数理解析研究所講究録 1293 巻 2002 年 42-54

42Function spaces and stochastic processes on fractals
Takashi Kumagai

熊谷

1

*

(京都大学数理解析研究所)隆

Introduction

Since the late 80’s of the last century, there has been alot of development in
the mathematical study of stochastic processes and the corresponding operators

on fractals (see, for instance, [2], [12], [13], [17]). On the other hand, there has
been intensive study of Besov spaces, (which are roughly speaking, fractional
extensions of Sobolev spaces) on -sets, which correspond to regular fractals$d$(see [11], [24], [25]). In this survey paper, we summarize several recent works toconnect these two research areas, i.e., functional spaces and stochastic processes.
In Section 2, we first review basic facts in the Dirichlet form theory whichconnects functional spaces and stochastic processes. We will also give adefinition of -sets, the state space we work on. In Section 3, we discuss on function$d$spaces appear as domains of local regular Dirichlet forms on fractals, whose corresponding generators are so called Laplacians on fractals. The characterizationof the domain is possible through heat kernel estimates (3.1) of the Laplacian.

In Section 4, we introduce other types of function spaces appear as domains of. Research Institute for Mathematical Sciences, Kyoto University,
Kyoto 606-8502, Japan.

$\mathrm{E}$-mail:kumagai@kurims.ky0t0-u.ac.jp

URL http://www.kurims.ky0t0-u.ac.jp/”$\mathrm{k}\mathrm{u}\mathrm{m}\mathrm{a}\mathrm{g}\mathrm{a}\mathrm{i}/\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{x}.\mathrm{h}\mathrm{t}\mathrm{m}\mathrm{l}$43non-local Dirichlet forms on -sets, whose corresponding processes are stablelike jump processes. Three natural jump-type processes are introduced on d-sets$d$and the corresponding three forms are shown to be equivalent. It turns out that
one of the forms (and the corresponding operator) corresponds to the form (andthe operator) studied by Triebel in [24]. In Section 5, we will summarize theresults on heat kernel estimates for the stable-like jumP processes.
Throughout the paper, we only consider compact fractals, but most of theresults hold for unbounded fractals with suitable modifications of the statements.
We do not give any proof except Theorem 3.2. The proof and further details

axe given in the references cited.

2Dirichlet forms and d-sets
In this section, we will briefly review the definition of Dirichlet forms and thed-sets,correspondence to processes following [6]. We will also introduce
Let $X$ be alocally compact separable metric space andwhose support is $X$ . Let

measure on

$X$

on

with domain$\mathrm{L}^{2}(X, \nu)$i.e., for each$(\mathcal{E},\mathcal{F})$$u\in \mathcal{F}$,$\mathcal{F}$.

$(\mathcal{E},\mathcal{F})$$v:=(0\mathrm{V}\mathrm{r}\mathrm{x})$is regular if there exists

A$\mathcal{E}$$\nu$de apositive Radonbe asymmetric bilinear closed form

is called aDirichlet form if it is Markovian,$1\in \mathcal{F}$and$\mathcal{E}(v,v)\leq \mathcal{E}(u,u)$$C\subset \mathcal{F}\cap C_{0}(X)$such that$C$. ADirichlet form

is dense in$\mathcal{F}$with$\mathcal{E}_{1^{-}}$

is dense in $C_{0}(X)$ under the uniform norm, where $C_{0}(X)$ is aspace.
of continuous compact supported functions on $X$ andnorm and$C$$\mathcal{E}_{1}(\cdot, \cdot)=\mathcal{E}(\cdot, \cdot)+||\cdot$$(\mathcal{E},\mathcal{F})$is local if for each$u,v\in \mathcal{F}$$||_{\mathrm{L}^{2}}^{2}$

whose supports axe disjoint compact sets,. There is aone to one correspondence between aregular Dirichlet
and a -symmetric Hunt process (i.e., astrong Markov process
form onwhose paths are right continuous and quasi-left continuous w.r.t. some filtration$\mathcal{E}(u, v)=0$$\mathrm{L}^{2}(X, \nu)$

$\nu$

44
on

$X$

except for some exceptional set of starting points. Further, if the regular

Dirichlet form is local, then the corresponding process is adiffusion process (i.e.
Hunt process with continuous paths).

