---
abstract: 'We consider generalised Kreĭn-Feller operators $\Delta_{\nu, \mu}  $ with respect to compactly supported Borel probability measures $\mu$ and $\nu$ under the natural restrictions ${\operatorname{supp}}(\nu)\subset{\operatorname{supp}}(\mu)$ and $\mu$ atomless. We show that the solutions of the eigenvalue problem for $\Delta_{\nu, \mu}  $ can be transferred to the corresponding problem for the classical Kreĭn-Feller operator $\Delta_{\nu, \Lambda}=\partial_{\mu}\partial_{x}$ with respect to the Lebesgue measure $\Lambda$ via an isometric isomorphism of the underlying Banach spaces. In this way we reprove the spectral asymptotic on the eigenvalue counting function obtained by Freiberg. Additionally, we investigate infinitesimal generators of generalised Liouville Brownian motions associated to generalised Kreĭn-Feller operator $\Delta_{\nu, \mu}$ under Neumann boundary condition. Extending the measure $\mu$ and $\nu$ to the real line allows us to determine its walk dimension.'
address:
- 'FB 3 – Mathematik, Universität Bremen, Bibliothekstr. 1, 28359 Bremen, Germany'
- 'School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK'
author:
- Marc Kesseböhmer
- Aljoscha Niemann
- Tony Samuel
- Hendrik Weyer
title: 'Generalised Kreĭn-Feller operators and Liouville Brownian motion via transformations of measure spaces'
---

[^1]

Introduction and motivation
===========================

The classical Kreĭn-Feller differential operator $\Delta_{\nu, \Lambda}$, introduced in [@Fe57; @KK68], where $\nu$ denotes a non-atomic compactly supported Borel probability measure on $\mathbb{R}$ and where $\Lambda$ denotes the one-dimensional Lebesgue measure, has been investigated with respect to its spectral properties first by Fujita [@Fu87], Küchler [@MR574035], Langer [@MR0314125] and Kotani and Watanabe [@MR661628] and more recently by Arzt [@A15b], Ehnes [@Ehnes2019] and Freiberg [@MR2017701; @MR2030736; @Fr05]. The case when $\nu$ is purely atomic has also been studied in [@MR2513598]; where it was shown that the eigenvalues of $\Delta_{\nu, \Lambda}$ have a dependence not only on the positions of the atoms of $\nu$ but also on the weights of the atoms. Returning to the case when $\nu$ is a non-atomic, it has been established that $\Delta_{\nu, \Lambda}$ is the infinitesimal generator of a Liouville Brownian motion (also known as gap diffusion, skip-free diffusion, quasi-diffusion or generalised diffusion), see [@Burkhardt1983; @MR2817342; @MR3034785; @X_Jin_2017; @MR574035; @MR0314125; @MR3005002; @MR3272822].

Here, we investigate generalised Kreĭn-Feller operators $\Delta_{\nu, \mu}$ for Borel measures $\nu$ and $\mu $ on the real line under the natural assumptions ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}( \mu)$ and $\mu$ atomless. In the case that $\nu=\mu =\Lambda$ the operator coincides with the classical second order weak derivative. For arbitrary $\mu=\nu$, atomless and compactly supported, a harmonic calculus for $\Delta_{\mu, \mu}$ was developed in [@FZ02] and, when $\mu$ is a self-similar measure supported on a Cantor set, it is now well established that the eigenvalue counting function of $\Delta_{\mu, \mu}$ is comparable to the square-root function. In [@KSW16] the exact eigenvalues and eigenfunctions of $\Delta_{\mu, \mu}$ were obtained and it was shown that the eigenvalues do not depend on the given measure. Moreover, the eigenfunctions are given by a composition of the appropriated classical trigonometric functions with a phase space transformation induced by the distribution function of $\mu$. The case when the measure $\mu$ has a continuous as well as an atomic part, was the subject of [@KSW17; @KSW2019b]. Here, it has been shown that, if $\mu$ has a continuous part, then the eigenvalues may depend on the position of the atoms, and otherwise not.

In the present article, we elaborate on the connections between the generalised and the classical Kreĭn-Feller operators by establishing a suitable phase space transformation determined by the distribution function of $\mu$. As a first application of this observation we show that the spectral properties of the generalised Kreĭn-Feller operators can be reduced to those of the to the classical ones; and as a second application, we connect properties of the associated Liouville Brownian motions for generalised Kreĭn-Feller operators to that of the classical Kreĭn-Feller operators with a special focus on the concept of walk dimension. This complements and partially resembles the general framework established by Dynkin [@DynkinI_II Vol.I,6].

Setup and statement of main results
===================================

Our setting
-----------

Let $\mu$ and $\nu$ denote two Borel probability measures on $[0,1]$ with ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$, $\nu(\{0,1\})=0$ and $\mu$ atomless. Denote the distribution function of $\mu$ and $\nu$ by $F_{\mu}$ and $F_{\nu}$, respectively. Let $(C_{\nu, \mu}, \lVert \cdot \rVert_{\infty} )$ denote the Banach space of continuous function on $[0,1]$ which are linear in scale $F_{\mu}$ on intervals where $F_{\nu}$ is constant. Namely, on each connected component $J$ of $[0,1] \setminus {\operatorname{supp}}(\nu)$ the function $f$ is linear in $F_{\mu}$, that is $f(x)=a_J F_{\mu}(x)+b_J$ for all $x \in J$ and some $a_J, b_J \in  \mathbb{R}$. As indicated above, we let $\Lambda$ denote the one-dimensional Lebesgue measure restricted to $[0,1]$.

