _0^\infty w(t,x) dt, \end{aligned}$$ 
if \(\Phi ^\prime (0) = 1\), then
$$\begin{aligned} \int _0^\infty u(t,x)dt = \int _0^\infty w(t,x) dt. \end{aligned}$$
Proof
The solution to (5.12) has the following probabilistic representation
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
is the solution to the elliptic problem on E
From Theorem 4.1, we have that the timechanged process \(X^L\) can be considered in order to solve the problem (5.13), that is
where \(\zeta ^L\) is the lifetime of \(X^L\). As before, we introduce the
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 \(P_sf(x)\) here is given by w(s, x) with \(w(s,x) \rightarrow f(x)\) as \(s\rightarrow 0\). Moreover (see [12]),
We have that
That is
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. \(\square \)
The characterization given in the previous result admits a probabilistic interpretation in terms of mean lifetime of the base and timechanged 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
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 nontrap 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 \(\varvec{\xi }\), the domains \(\Omega ^{(\varvec{\xi } n)}\), \(n \in {\mathbb {N}}\) are \(\varvec{\xi }\)a.s. nontrap for the Brownian motion.
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Open access funding provided by Università degli Studi di Roma La Sapienza within the CRUICARE Agreement. The authors are members of GNAMPA (INdAM) and are partially supported by Grants Ateneo “Sapienza” 2018.
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Capitanelli, R., D’Ovidio, M. Fractional Cauchy problem on random snowflakes. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028021006737
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Keywords
 Time changes
 Fractional operators
 Timefractional equations
 Asymptotics
Mathematics Subject Classification
 26A33
 35R11
 60J50
 58J37