We next introduce our state space. Let$2,0<d\leq n)$ .

That is,$G\subset \mathrm{R}^{n}$where $B(x, r)$ is aball centered at
$d$

$x$

is the Hausdorff dimension of

normalize the size of$G$be acompact

and there exists$c_{2.1}$$d$

set in$\mathrm{R}^{n}(n\geq$, $c_{2.2}>0$ such thatfor all $x\in G$ , $0<r<1$ ,$c_{2.1}r^{d}\leq\mu(B(x,r))\leq c_{2.2}r^{d}$

Thus

$G$

and radius
$G$

and

$\mu$

$r$

(2.1)

w.r.t. the Euclidean metric.

the Hausdorff measure on

so that the diameter of

$G$

$G$

. We

is 1.

3Lipschitz spaces and domains of Dirichlet forms
For several sub-cla ses of -sets, diffusion processes, corresponding Laplace op$d$

erators and Dirichlet forms have been studied extensively (see [2], [12], [13], [17]etc). The most typical example is the Sierpinski gasket which we will definelater. In this section, we will consider aclass of -sets which has afractional$d$diffusion in the sense of Barlow [2] and show that the domain of the Dirichletform is the Lipschitz space. We first give adefinition of the fractional diffusion.
Definition 3.1 Let$a$$(G, \rho)$

be a complete compact metric space.

fractional metric space and$\{B_{t}^{G}\}_{t\geq 0}$

is called$a$$(G, \rho)$

is called

fractional diffusion if the fol-lowing holds.
1)$\rho$

has the midpoint property; for each , $y\in G$ , there exists $z\in G$ such that$x$$\rho(x,y)=\rho(x, z)/2=\rho(z, y)/2$ ,Further, there exists a Borel measure

$\mu$

which

satisfies (24) $w.r.t$ . .
$\rho$

2)$\{B_{t}^{G}\}_{t\geq 0}$

is a -symmetric conservative Feller diffusion on
$\mu$

$G$

which has a syrn-

45
metric jointly continuous transition density (fundamental solution of the heat
equation) $p_{t}(x,y)(t>0, x,y\in G)$ satisfying the following estimate,(3.1)$c_{3.1}t^{-d_{s}/2}\exp(-c_{3.2}(\rho(x, y)^{d_{w}}t^{-1})^{1/(d_{w}-1)})\leq p_{t}(x, y)$$\leq$$c_{3.3}t^{-d_{l}/2}\exp(-c_{3.4}(\rho(x,y)^{d_{w}}t^{-1})^{1/(d_{w}-1)})$

with some constants $d_{s}>0$ ,$d_{w}\geq 2$

for all $0<t<1$ , $x,y\in G$ ,.

We note that in the original definition of the fractional diffusions in [2], is not, is equivalentnecessarily compact. For simplicity, we further assume that$G$$\rho(\cdot$$\cdot$$)$to the Euclidean metric, i.e.,$c_{3.5}|x-y|\leq\rho(x,y)\leq c_{3.6}|x-y|$(3.2)$x,y\in G$ .

for allIn this case, $d_{s}/2=d/d_{w}$ holds.
Example: Sie pinski gasketLet$D_{n}$$i=1,2$ ,be a$\cdots$$n$-dimensional simplex whose vertices are, $n+1$ , let $F_{i}(z)=(z-p_{i})/2+p_{\acute{t}}$ ,unique non-void compact set$G$such that$z\in \mathrm{R}\mathrm{n}$$\{p_{0},p_{1}, \cdots,p_{n}\}$. For. Then, there exists a$G= \bigcup_{i=1}^{n+1}F_{i}(G)$. This$G$

is called a( -dimensional Sierpinski gasket It is known that the Sierpinski gasket has$n$afractional diffusion with $d=\log(n+1)/\log 2$ , $d_{s}=2\log(n+1)/\log(n+3)$ ,$d_{w}=\log(n+3)/\log 2$. (Note that $d_{s}<2$ for this example.)