Set $\S^w \coloneqq L^2(\nu)$ and $\S^s \coloneqq C_{\nu,\mu}$, where $w$ stands for [*weak*]{} and $s$ stands for [*strong*]{}; we sometimes write $\S^*(\mu,\nu)$ instead of $\S^*$ to stress the dependence of the underlying measure spaces for $* \in \{s,w\}$. In what follows we will mainly be concerned with the Banach spaces $(\S^{w},\Vert\cdot\Vert_{2})$ and $(\S^{s},\Vert\cdot\Vert_{\infty})$. Now fix $* \in \{s,w\}$; a function $f$ belonging to the set $C([0,1] )$ of continuous function on $[0,1] $ is said to lie in $\mathcal{D}^*(\Delta_{\nu,\mu})$ if there exist $a,b \in \R$ and $g \in \S^*$ with $$\begin{aligned}
\label{KreinFeller}
 f(x)=a+bF_{\mu}(x)+\int_{[0,x]}( F_{\mu}(x)-F_{\mu}(y) ) g(y)\;{\mathrm{d}}\nu(y)
 \end{aligned}$$ for all $x \in [0,1]$. Alternatively, by Fubini we can write $ f(x)=a+bF_{\mu}(x)+\int_{[0,x]}\int_{[0,y]}  g(s)\;{\mathrm{d}}\nu(s)\;{\mathrm{d}}\mu(y)$. By the uniqueness of densities we observe that $f$ determines $a$, $b$ and $g$ uniquely and by setting $ \Delta_{\nu,\mu}f \coloneqq g$ we define an injective linear operator $\Delta_{\nu,\mu}:\mathcal{D}^*(\Delta_{\nu,\mu})\to \S^*$. The first derivative of $ f \in \mathcal{D}^{\text{*}}(\Delta_{\nu,\mu})$ is defined by $$\begin{aligned}
\label{eq:first_derivative}
\nabla_{\mu}f(x) \coloneqq \nabla_{\mu}f(0)+\int_{[0,x]} \Delta_{\nu,\mu}f(y)\;{\mathrm{d}}\nu(y),
\;\; \text{where} \;\;
\nabla_{\mu}f(0) \coloneqq \lim_{x \downarrow 0^+} \dfrac{f(x)-f(0)}{F_{\mu}(x)-F_{\mu}(0)}
\;\;
\text{and}
\;\;
 0^+ \coloneqq \inf ({\operatorname{supp}}(\mu)).\end{aligned}$$ The existence of the above limit follows from and the assumption that ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}( \mu)$. Additionally, from together with Lebesgue’s dominated convergence theorem, we have that $f$ is constant on every interval of constancy of $F_{\mu}$, $a=f(0)$, $b= \nabla f_{\mu}(0)$ and $\mathcal{D}^s ( \Delta_{\nu,\mu} ) \subset \mathcal{D}^w (\Delta_{\nu,\mu}) \subset C_{\nu,\mu}$. Moreover, for all $f \in\mathcal{D}^*(\Delta_{\nu,\mu})$ and all $x \in (0,1)$, $$\begin{aligned}
    \nabla_{\mu}(f)(x)
    = \lim_{ \substack{ y \downarrow x^+}} \dfrac{f(y)-f(x)}{F_{\mu}(y)-F_{\mu}(x)}   +  \Delta_{\nu,\mu} f (x)\nu(\{ x^+ \} ) (1 - \mathds{1}_{\{x^+\}}(x) )
    = \lim_{ \substack{ y \uparrow x^-}} \dfrac{f(x)-f(y)}{F_{\mu}(x)-F_{\mu}(y)}   +  \Delta_{\nu,\mu} f (x)\nu(\{ x^- \} ).
    \end{aligned}$$ Here, $x^+ \coloneqq \inf\{ y \in [0,1] \colon F_{\mu}(x) < F_{\mu}(y) \}$ and $x^- \coloneqq \sup\{ y \in [0,1] \colon F_{\mu}(x)>F_{\mu}(y) \}$. Observe, if $\nu$ is atomless, then $\nabla_{\mu}(f)$ is independent of $\nu$ and given by an ‘ordinary’ differential quotient.

For $ \gamma =(\alpha,\beta) \in [0,{\pi/2}]^{2}$ we consider the following eigenvalue problem for $\Delta_{\nu,\mu}$, see [@Fu87], $$\begin{aligned}
\label{EWC}
    \Delta_{\nu,\mu}f= \lambda f  
    \end{aligned}$$ with [*Robin boundary conditions*]{} $$\begin{aligned}
\label{BC}
    f(0)\cos(\alpha)-\nabla_{\mu}f(0)\sin(\alpha)=0
    \quad \text{and} \quad
    f(1)\cos(\beta)+\nabla_{\mu}f(1)\sin(\beta)=0.
    \end{aligned}$$ We refer to the particular case $\gamma=(\pi/2,\pi/2)$ as the [*Neumann case*]{} and the case $\gamma=(0,0)$ as the [*Dirichlet case*]{}. We denote by $\mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\mu} )$ the set of all $f \in \mathcal{D}^{*}( \Delta_{\nu,\mu} )$ which satisfy and . Combining and with Fubini’s theorem and the assumptions on $\nu,\mu$ one obtains a [*Gauss-Green formula*]{}; namely, for $f,g \in \mathcal{D}_{\gamma}^s( \Delta_{\nu,\mu} )$, $$\begin{aligned}
    \int (\Delta_{\nu,\mu}f )g\mathrm{d} \nu
        &= \left(
        \nabla_{\mu}f (1)-\nabla_{\mu}f (0)\right)g(0)+ \int\nabla_{\mu}g(y)\left(
\nabla_{\mu}f (1)+\Delta_{\nu,\mu}f (y)\nu \left(\{y\} \right)-\nabla_{\mu}f (y)        
         \right)
{\mathrm{d}}\mu(y)
    \\
    &= \nabla_{\mu}f (1) g(1) - \nabla_{\mu}f (0) g(0) - \int \nabla_{\mu}f \nabla_{\mu} g \;\mathrm{d} \mu\end{aligned}$$ where $\int  \nabla_{\nu,\mu}g(y)\Delta_{\nu,\mu}f (y)\nu \left(\{y\} \right) \,{\mathrm{d}}\mu(y)=0$ since $\mu$ is atomless. Considering $g = f$ we obtain that $\Delta_{\mu,\nu}$ with domain $\mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\mu} )$, for $\gamma\coloneqq (\alpha,\beta) \in [0,\pi/2]^{2}$, is non-positive and since $$\begin{aligned}
    &\int (\Delta_{\nu,\mu}f )gd \nu-\int (\Delta_{\nu,\mu}g )f \mathrm{d} \nu\\
    &=-\tan(\beta)  \nabla_{\mu}f(1) \nabla_{\mu}g(1)-\tan(\alpha)  \nabla_{\mu}f(0) \nabla_{\mu}g(0)+\tan(\beta)  \nabla_{\mu}f(1) \nabla_{\mu}g(1)+\tan(\alpha)  \nabla_{\mu}f(0) \nabla_{\mu}g(0)
    =0,\end{aligned}$$ the boundary conditions force $\Delta_{\mu,\nu}$ also to be symmetric.

Main results
------------

In we establishes a strong connection between $\Delta_{\nu,\mu}$ and $\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}$. Indeed, by utilising the *pseudo-inverse* $$\begin{aligned}
    \check{F}^{-1}_{\mu}(x) \colon x \mapsto \inf\{ y \in [0,1] \colon F_{\mu}(y) \geq x \}
    \end{aligned}$$ of $F_{\mu}$, we prove, for $* \in \{s,w\}$ and $\gamma \in [0,\pi/2]^{2}$, that $\varphi: f \mapsto f \circ \check{F}_{\mu}^{-1}$ is an isometric isomorphism on $\S^{*}$ with $$\begin{aligned}
  \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda }\circ\varphi=\varphi\circ \Delta_{\nu,\mu }\quad \text{and}\quad\varphi ( \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu}))=\mathcal{D}^*_{\gamma}(\Delta_{\nu \circ F_{\mu}^{-1},\Lambda}).\end{aligned}$$ With this at hand, one may conclude that the spectral properties of $\Delta_{\nu \circ F_{\mu}^{-1},\Lambda}$ will be inherit from $\Delta_{\nu ,\mu}$ and vice versa. For instance, using our correspondence theorem (), we obtain that $\Delta_{\nu ,\mu}$ is a densely defined, self adjoint operator with compact resolvent. Further, we are able the prove the following.

1.  We obtain concerning the asymtotic growth rate of the eigenvalue counting function of $\Delta_{\nu,\mu}$, first provided in [@Fr05] for a certain class of self-similar measures. This is achieved via an application of and the identification of $\eta =  \nu \circ F_{\mu}^{-1}$ as a certain self-similar measure, in tandem with the corresponding result for the classical Kreĭn-Feller operator $\Delta_{\eta,\Lambda}$, see [@Fu87]. This generalises several of the results of [@KSW16], where the case $\nu=\mu$ was considered.

2.  Letting $(X_{t})_{t \geq 0}$ denote the a Liouville Brownian motion with speed measure $\nu \circ F_{\mu}^{-1}$ (see ), utilising our correspondence theorem (), we show that the infinitesimal generator of $(\check{F}^{-1}_{\mu}( X_t ))_{ t\geq 0}$ is the generalised Kreĭn-Feller operator $\Delta_{\nu,\mu}$ with Neumann boundary condition. Additionally, we compute the walk dimension of $(X_t)_{t \geq 0}$ and $(\check{F}^{-1}_{\mu}( X_t ))_{ t\geq 0}$.

Kreĭn-Feller operators
======================

Properties of classical Kreĭn-Feller operators
----------------------------------------------

In this section we collect important properties of classical Kreĭn-Feller operators, that is we consider the case $\mu= \Lambda$, with respect to weak and strong solutions. Most of these results are nowadays folklore and can be found for instance in [@MR0314125 Behauptung 2.2, Satz 2.1, Satz 3.1] with more or less detailed proofs. Since we could not find references where all the facts are proven in detail we decided to give here a quick overview and reduce all properties essentially on two key observations, namely the symmetry as a consequence of the Gauss-Green formula and surjectivity as stated in the following lemma.

\[Surjec\] For $\gamma=(\alpha, \beta)  \in [0,{\pi/2}] ^{2}$ and $* \in \{s,w\}$ we have that the map $\Delta_{\nu,\Lambda}: \mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\Lambda} ) \rightarrow \S_{\gamma}^* $ is surjective, where $\S_{(\pi/2,\pi/2)}^*\coloneqq\left\{ g \in \S^* : \int g {\mathrm{d}}\nu=0 \right\}$ and $\S_{\gamma}^* \coloneqq \S^
*$ for $\gamma\not=(\pi/2,\pi/2)$.

For $\gamma=(\alpha, \beta)  \in [0,{\pi/2}]^{2}\setminus\{(\pi/2,\pi/2)\}$ we have that $\Delta_{\nu,\Lambda}$ is also injective and for its inverse $\Delta_{\nu,\Lambda}^{-1}: \S_{\gamma}^* \rightarrow   \mathcal{D}^{*}_{\gamma}( \Delta_{\nu,\Lambda} )$ we have the following kernel representation $$\begin{aligned}
 \Delta_{\nu,\Lambda}^{-1}g:x\mapsto \int K_{\alpha,\beta}(x,y) g(y) \,{\mathrm{d}}\nu(y)\end{aligned}$$ with continuous kernel given by $$\begin{aligned}
 K_{\alpha,\beta}(x,y)\coloneqq 
 \begin{cases}
 A_{\alpha,\beta}(1+\tan(\beta)-y)( \tan(\alpha)+x)+{\mathbbm{1}}_{[0,x]}(y)(x-y) &\mbox{for $\alpha,\beta\in [0,\pi/2)$},\\
 (y-1)(x+\tan(\alpha) )+{\mathbbm{1}}_{[0,x]}(y)(x-y)   &\mbox{for $\beta=\pi/2$ and $\alpha \in [0,\pi/2)$},\\
 (y-1-\tan(\beta))+{\mathbbm{1}}_{[0,x]}(y)(x-y)   &\mbox{for  $\alpha= \pi/2$ and $\beta \in [0,\pi/2)$}.
 \end{cases}\end{aligned}$$ with $A_{\alpha,\beta}:= {-1}/(1+\tan(\alpha)+\tan(\beta))$, for $\alpha,\beta\in (0,\pi/2)$. For the Neumann case $\alpha=\beta=\pi/2$ the operator $\Delta_{\nu,\Lambda}$ is not injective with kernel $ \Delta_{\nu,\Lambda}^{-1}(\{0\})=\R {\mathbbm{1}}$.

We only consider the case $\alpha,\beta \in [0,\pi/2)$ the other cases can be proved along the same lines. For fixed $ g \in \S^{*}$ and $x \in [0,1]$ we set $$f(x):=b\tan(\alpha)+bx+\int_{[0,x]} (x-y)g(y) {\mathrm{d}}\nu(y),$$ with $$b\coloneqq\dfrac{-1}{(1+\tan(\alpha)+\tan(\beta))} \left(\tan(\beta)\int_{[0,1]}g(y) {\mathrm{d}}\nu(y)+
\int_{[0,1]} (1-y)g(y) {\mathrm{d}}\nu(y)\right).$$ This imposes the right boundary conditions on $f$ and consequently we have $f \in  \mathcal{D}^{w}_{\gamma}( \Delta_{\nu,\Lambda} ) $ with $\Delta_{\nu, \Lambda}f=g$.

As a direct consequence it follows that only in the Neumann case one has an eigenfunction with eigenvalue equal to zero.

Recall the following abstract facts on linear operators: Let us assume that $A: \operatorname{dom}(A) \subset H \rightarrow
H$ is a linear, symmetric and surjective operator on a Hilbert space $H$. Then one easily verifies that the annihilator of $\operatorname{dom}(A)$ is trivial. i.e. $\operatorname{dom}(A)^\perp=\{0\}$, and equivalently, $\operatorname{dom}(A)$ is dense in $H$. We can deduce further that $A$ is also self-adjoint: The inclusion $\operatorname{dom}(A) \subset \operatorname{dom}(A^*)$ holds by symmetry of $A$, where $A^*$ denotes the adjoint of $A$. For the reverse inclusion note that for fixed $f \in \operatorname{dom}(A^*)$ there exists, by surjectivity of $A$, an element $g \in \operatorname{dom}(A)$ such that $A^*f=Ag$. Then, using symmetry again, for each $h \in \operatorname{dom}(A)$, we have $\langle f ,A h \rangle = \langle  A^* f,h\rangle =
 \langle  A g,h \rangle =  
 \langle g ,  A h\rangle  $ and by surjectivity of $A$ we deduce $f=g \in \operatorname{dom}(A)$.