In general, (affine) nested fractals, (which is aclass of fractals including
Sierpinski gaskets) and Sierpinski carpets have fractional diffusions with $c_{1}|x-$) $\leq c_{2}|x-y|^{d_{e}}$ for some$y|^{d_{\mathrm{C}}}\leq \mathrm{p}\{\mathrm{x}9\mathrm{y}$$d_{c}\geq 1$

instead of (3.2).

We now introduce Lipschitz spaces. For $1\leq p<\infty$ , $1\leq q\leq\infty$ ,$m\in \mathrm{N}\cup\{0\}$$\beta\geq 0$

and

, set

$a_{m}( \beta, f):=L^{m\beta}(L^{md}\int\int_{|x-y|<c\mathrm{o}L^{-m}}|f(x)-f(y)|^{p}d\mu(x)d\mu(y))^{1/p}f\in \mathrm{L}^{p}(G,\mu)$,46
where $1<L<\infty$ , $0<c_{0}<\infty$ . Define aLipschitz space Lip $(\beta,p, q)(G)$ as aset of $f\in L2(G, \mu)$ such that

. Lip(7J, , ) $(G)$ is$\overline{a}(\beta, f):=\{a_{m}(\beta, f)\}_{m=0}^{\infty}\in l^{q}$aBanach space with the norm$p$$||f||\mathrm{L}\mathrm{i}\mathrm{p}:=||f||_{\mathrm{L}^{p}}+||\overline{a}(\beta, f)||_{l^{q}}$Lipschitz space is determined independently of the choice of

$L$

$q$

. Note that theand$c\mathit{0}$as long as

the former is greater than 1and the latter is positive.
Theorem 3.2 ([9], [14], [19], [8]) Let$G$$(\mathcal{E}, \mathcal{F})$be a local regular DiricUet form on

which corresponds to a fractional diffusion. Then, the following holds.
$\mathcal{F}=Lip(\frac{d_{w}}{2}, 2, \infty)(G)$.

Proof. We will follow the argument in [19]. We first prove$\mathrm{L}^{2}(G,\mu)$

, let

to

. Then,$(\mathcal{E},\mathcal{F})$$\mathcal{E}_{t}(f, f):=(f-P_{t}f, f)_{\mathrm{L}^{2}}/t$$\mathcal{E}_{t}(f, f)$, where$P_{t}$$\mathcal{F}\subset \mathrm{L}\mathrm{i}\mathrm{p}$. For $f\in$

is asemigroup corresponding$=$$\frac{1}{2t}\int\int_{G\mathrm{x}G}(f(x)-f(y))^{2}p_{\mathrm{t}}(x, y)\mu(dx)\mu(dy)$$\geq$$\frac{1}{2t}\int\int_{|x-y|\leq c\mathrm{o}t^{1/d_{w}}}(f(x)-f(y))^{2}p_{t}(x,y)\mu(dx)\mu(dy)$$\geq$$\frac{c_{1}}{2t}\int\int_{|x-y|\leq c_{0}t^{1/d_{w}}}t^{-d./2}(f(x)-f(y))^{2}\mu(dx)\mu(dy)$

,

(3.3)

where we use the lower bound of (3.1) in the last inequality. Taking $t=L^{-md_{w}}$and using the fact $d_{s}/2=d/d_{w}$ , we see that (3.3) is equal to
It is well known thatobtain$\mathcal{E}_{t}(f, f)\nearrow \mathcal{E}(f, f)$$\sup_{m}a_{m}(d_{w}/2, f)^{2}\leq c_{2}\sqrt{\mathcal{E}(f,f)}$as$t\downarrow 0$$c_{1}a_{m}(d_{w}/2, f)^{2}$([6], Lemma 1.3.4). We thus

and the result holds.