Now, we can apply this observation to our special setting, namely, for $\gamma \in [0,\pi/2]^2$ we set $H=\S_{\gamma}^w$, $A= \Delta_{\nu, \mu}$ and $\operatorname{dom}(A)={\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )\cap \S_{\gamma}^w$. In the case of $\gamma \in [0, \pi/2] \setminus \{ (\pi/2,\pi/2)\}$ it follows immediately that $ \Delta_{\nu, \mu}$ with domain ${\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )$ is a densely defined, self-adjoint operator on $L^{2}(\nu)$. In the case of Neumann boundary condition, i.e. $\gamma=(\pi/2,\pi/2)$, note that for the kernel we have $\Delta_{\nu,\ \mu}^{-1}(\{0\})=  \R\mathds{1}$ and hence $L^2(\nu) = \S_{\gamma}^w \oplus   \R\mathds{1}$. Therefore we have $ {\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )=\{f +a:f\in {\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )\cap \S_{\gamma}^w, a \in \R\}$ is dense in $L^2(\nu) $. Using this oberservation it follows again that $\Delta_{\nu, \mu} $ with domain ${\mathcal{D}}^w_{\gamma}(\Delta_{\nu, \mu} )$ is a densely defined and self-adjoint operator on $L^2(\nu) $. The following proposition summarizes this observation.

\[adjoint\] The partially defined operator $\Delta_{\nu,\Lambda}:L^{2}(\nu)\to L^{2}(\nu) $ with domain $ \mathcal{D}^{w}_{\gamma}( \Delta_{\nu,\Lambda}) $ for $\gamma  \in [0,{\pi/2}]^{2}$ is self-adjoint, non-positive and, in particular, closed.

Fix $\gamma=(\alpha,\beta) \in [0,\pi/2]^2\setminus\{ (\pi/2, \pi/2)\}$ then the operator $R_0\coloneqq  -\Delta_{\nu,\Lambda}^{-1}: \S_{\gamma}^w \to \S_{\gamma}^w$ is compact and self-adjoint.

Lemma \[Surjec\] shows that $R_0$ is a Hilbert-Schmidt operator with continuous (bounded) kernel function, therefore the compactness follows. Furher, the operator $R_0$ is self-adjoint this follows from symmetry of $\Delta_{\nu, \Lambda}$ in tandem with the fact that $R_0$ is also bounded.

\[Spectral\] Let $(\alpha,\beta) \in [0,\pi/2]^2$ , then the operator $\Delta_{\nu,\mu}$ with domain ${\mathcal{D}}_{\gamma}^{w}(\Delta_{\nu,\mu} )$ gives rise to an orthonormal (possibly finite) basis of eigenfunctions with eigenvalues $\lambda_{n}\leq 0$. If $L^{2}(\nu)$ is not finite dimensional then we have a countable number of eigenvalues with $\lim_{n \rightarrow \infty} -\lambda_n=\infty$, in particular $\Delta_{\nu,\Lambda}$ is an unbounded operator, otherwise there are only finitely many eigenfunctions and $\Delta_{\nu,\Lambda}$ is bounded.

At first we consider the case $\gamma=(\alpha,\beta) \in [0,\pi/2]^2\setminus\{ (\pi/2, \pi/2)\}$. Note that if $f \in {\mathcal{D}}^w_{\gamma}\left(\Delta_{\nu, \Lambda} \right)$ is an eigenfunction of $\Delta_{\nu, \Lambda}$ with eigenvalue $\lambda<0$, then applying Lemma \[Surjec\] gives $$\Delta_{\nu, \Lambda}f=\lambda f \Longleftrightarrow  \lambda^{-1} f = \Delta_{\nu, \Lambda}^{-1}f.$$ Then the statement follows directly from the spectral theorem for linear, compact and self-adjoint operators applied to $ \Delta_{\nu, \Lambda}^{-1}$.

For the case $\gamma=(\alpha,\beta)=(\pi/2,\pi/2)$ we have to consider the resolvent operator $R^{\lambda}_{\nu,\Lambda}\coloneqq(\lambda I -\Delta_{\nu,\Lambda})^{-1}$ with $\lambda>0$ , from the integral representation of the resolvent operator $R^{\lambda}_{\nu,\Lambda}\coloneqq(\lambda I -\Delta_{\nu,\Lambda})^{-1}$ of $\Delta_{\nu,\Lambda}$ with domain ${\mathcal{D}}^w_{\gamma}\left(\Delta_{\nu, \Lambda} \right)$ given in [@MR0314125 1.2], one may conclude that $R^{\lambda}_{\nu,\Lambda}$ is compact and self-adjoint, see also [@MR574035 Theorem 1, p. 251]. Again applying the spectral theorem proves the statement.

Clearly, $L^{2}(\nu)$ is finite dimensional if and only if the support of $\nu$ is a finite set.

\[densecm\] For $\gamma=(\alpha,\beta )\in [0,{\pi/2}]^{2} $, set $$\begin{aligned}
    C_{\nu,\Lambda}^{\gamma} 
    \coloneqq \begin{cases}
    \left\{ f \in C_{\nu,\Lambda}  \colon f(0)=0, \ f(1)=0\right\}& \text{ if } \alpha=\beta=0, \\
    \left\{ f \in C_{\nu,\Lambda}  \colon f(0)=0\right\}& \text{ if }  \alpha=0, \ \beta \in (0,\pi/2],\\
    \left\{ f \in C_{\nu,\Lambda}  \colon f(1)=0\right\} &\text{ if } \alpha \in (0,\pi/2], \ \beta=0, \\
     C_{\nu,\Lambda
     }& \text{ if } \alpha,\beta \in (0,\pi/2].
    \end{cases}
    \end{aligned}$$ The set $\mathcal{D}^s_{\gamma}(\Delta_{\nu, \, \Lambda})$ is dense in $\left(C_{\nu,\Lambda}^{\gamma}, \lVert \cdot \rVert_{\infty}\right)$.

The result can be found in [@MR0314125 Behauptung 2.4] without a detailed proof. We will sketch a proof of this result for the case $\gamma=(\alpha,\beta) \in (0,\pi/2) \times [0,\pi/2)$; the other cases follow in a similar fashion. Fix $\Phi \in C_{\mu,\Lambda}' $ such that $$\forall f \in \mathcal{D}^s_{\gamma}(\Delta_{\nu, \, \Lambda}): \
\Phi(f)=\int f(x) \,{\mathrm{d}}\phi(x)=0,$$ where $\phi$ is the signed distribution function representing $\Phi$. By definition $\phi$ is of bounded variation and local constant on the complement of ${\operatorname{supp}}(\nu)$. With $E:=-(1+\tan(\beta))A_{\alpha,\beta} $ it follows from the proof of Lemma \[Surjec\] for all $g \in C_{\nu,\Lambda}^{\gamma}$ $$\begin{aligned}
\int  (E+s\cdot A_{\alpha,\beta})g(s)\,{\mathrm{d}}\nu(s) \cdot \int (\tan(\alpha)+x) \,{\mathrm{d}}\phi(x)&= \int \int  (x-s)g(s) \,{\mathrm{d}}\nu (s)\,{\mathrm{d}}\phi(x) \label{eq:dual1}\\
&= \int  \left(\phi(1)(1-s)- \int_{[s,1]}\phi(s) \,{\mathrm{d}}s \right)g(s)\,{\mathrm{d}}\nu(s).\nonumber\end{aligned}$$ Furher, we have $$\int (\tan(\alpha)+x) \,{\mathrm{d}}\phi(x)=\tan(\alpha)\left( \phi(1)-\phi(0)\right)+\phi(1)-\int  \phi(s) \,{\mathrm{d}}s\eqqcolon B_{\phi}.$$ Combining these identities gives $$\begin{aligned}
\int  \left(E \cdot B_{\phi} +s \cdot  A_{\alpha,\beta} \cdot B_{\phi} -\phi(1)(1-s)+ \int_{[s,1]}\phi(s) \,{\mathrm{d}}s\right)g(s) \,{\mathrm{d}}\nu(s)=0\end{aligned}$$ If we consider $g(s):=E \cdot B_{\phi} +s \cdot  A_{\alpha,\beta} \cdot B_{\phi} -\phi(1)(1-s)+ \int_{[s,1]}\phi(x) \,{\mathrm{d}}x \in \mathcal{C}_{\nu,\Lambda}^{\gamma}$ it follows, that for all $s \in [0,1]$ $$\int_{[s,1]}\phi(x) \,{\mathrm{d}}x=\phi(1)-E \cdot B_{\phi} -s(   A_{\alpha,\beta} \cdot B_{\phi}+\phi(1)),$$