We next prove 70) Lip. Set $\gamma=1/(d_{w}-1)$ . Since the diameter of

we have for each$g\in \mathrm{L}\mathrm{i}\mathrm{p}$

$G$

is 1,

,

$\mathcal{E}_{t}(g,g)=\frac{1}{2t}\int\int_{\mathrm{I}\cdot-u\mathrm{I}\leq 1}x,y\in G(g(x)-g(y))^{2}p_{t}\langle x,y)\mu(dx)\mu(dy)$$\leq$.

$\frac{1}{2t}\sum_{m=\mathrm{I}}^{\infty}c_{3}t^{-d_{*}/2}e^{-c_{4}(tL^{md_{w}})^{-\gamma}}\int\int_{L^{-m}<|x-y|\leq L^{-m+1}}(g(x)-g(y))^{2}\mu(dx)\mu(dy)$47$c_{3}t^{-(1+d_{\epsilon}/2)} \sum_{m=1}^{\infty}e^{-c_{4}(tL^{md_{w}})^{-\gamma}}L^{-m(d_{w}+d)}a_{m-1}(d_{w}/2, g)^{2}$$\leq$(3.4),

where we use the upper bound of (3.1) in the first inequality. For $0<t$ and$0\leq x$, let $t (0)$t(0)>0,$x_{t}>0$and$=e^{-c_{4}(tL^{xd_{w}})^{-\gamma}}L^{-x(d_{w}+d)}$$\lim_{xarrow\infty}\Phi_{t}(x)=0$

such that$\Phi_{t}(x)$$\Phi_{t}(x_{t})=c_{6}t^{1+d_{s}/2}$

and$\int_{0}^{\infty}\Phi_{t}(x)dx=c_{5}t^{1+d_{s}/2}$. Further, there exists

is increasing for $0\leq x<x_{t}$ , decreasing for $x_{t}<x<\infty$. Thus,.

$\Sigma_{m=1}^{\infty}\Phi_{t}(m)\leq f_{0}^{\infty}\Phi_{t}(x)dx+2\Phi_{t}(x_{t})\leq c_{7}t^{1+d./2}$

Since (3.4) is less than or equal tothat. By elementary calculation, we see that$c_{3}\mathrm{t}^{-(1+d_{*}/2)}||g||_{\mathrm{L}\mathrm{i}\mathrm{p}}^{2}\Sigma_{m=1}^{\infty}\Phi_{t}(m)$, we conclude

and the result holds.

$\sup_{t>0}\mathcal{E}_{t}(g, g)=\lim_{tarrow 0}\mathcal{E}_{t}(g,g)\leq c_{8}||g||_{\mathrm{L}\mathrm{i}\mathrm{p}}^{2}$

4Dirichlet forms and jump type processes on d-sets
In [15], three natural non-local regular Dirichlet forms are introduced, whose

corresponding processes are stable-like jump type processes on compact d-sets.
We will survey the results here.
4.1

Jump process as aBesov space on ad-set

We first introduce Besov spaces on$G$and their$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$theory within the scope of

our use (see [11], [24] etc. for details).
For $0<\alpha<1$ , we introduce aBesov space$B_{\alpha}^{2,2}(G)$afollows,$||u|B_{\alpha}^{2,2}(G)||$$=$$||u||_{\mathrm{L}^{2}(G,\mu)}+( \int\int_{G\mathrm{x}G}\frac{|u(x)-u(y)|^{2}}{|x-y|^{d+2\alpha}}\mu(dx)\mu(dy))^{1/2}(4.1)$$B_{\alpha}^{2,2}(G)$$=${ $u:u$ is measurable,

In [11], it is shown that for $0<\alpha<1$ ,$||u|B_{\alpha}^{2,2}(G)||<\infty$$B_{\alpha}^{2,2}(G)=\mathrm{L}\mathrm{i}\mathrm{p}(\alpha, 2,2)(G)$

norms are equivalent (Chapter , Proposition 3)$\mathrm{V}$

}.