which is only possible if $\phi(s)=A_{\alpha,\beta} B_{\phi}+\phi(1)$, for all $s \in [0,1]$. Therefore $\Phi$ is a Dirac-measure in $\{0\}$ with weight $\phi(0)$ and we have by , or all $f \in \mathcal{C}_{\nu,\Lambda}^{\gamma}$, $$\phi(0) \ \tan(\alpha) \ \int\left(E+A_{\alpha,\beta} \cdot s\right)f(s) \,{\mathrm{d}}\nu(s)=0$$ For the particular choice $f \in \mathcal{C}_{\nu,\Lambda}^{\gamma}$ given by $f:s\mapsto E+A_{\alpha,\beta} \cdot s$ the above integral is positive and hence $
\phi(0)=0$. Consequently, $\Phi=0$ and we have shown that the annihilator of $  \mathcal{D}^s_{\gamma}(\Delta_{\nu, \Lambda}
)$ is trivial and therefore $  \mathcal{D}^s_{\gamma}(\Delta_{\nu, \, \Lambda})$ is dense in $ \mathcal{C}_{\nu,\Lambda}^{\gamma}$.

Generalized Kreĭn-Feller operators and transformations of measure spaces
------------------------------------------------------------------------

Let us first state two basic key observations. three key observation.

\[identity\] The function $\check{F}^{-1}_{\mu} \circ F_{\mu}$ equals the identity $\nu$-almost everywhere.

Note, $\check{F}^{-1}_{\mu}( F_{\mu}(x) )\neq x$ if and only if there exists $\varepsilon>0$ with $F_{\mu}(x-\varepsilon)=F_{\mu}(x)$. This means, if $\check{F}^{-1}_{\mu}( F_{\mu}(x) )\neq x$, then $x$ belongs to an interval of constancy for $F_\mu$. This in tandem with our hypothesis ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$, implies that the countable union of these intervals have $\nu$-measure zero.

\[lem: isomorphism\] The mapping $\varphi: \S^{*}(\nu,\mu) \to \S^{*}(\nu\circ {F}^{-1}_{\mu} ,\Lambda))$ defined by $\varphi(f) \coloneqq f \circ \check{F}^{-1}_{\mu}$, is an isometric isomorphism with inverse $\varphi^{-1}(f) \coloneqq f \circ F_{\mu}$, $*\in \{s,w\}$.

This is a consequence of together with the push-forward formula for measures in the strong case and the definition of $C_{\nu,\mu}$ in the weak case.

The following theorem is the main observation in this section needed for all subsequent corollaries.

\[prop:laplace\_backward\] For $\gamma \in [0,{\pi/2}]^{2}$, we have that $\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda }\circ\varphi=\varphi\circ \Delta_{\nu,\mu }$ and $\varphi ( \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu}))=\mathcal{D}^*_{\gamma}(\Delta_{\nu \circ F_{\mu}^{-1},\Lambda})$.

If $f \in  \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu})$, then, for all $x \in  {\operatorname{supp}}(\mu)$, $$\begin{aligned}
f(x)=a+bF_{\mu}(x)+
\int \mathds{1}_{[0, x]}(y)(F_{\mu}(x)-F_{\mu}(y) ) \Delta_{\nu, \mu } f(y) \;{\mathrm{d}}\nu(y),\end{aligned}$$ where $a=f(0)$ and $b= \nabla_{\mu} f(0)$. Using and replacing $x$ with $\check{F}_{\mu}^{-1}(x)$ gives $$\begin{aligned}
    f(\check{F}_{\mu}^{-1}(x))
    &=a+bF_{\mu}(\check{F}_{\mu}^{-1}(x))+
\int \mathds{1}_{[0, \check{F}_{\mu}^{-1}(x)]}(y)(F_{\mu}(\check{F}_{\mu}^{-1}(x))-F_{\mu}(y) ) \Delta_{\nu, \mu }(f )(y)\;{\mathrm{d}}\nu(y) \\
&=a+b x+
\int \mathds{1}_{[0, \check{F}_{\mu}^{-1}(x)]}(y)(x-F_{\mu}(y) ) \Delta_{\nu, \mu }(f )(\check{F}_{\mu}^{-1}(F_{\mu}(y)))\;{\mathrm{d}}\nu(y) \\
&=a+bx+
\int \mathds{1}_{[0, x]}(F_{\mu}(y))(x-F_{\mu}(y) ) \Delta_{\nu, \mu }(f )(\check{F}_{\mu}^{-1}(F_{\mu}(y)))\;{\mathrm{d}}\nu(y)  \\
&=a+bx+
\int \mathds{1}_{[0, x]}(y)(x-y ) \Delta_{\nu, \mu }(f ) \circ \check{F}_{\mu}^{-1}(y) \;{\mathrm{d}}(\nu \circ F_{\mu}^{-1})(y).\end{aligned}$$ Therefore, $a=f ( \check{F}^{-1}_{\mu}(0) )$ and $b=\nabla_{\mu}f ( \check{F}^{-1}_{\mu}(0) )$. This shows $$\begin{aligned}
    f \circ \check{F}_{\mu}^{-1} \in \mathcal{D}^*(\Delta_{\nu \circ F_{\mu}^{-1}, \, \Lambda})
    \quad \text{and} \quad
    \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} (f \circ \check{F}_{\mu}^{-1}) = \Delta_{\nu, \, \mu}f\circ \check{F}^{-1}_{\mu}.
    \end{aligned}$$ If $f \circ \check{F}_{\mu}^{-1} \in \mathcal{D}^*_{\alpha,\beta}(\Delta_{\nu \circ F_{\mu}^{-1}, \, \Lambda})$, then, as above, for $x \in {\operatorname{supp}}(\mu)$ with $F_{\mu}(x-\varepsilon)<F_{\mu}(x)$ for all $\varepsilon>0$, $$\begin{aligned}
    f( x )
    =f(\check{F}_{\mu}^{-1}( F_{\mu}( x)) )
    &= c+dF_{\mu}(x) + \int \mathds{1}_{[0, F_{\mu}(x)]}(y)(F_{\mu}(x)-y ) \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}(f   \circ \check{F}_{\mu}^{-1})(y) \;{\mathrm{d}}(\nu \circ F_{\mu}^{-1})(y)\\
    &=
c+dF_{\mu}(x)+
\int \mathds{1}_{[0, F_{\mu}(x)]}(F_{\mu}(y))(F_{\mu}(x)-F_{\mu}(y) ) \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}
(f   \circ \check{F}_{\mu}^{-1})
 \circ F_{\mu}(y)\;{\mathrm{d}}\nu (y)  \\ &=
c+dF_{\mu}(x)+
\int \mathds{1}_{[0, x]}(y)(F_{\mu}(x)-F_{\mu}(y) ) \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}
(f   \circ \check{F}_{\mu}^{-1})
 \circ F_{\mu}(y)\;{\mathrm{d}}\nu (y),\end{aligned}$$ with $c=f (\check{F}^{-1}_{\mu}(0 ))$ and $d =\nabla f (\check{F}^{-1}_{\mu} (0 ) )$. The case $x  \in {\operatorname{supp}}(\mu)$ and $F_{\mu}(x-\varepsilon)=F_{\mu}(x)$ for some $\varepsilon>0$ implies that $x$ lies in an interval of constancy. Notice, $$\begin{aligned}
    \check{F}_{\mu}^{-1}( [0,1])=\{0\} \cup {\operatorname{supp}}(\mu) \cap \{ x \in [0,1] \text{ is not a right endpoint of an interval of constancy}\},       \end{aligned}$$ thus we can consider a modification of $f$ such that $f$ is constant on each interval of constancy of $F_{\mu}$. Thus, we have $c=f(0), \ d=\nabla_{\mu}f(0)$. Therefore, we have $$\begin{aligned}
    \Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}(f   \circ \check{F}_{\mu}^{-1} ) \circ F_{\mu}
    = \Delta_{\nu,\mu} ( f ) \quad \text{and} \quad f\in  \mathcal{D}^{*}(\Delta_{\nu, \mu}).
    \end{aligned}$$ It remains to verify that the boundary conditions are preserved. Combining the above with gives $$\begin{aligned}
 \nabla_{\mu}f(1)= \nabla_{\mu}f(0)+\int_{[0,1]} \Delta_{\nu,\mu}f(y)\;{\mathrm{d}}\nu(y) 
 &= \nabla_{\Lambda}f(0)+\int_{[0,1]} \Delta_{\nu,\mu}f ( \check{F}_{\mu}^{-1}(y) )\;{\mathrm{d}}\nu \circ F_{\mu}^{-1}(y) \\
 &= \nabla_{\Lambda}f(0)+\int_{[0,1]} 
 \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} (f \circ \check{F}_{\mu}^{-1})
 \;{\mathrm{d}}\nu \circ F_{\mu}^{-1}(y)
 = \nabla_{\Lambda} (f \circ \check{F}_{\mu}^{-1} )(1).  \qedhere\end{aligned}$$

For $\gamma=(\alpha,\beta )\in [0,{\pi/2}]^{2} $, set $$\begin{aligned}
    C_{\nu,\mu}^{\gamma} 
    \coloneqq \begin{cases}
    \left\{ f \in C_{\nu,\mu}  \colon f(0)=0, \ f(1)=0\right\}& \text{ if } \alpha=\beta=0, \\
    \left\{ f \in C_{\nu,\mu}  \colon f(0)=0\right\}& \text{ if }  \alpha=0, \ \beta \in (0,\pi/2],\\
    \left\{ f \in C_{\nu,\mu}  \colon f(1)=0\right\} &\text{ if } \alpha \in (0,\pi/2], \ \beta=0, \\
     C_{\nu,\mu
     }& \text{ if } \alpha,\beta \in (0,\pi/2].
    \end{cases}
    \end{aligned}$$ The set $\mathcal{D}^s_{\gamma}(\Delta_{\nu, \, \mu})$ is dense in $(C_{\nu,\mu}^{\gamma}, \lVert \cdot \rVert_{\infty})$.

This follows from , Lemma \[densecm\] and Theorem \[prop:laplace\_backward\].

For each $\gamma \in [0,{\pi/2}]^{2}$, the operator $\Delta_{\nu, \, \mu}$ with domain ${\mathcal{D}}_{\gamma}^w(\Delta_{\nu,\mu})$ is densely defined and self-adjoint.

The denseness follows from , Lemma \[adjoint\] and Theorem \[prop:laplace\_backward\]. Furhermore, for $f \in \operatorname{dom}(\Delta_{\nu,\mu}^*)$ by Theorem \[prop:laplace\_backward\] and Lemma \[identity\], $$\begin{aligned}
g\mapsto
\left\langle  f   ,\Delta_{\nu,\mu} g  \right\rangle_{L^2(\nu)}
=\left\langle f\circ \check{F}^{-1}_{\mu} ,  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda} (g \circ \check{F}^{-1}_{\mu})   \right\rangle_{L^2(\nu \circ F_{\mu}^{-1})}
 \end{aligned}$$ defines a continuous linear functional on $ \operatorname{dom}(\Delta_{\nu,\mu})$. Combining Proposition \[adjoint\] and Theorem \[prop:laplace\_backward\] we deduce $ f  \circ \check{F}^{-1}_{\mu} \in \operatorname{dom}\big(  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda}^{*}\big)= \operatorname{dom}\big(  \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda}\big)={\mathcal{D}}_{\gamma}^w( \Delta_{\nu \circ \check{F}^{-1}_{\mu},\Lambda})$ and consequently $f  \in {\mathcal{D}}_{\gamma}^w(\Delta_{\nu,\mu})=\operatorname{dom}(\Delta_{\nu,\mu})$.

There is an analogous theorem in general theory of Markov processes \[[@DynkinI_II], p. 325, Theorem 10.13\]. Taking into account that $\Delta_{\nu,\mu}$ with domain ${\mathcal{D}}_{\gamma}^{*}(\Delta_{\nu,\mu})$ has a probabilistic interpretation as infinitesimal generator of a Markov process, thus the result above is not surprising.

Corollary \[Spectral\] implies that $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$ with domain ${\mathcal{D}}_{\gamma}^w \left( \Delta_{\nu, \Lambda} \right)$ gives rise to an orthonormal basis of eigenfunctions with non-positive eigenvalues. In the case of ${\operatorname{supp}}(\nu)$ is infinite we have $(\lambda_n)_{n \in \mathbb{N}}$ with $\lim_{n \to \infty}-\lambda_n = \infty$ otherwise there are only finitely many eigenvalues. Using the one-to-one correspondence established in to relate the spectral properties of $\Delta_{\nu,\mu}$ with those of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$, we obtain the following.