(4.2)

and the two

48
For each$f\in \mathrm{L}_{loc}^{1}(\mathrm{R}^{n})$and$x\in \mathrm{R}^{n}$, define$Rf(x)= \lim_{f\downarrow 0}\frac{1}{m(B(x,r))}\int_{B(x,r)}f(y)dy$

if the limit exists, where $m$ is the Lebesgue measure in,$\mathrm{R}^{n}$. It is well-known

that the limit exists quasi-everywhere (i.e. except aset of zero capacity) in$\mathrm{R}^{\mathrm{n}}$with respect to the Newtonian capacity if $n\geq 3$ or logarithmic capacity

if $n=2$ and coincides with $f(x)$ almost everywhere indenote by$B_{\beta}^{2,2}(\mathrm{R}^{n})$the classical Besov space on

its definition). The following$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$\mathrm{R}^{n}$$\mathrm{R}^{n}$. For each $\beta>0$ ,(see Remark 4.2 below fortheorem plays an important role in the studyof Besov spaces on -sets(see, for instance, Chapters$d$$\mathrm{V}$and VI in [11] or Section

20 in [24] .
$)$Proposition 4,1 For $0<s<1$ , the trace operator

linear surjection from

$B_{s+(n-d)/2}^{2,2}(\mathrm{R}^{n})$right inverse operator$E_{G}$

so that$\mathit{2}\}_{G}\circ E_{G}$onto$B_{s}^{2,2}(G)$$\mathfrak{R}c$: $f\mapsto Rf$ is a bounded

and it has a bounded linear(which is called the extension operator in literature)is the identity map on$B_{s}^{2,2}(G)$.

Remark 4.2 Note that for $\beta>0$ with integer $k<\beta\leq k+1$ , the classicalBesov spaceis defined to be$B_{\beta}^{2,2}(\mathrm{R}^{n})$$B_{\beta}^{2,2}( \mathrm{R}^{n})=\{u\in C^{k}(\mathrm{R}^{n}):||u||_{B_{\beta}^{2_{1}2}}:=\sum_{0\leq|j|\leq k}||D^{j}u||_{2}+\sum_{|j|=k}(\int_{\mathrm{R}^{n}}\frac{||\Delta_{h}D^{j}f||_{2}^{2}}{|h|^{n+2(\beta-k)}}dh)^{1/2}<\infty\}$where for$j=(j_{1},j_{2}, \cdots,\mathrm{j}\mathrm{n})\in \mathrm{Z}_{+}^{n}$the difference operator so that for$||\cdot$$||_{2}$

denotes the$\mathrm{L}^{2}$

norm in,$|j|=\Sigma_{k=1}^{n}j_{k}$$h\in \mathrm{R}^{n}$$\mathrm{L}^{2}(\mathrm{R}^{n}, m)$and$D^{j}= \frac{\partial^{|\mathrm{j}|}}{\partial x_{1}^{\mathrm{J}1}\cdots\partial x_{n}^{\mathrm{j}n}}$$B_{\beta}^{2,2}(\mathrm{R}^{n})$$||u|B_{\beta}^{2,2}(\mathrm{R}^{n})||$$\Delta_{h}$is, (Ahf)(x) $=f(x+h)-f(x)$ , and(see, for instance, section 1.1.5 in [11]).