\[thm Spec\] For fixed $ \gamma \in [0,{\pi/2}]^{2}$, the operators $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $ with domain $\mathcal{D}^w_{\gamma}( \Delta_{\nu \circ F_{\mu}^{-1},\Lambda} )$ and $ \Delta_{\nu,\mu}  $ with domain $\mathcal{D}^w_{\gamma}( \Delta_{\nu,\mu})$ have the same eigenvalues $(\lambda_n)_{n \in \mathbb{N}}$. Further, if $f$ is an eigenfunction of $ \Delta_{\nu,\mu} $, then $f \circ \check{F}^{-1}_{\mu}$ is an eigenfunction of $ \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $, and if $f$ is an eigenfunction of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$, then $f \circ F_{\mu}$ is an eigenfunction of $ \Delta_{\nu,\mu}$. In particular, if $(f_n)_{n \in \mathbb{N}}$ denotes the orthonormal basis consisting of eigenfunctions of $ \Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda}$ then $(f_n \circ F_{\mu})_{n \in \mathbb{N}}$ forms an orthonormal basis consisting of $ \Delta_{\nu,\mu}$-eigenfunctions.

can be seen as a generalisation of [@KSW16] where the $\Delta_{\mu,\mu}$-Laplacian has been considered.

\[Reso\] Assume that ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$ and $\gamma \in[0,{\pi/2}]^{2}$. For $\lambda>0$, letting $R_{\nu , \mu}^{\lambda}$ denote the resolvent operator of $ \Delta_{\nu, \mu}$ with domain ${ \mathcal{D}^*_{\gamma}(\Delta_{\nu,\mu})}$, for all $f \in \S^*$ $$\begin{aligned}
R_{\nu,\mu}^{\lambda}(  f ) \circ \check{F}_{\mu}^{-1}=R_{\nu \circ F_{\mu}^{-1},\Lambda}^{\lambda}(f \circ \check{F}_{\mu}^{-1}).\end{aligned}$$ In particular, we have $$\begin{aligned}
\big \Vert R_{\nu \circ F_{\mu}^{-1},\Lambda}^{\lambda} \big\Vert_{L^2(\nu \circ F_{\mu}^{-1} ) }=\big \Vert R_{\nu,\mu}^{\lambda} \big\Vert_{L^2 (\nu) },\end{aligned}$$ and $$\begin{aligned}
\big \Vert R^{\lambda}_{\nu \circ F_{\mu}^{-1},\Lambda} \big\Vert_{C_{\nu \circ F_{\mu}^{-1},\Lambda}}
=
\big\Vert R^{\lambda}_{\nu,\mu}\big\Vert_{C_{\nu,\mu}}\end{aligned}$$

Note that the resolvent $R^{\lambda}_{\nu,\mu}$ is well defined for all $\lambda>0$ (see [@MR2017701]) and for $f \in \S^*$ we have $$\begin{aligned}
\Delta_{\nu,\mu} R_{\nu ,\mu}^{\lambda}f
=\lambda R_{\nu, \mu}^{\lambda}f-f, \  R_{\nu ,\mu}^{\lambda}f \in \mathcal{D}^*_{ \gamma}( \Delta_{\nu,\mu} ) .\end{aligned}$$ gives $$\begin{aligned}
\Delta_{\nu\circ F_{\mu}^{-1},\Lambda} (R_{\nu ,\mu}^{\lambda}f\circ \check{F}^{-1}_{\mu}) \circ {F}_{\mu}
=\lambda R_{\nu,\mu}^{\lambda}f-f\end{aligned}$$ and hence $$\begin{aligned}
\Delta_{\nu\circ F_{\mu}^{-1},\Lambda} (R_{\nu ,\mu}^{\lambda}f\circ \check{F}^{-1}_{\mu}) 
=\lambda R_{\nu,\mu}^{\lambda}f\circ \check{F}_{\mu}^{-1} -f \circ \check{F}_{\mu}^{-1}\end{aligned}$$ proving the first part. The statement on the norms now follows from .

\[Feller\] Let $\gamma \in[0,{\pi/2}]^{2}$. Then the operator $\Delta_{\mu,\nu}$ with domain $ {\mathcal{D}}_{\gamma}^s(\Delta_{\nu,\mu})$ is an infinitesimal generator of a strongly continuous semi-group of contraction on $C_{\nu,\mu}^{\gamma}$.

It is well known that this holds true for the classical Krein-Feller operator [@MR0314125 Behauptung 4.1]. This in tandem with the Yosida-Hille Theorem [@ma1992introduction p. 11, Theorem 1.12] and , yields $$\begin{aligned}
 \big\Vert R^{\lambda}_{\nu \circ F_{\mu}^{-1},\Lambda} \big\Vert_{C^{\gamma}_{\nu \circ F_{\mu}^{-1},\Lambda}}
=
 \big\Vert R^{\lambda}_{\nu,\mu} \big\Vert_{C_{\nu,\mu}^{\gamma}} \leq 1/\lambda ,\end{aligned}$$ for all $\lambda>0$. Further $\Delta_{\nu, \mu}$ is densely defined and closed operator in $C_{\nu,\mu}^{\gamma}$. Applying again the theorem of Yosida-Hille gives the statement.

Applications
============

Spectral asymptotics
--------------------

In this section we review the asymptotic spectral properties of $\Delta_{\nu,\mu}$ for certain classes of self-similar measures. We will show how the results in [@Fr05] can be deduced from [@Fu87] with the help of the above established isomorphism. Let us give general assumption on the self-similar measures which are clearly fulfilled under the assumptions (A.1) – (A.4) of [@Fr05].

\[StrongAssumption\] Let $S_i:[0,1]\to [0,1]$, $i=1,\dots,M$, denote a family of affine contractions fulfilling the open set condition (OSC), that is, $S_i((0,1)) \cap S_j((0,1))= {\varnothing}$, for all $j \neq i$, and let $\nu$ denote the associated self-similar measure with probability weight vector $(p_{1},\ldots ,p_{M}) \in (0,1)^M$ uniquely determined by $$\begin{aligned}
\nu(A)=\sum_{i=1}^{M} p_i \nu(S_i^{-1}(A))\end{aligned}$$ for $A \in \mathfrak{B}([0,1])$. For fixed $(\sigma_{1},\ldots ,\sigma_{M})\in (0,1)^{M}$ with $\sum_{i=1}^M \sigma_i \leq 1$ let $\mu$ denote an atomless probability measure with ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu)$ such that for all $A \in \mathfrak{B}([0,1])$, $i=1,\dots ,M$, we have $\mu(S_i(A))=\sigma_i \mu(A)$.

\[Freiberg\] For the decreasing sequence of eigenvalues $( \lambda_n )_{n \in \mathbb{N}}$ of $\Delta_{\nu,\mu}$ with domain $ \mathcal{D}^{w}_{\gamma}(\Delta_{\nu ,\mu})$, $\gamma\in [0,\pi/2]^{2}$, we have $$\begin{aligned}
  -\lambda_n\asymp n^{1/ u} \quad \text{and} \quad N_{\Delta_{\nu,\mu}}(x)\asymp x^{ u},\end{aligned}$$ where $ u\in (0,1)$ denotes the unique number with $\sum_{i=1}^M ( \sigma_i p_i )^{ u}=1.$