It is known (cf. Section V.I.I in [11]) that when $0<\beta<1$ , the norm
is equivalent to,defined by (4.2) with

$G=\mathrm{R}^{n}$is the same as the space defined by (4.1) with$||u||_{B_{\beta}^{2,2}}$, and therefore$G=\mathrm{R}^{n}$. Furthermore

49
the space$B_{\beta}^{2,2}(\mathrm{R}^{n})$coincides with the classical Bessel potential space on$\mathrm{R}^{n}$(also

called the fractional Sobolev space or the Liouville space); see, for instance, .
$\mathrm{p}$8 in Section 1.1.5 of [11],Now, for $0<\alpha<1$ and$u,v\in B_{\alpha}^{2,2}(G)$, define$\mathcal{E}_{Y^{(\alpha)}}(u, v)=\int\int_{G\mathrm{x}G}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^{d+2\alpha}}\mu(dx)\mu(dy)$By standard properties of Besov spaces, it is easy to check thatis aregular Dirichlet space on$\mathrm{L}^{2}(G, \mu)$in Theorem 3of [21] . We denote
$)$

$G$

. We note that when

4.2

$G=\mathrm{R}^{n}$.

$(\mathcal{E}_{Y(\alpha)}, B_{\alpha}^{2,2}(G))$(a detailed proof is given, for instance,$\{Y_{t}^{(\alpha)}\}_{t\geq 0}$, this is athe corresponding Hunt process on$(2\alpha)$-stable process on$\mathrm{R}^{n}$.

Jump process as asubordination of adiffusion

In this subsection, we assume that there exists afractional diffusion on .
be the strictly stable subordinator, i.e., it is
For $0<\alpha<1$ , let$G$$\{\xi_{t}\}_{t>0}$$\mathrm{o}$aone dimensional non-negative Levy process independent of$\{B_{t}^{G}\}_{t\geq 0}$

with thegenerating function $E[\exp(-u\xi_{t})]=\exp(-tu^{\alpha})$ . Let $\{\eta_{t}(u) : t>0,u\geq 0\}$ be. Using $p_{t}(x, y)$ in (3.1), we define
the distribution density of$\{\xi_{t}\}_{t>0}$$q_{t}(x, y)= \int_{0}^{\infty}p_{u}(x, y)\eta_{t}(u)du$for all $t>0$ , $x,y\in G$ .

Then, by ageneral theory, $q_{t}(x, y)$ is atransition density of some Markov pr0-cess which we denote by
In our case,$\{X_{t}^{(\alpha)}\}_{t\geq 0}$$\{X_{t}^{(\alpha)}\}_{t\geq 0}$, called the subordinate process (see [3], [20]).

is a -symmetric Hunt process and we denote the corre-

sponding Dirichlet form on$\mu$$\mathrm{L}^{2}(G, \mu)$as$(\mathcal{E}_{X(\alpha)},\mathcal{F}_{X(\alpha)})$.

Remark 4.3 The argument here can be extended to a class of diffusions iderthan fractional diffusions. Indeed, by checking the proof of [15], [21] carefully

50
we see that the same results hold for diffusions whose transition densities satisfy
estimates similar to (3.1), but with different orders
shoulders of$\rho(x, y)^{d_{w}}t^{-1}$$0<\gamma_{1}$,$\gamma_{2}<\infty$on the{instead of $1/(d_{w}-1))$ . If we weaken the condition

as above, then we can include all diffusions on

self-similar sets (which$p.c.f.\cdot$roughly corresponds to finitely ramified fractals) as mentioned in [16].
Jump process as atime change of astable process on

4.3$\mathrm{R}^{n}$

We first briefly give aresult by Triebel. In [24], hiebel define aBesov space on
$d$

-set

as follows.

$G$$||u|\hat{B}_{\alpha}^{2,2}(G)||$

inf

$=$

$||g|B_{\alpha+(n-d)/2}(\mathrm{R}^{n})||$$Tr_{G}g=u$$\hat{B}_{\alpha}^{2,2}(G)$where $\alpha>0$ . Here $Trc$ is thenorm of$B_{\alpha+(n-d)/2}(\mathrm{R}^{n})$$=Tr_{G}B_{\alpha+(n-d)/2}(\mathrm{R}^{n})$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$,,operator given in Proposition 4.1 and theis the one given in Remark 4.2. In general, this Besov

space is different from the one defined by Jonsson-WaJlin ([11]), but it is known
that for $0<\alpha<1$ , the two spaces coincide and the two norms are equivalent.
Let$H_{\alpha}$be the corresponding self-adjoint operator on$(H_{\alpha}^{1/2}u,H_{\alpha}^{1/2}u)_{\mathrm{L}^{2}(G,\mu)}=||u|\hat{B}_{\alpha}^{2,2}(G)||^{2}$

Theorem 4.4 ([24]; Theorem 25.2)

tor on$\mathrm{L}^{2}(G, \mu)$$H_{\alpha}$,$c_{4.1}$

so that$(H_{\alpha}^{1/2})=\hat{B}_{\alpha}^{2,2}(G)$.

is a positive definite self-adjoint opera-with pure point spectrum. Let

multiplicities). Then there exist

Dom

$G$

$\mu_{k}$be its -th eigenvalue (including$k$, $c_{4.2}>0$ such that the following holds.

$c_{4.1}k^{2\alpha/d}\leq\mu_{k}\leq c_{4.2}k^{2\alpha/d}$

for all$k\in \mathrm{N}$.

We now give the third jump process on -sets([15]). It turns out that$d$the corresponding operator of the process is the one given by Triebel then$0<\alpha<1$.

51
We construct ajump process though atime change ofon$\mathrm{R}^{n}$Also, let$\mathrm{R}^{n}$$\mathrm{R}^{n}$with aRadon measure$\{B_{t}^{(\alpha)}\}_{t\geq 0}(0<\alpha\leq 1)$

be arotation invariant

. Let

$G$

be a -set on
$d$

$\mu$

$(2\alpha)$-stable process

which satisfies (2.1).

$(2\alpha)$-stable process on(when $\alpha=1$ it is aBrownian motion). Then, it is proved in [15] that when$2\alpha>n-d$ , then$\mu$

is asmooth measure w.r.t.

$\{B_{t}^{(\alpha)}\}_{t}$of zero capacity w.r.t. the form corresponding to, i.e.

$\{B_{t}^{(\alpha)}\}_{t}$$\mu$charges no set. Thus, by ageneral

theory (see [6]), there exists aunique positive continuous additive functional$\{A_{t}^{(\alpha)}\}_{t\geq 0}$

whenwhich is in Revuz correspondence with

$B_{t}^{(\alpha)}\in G$$\mu$$A_{t}^{(\alpha)}$increases only). Set $\tau_{t}=\inf\{s>0 : A_{s}^{(\alpha)}>t\}$ and define

Then, again by ageneral theory,$\{Z_{t}^{(\alpha)}\}_{t\geq 0}$

corresponding form is a$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$$Z_{t}^{(\alpha)}=B_{\tau_{t}}^{(\alpha)}$.

is a -symmetric jump process whose$\mu$

corresponding regular Dirichlet form we denote by$\mathrm{R}^{n}$(thus,$(\mathcal{E}_{Z(\alpha)},\mathcal{F}_{Z(\alpha)})$of the Dirichlet form of the$(2\alpha)$. Since the-stable process on(see Theorem 6.2.1 in [6]), we can check that the corresponding operator is$H_{\alpha-(n-d)/2}$

given above.

Comparison of the forms and heat kernel bounds

4.4

Define$\overline{\alpha}=\alpha d_{w}/2$

and $\hat{\alpha}=\alpha-(n-d)/2$ . We then have the following.

Proposition 4.5 $([21],[15])$
Let

$G$

be a -set For $(n-d)/2<\alpha<1$ or $\alpha=1,n-2<d<n$ ,$d$$c_{4.3}\mathcal{E}_{Y^{(\dot{\alpha})}}(f, f)\leq \mathcal{E}_{Z^{(\alpha)}}(f, f)\leq c_{4.4}\mathcal{E}_{Y^{(\ )}}(f, f)$

for all$f\in \mathrm{L}^{2}(G,\mu)$

.

(4.3)

Assume further that there exists a fractional diffusion on G. For $0<\alpha<1$ ,$c_{4.5}\mathcal{E}_{Y^{(\overline{\alpha})}}(f, f)\leq \mathcal{E}_{X^{(\alpha)}}(f, f)\leq c_{4.6}\mathcal{E}_{Y^{(\overline{\alpha})}}(f, f)$

for all

In particular, under the conditions,$\mathcal{F}_{Z^{(\alpha)}}=B_{\hat{\alpha}}^{2,2}(G)$,$\mathcal{F}_{X^{(a)}}=B\frac{2}{\alpha}’(2G)$.

$f\in \mathrm{L}^{2}(G,\mu)$

.

(4.4)

52
Remark 4.61) In [15], (4.4) is stated for $0<\alpha<2/d_{w}$ . But the proof there
shows that it actually holds for $0<\alpha<1$ . From this fact, we see that whena -set on which there exists a fractional diffusion,$d$$(\mathcal{E}_{Y(\alpha)}, B_{\alpha}^{2,2}(G))$

$G$

is

is a regular

Dirichlet form for $0<\alpha<d_{w}/2$ .
2) Note that in general the three-type Dirichlet foms introduced are different

and the corresponding processes cannot be obtained by time changes of others by
positive continuous additive functional (see).

$[\mathit{1}\mathit{5}f$

5Heat kernel estimates for stable-like processes on d-sets
In [4], detailed estimates of heat kernels for$\{\mathrm{Y}^{(\alpha)}\}$

are obtained. There exists a

non-negative bounded heat kernel $p_{t}(x,y)$ on $(t, x,y)\in(0, \infty)\mathrm{x}G\mathrm{x}G$ with
$P_{t}^{Y^{(\alpha)}}f(x)= \int_{G}p_{t}(x,y)f(y)\mu(dy)$(where$P_{t}^{Y^{(a)}}$is the heat semigroup w.r.t.

for all$\mathcal{E}_{Y(\alpha)}$$x\in G$,

$f\in \mathrm{L}^{2}(G, \mu)$,), satisfying the following.

Theorem 5.1 ([4]) For $0<\alpha<1$ , the following holds.
1) For all , $y\in G$ , $0<t<1$ ,$x$$c_{5.1}(t^{-\frac{d}{2a}} \Lambda\frac{t}{|x-y|^{d+2\alpha}})\leq p_{t}(x, y)\leq c_{5.2}(t^{-\frac{d}{2\alpha}}\wedge\frac{t}{|x-y|^{d+2\alpha}})$.

2) $T/iere$ are constants $c_{5.3}>0$ and $\beta>0$ sucA that for any $0<t$ , $s<1$ and$(x:,y_{i})\in G\mathrm{x}G$with $i=1,2$ ,$|p_{s}(x_{1},y_{1})-p_{t}(x_{2},y_{2})|\leq c_{5.3}(t\Lambda s)^{-\frac{d+\beta}{2\alpha}}(|t-s|^{\frac{1}{2\alpha}}+|x_{1}-x_{2}|+|y_{1}-y_{2}|)^{\beta}$

Theorem 5.2 ([4]) For every $x\in F$ ,$\mathrm{Y}[0,1]:=\{\mathrm{Y}_{t} :0\leq t \leq 1\}$

is$d\wedge(2\alpha)$.

$\mathrm{P}^{x}- a.s.$, the Hausdorff dimension of

53
Note that above theorems hold under awider framework, i.e. when the form isexpressed as$\mathcal{E}(f, f)=\int\int_{G\mathrm{x}G}(f(x)-f(y))^{2}n(x,y)d\mu(x)d\mu(y)$,where $n(x, y)$ , $x,y\in G$ is ajointly measurable function such that $n(x,y)=$$n(y,x)$ for all $x,y\in G$ and satisfies$\frac{c_{5.4}}{|x-y|^{d+2\alpha}}\leq n(x,y)\leq\frac{c_{5.5}}{|x-y|^{d+2\alpha}}$.

In [4], it is also proved that all the parabolic functions (two variable functions

which satisfy the heat equation) satisfy the parabolic Harnadc inequality.

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