We have $\sigma_i F_{\mu}(x)=\sigma_i \mu([0,x])= \mu(S_i( [0,x] ) )=F_{\mu}(S_i(x))- F_{\mu}(S_i(0))$, for all $x \in [0,1]$, $i \in \{1,\dots,M\}$. Setting ${\widetilde}{S}_i(x)\coloneqq\sigma_i x+F_{\mu}(S_i(0))$ we obtain $F_{\mu} \circ S_i= {\widetilde}{S}_i \circ F_{\mu}$, and hence, for all $A \in \mathfrak{B}([0,1])$, $$\begin{aligned}
\sum_{i=1}^M p_i\nu( F_{\mu}^{-1} \circ {\widetilde}{S}_i^{-1}(A)) 
&=\sum_{i=1}^M p_i\nu(  S_i^{-1} \circ F_{\mu}^{-1}(A))
=\nu ( F_{\mu}^{-1}(A) ). \end{aligned}$$ This shows that $ \nu \circ F_{\mu}^{-1}$ is the unique self similar measure with contractions $( {\widetilde}{S}_i )_{i=1,\dots N}$ and weights $(p_i)_{i=1,...,M}$. Further, $$\begin{aligned}
{\widetilde}{S}_i( [0,1] )=[F_{\mu}(S_i(0)),\sigma_i+F_{\mu}(S_i(0))]=[F_{\mu}(S_i(0)),F_{\mu}(S_i(1))],\end{aligned}$$ which shows that the OSC is satisfied for $( {\widetilde}{S}_i \colon i=1,\dots ,M )$ given that $( {S}_i \colon i=1,\dots ,M )$ satisfies the OSC. Therefore, we can apply the classical result from Fujita [@Fu87] for spectral dimension of $\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} $ with domain ${\mathcal{D}^{w}_{\gamma} (\Delta_{\nu \circ {F}_{\mu}^{-1}, \, \Lambda} )}$. Combining this with completes the proof.

Let $S_i(x) \coloneqq s_ix+b_i$ for $x \in [0,1]$ with $s_i,b_i \geq 0$ and $s_i+b_i \leq 1$ for $i=1,\dots,M$ and assume the OSC. For $N\leq M$ let $\nu$ denote the self-similar measure with respect to $(S_{i}\colon i=1,\ldots ,N)$ and probability weight vector $(p_{1},\ldots , p_{N})\in (0,1)^{N}$ and support $L=\bigcup_{i=1}^N S_i(L)$, and $\mu$ the self-similar measure with respect to $(S_{i}\colon i=1,\ldots ,M)$ and probability weight vector $(\sigma_{1},\ldots , \sigma_{M})\in (0,1)^{M}$ and support $K=\bigcup_{i=1}^M S_i(K)$. Then we have $L \subset K$ and $$\begin{aligned}
\nu \circ F_{\mu}^{-1}(A)= \sum_{i=1}^N p_i \nu(S_i^{-1}\circ F_{\mu}^{-1}(A)).\end{aligned}$$ If for $\delta_{K}\coloneqq \dim_{H}(K)$ and $\delta_{L}\coloneqq \dim_{H}(L)$ we chose $\sigma_{m}\coloneqq s_{m}^{\delta_{K}}$, $m =1,\ldots, M$, and $p_{\ell}\coloneqq s_{\ell}^{\delta_{L}}$, $\ell =1,\ldots, L$, then we find the well-known relation $ u=\delta_{K}/(\delta_{K}+\delta_{L})$.

Liouville Brownian motion and walk dimension {#sec:LBM}
--------------------------------------------

In this section we will construct Liouville Brownian motion via a time change of a fixed Brownian motion. We will give a short overview about basic properties of this stochastic process. After which, we will compute the walk dimension for special classes of gap diffusions. Additionally, we show that the infinitesimal generator coincides with a generalised Kreĭn-Feller operator.

For the tuple $\left(\Omega, \mathcal{F}, ( \mathcal{F}_t)_{t \geq 0}, ( \theta_t)_{t \geq 0}, \mathbb{P}_x\right)$ where $\mathbb{P}_x$ denotes the probability measure such that, $( \mathcal{F}_t)_{t \geq 0}$ is the right-continuous completed natural filtration and $(\theta_t )_{t \geq 0}$ is the shift-operator. The expectation with respect to $\mathbb{P}_x$ is denoted by $\mathbb{E}_x$. We call the stochastic process $(B_t )_{t \geq 0}$ a Brownian motion if the following are satisfied.

1.  $\mathbb{P}_x(B_0=x)=1$

2.  For $0 \leq s_0< \dots <s_n$ with $n \in \N$ the increments $B_{s_1}-B_{s_0}, \dots ,B_{s_n}-B_{s_{n-1}}$ are stochastically independent.

3.  The distribution of the increment $B_{s}-B_{t}$ follows a Gaussian distribution with variance $s-t$ where $s>t \geq 0$ and mean $0$.

Let $(L_{x}^t)_{t \geq 0, x \in \mathbb{R}}$ the jointly continuous version of the local time of the Brownian motion, see [@Revuz2013 Chapter VI].

Let $m$ denote a non-zero Borel measure on $(\R,\mathcal{B})$.

\[LBM\] We define for $ t \geq 0$ the [*(inverse) time-change function*]{} $$\begin{aligned}
\Phi_t \coloneqq \int_{\R} L_x^t \;{\mathrm{d}m}(x),\;\; \ \hat{\Phi}^{-1}_t \coloneqq \inf\left\{s \geq 0 \colon \Phi_s>t\right \}\end{aligned}$$ where $\hat{\Phi}^{-1}_t$ is called right-continuous inverse of $\Phi_t$. Now, we define the new process for $x \in {\operatorname{supp}}(m)$ $$\begin{aligned}
\left((X_t)_{ t \geq 0} \coloneqq (B_{\hat{\Phi}^{-1}_t})_{t \geq 0}, (\mathcal{F}_{\hat{\Phi}^{-1}_t})_{t \geq 0},  \mathbb{P}_x\right )\end{aligned}$$ which will be called $m$-Liouville Brownian motion with speed measure $m$ starting in $x$.

\[DefWalkD\] Let $(X_t)_{t \geq 0}$ a Liouville Brownian motion with speed measure $m$. Then the *walk dimension* $d_w(x)$ in $x \in {\operatorname{supp}}(m)$ is defined by $$\begin{aligned}
\label{WalkD}
d_W(x) \coloneqq 
\lim_{R \to \infty} \dfrac{\log(\mathbb{E}_x[\tau_{(-R+x,R+x)}]  )}{\log(R)}\end{aligned}$$ assuming that the limit exists, where $\tau_{(-R+x,R+x)} \coloneqq  \inf \{ t \geq 0 \colon X_t \notin (-R+x,R+x)\}$

Assuming the support of $m$ is bounded, then the walk dimension does not exist. In this case it can be useful to consider the so-called *local walk dimension* $$\begin{aligned}
d_{LW}(x) \coloneqq    \lim_{R \downarrow 0} \dfrac{\log(\mathbb{E}_x[\tau_{(-R+x,R+x)}]  )}{\log(R)}.\end{aligned}$$ There are other definitions of the walk dimension; for example in [@KGOLMANKHANEH2018960] the walk dimension $d_{w}(x)$ is defined by $$\begin{aligned}
\mathbb{E}_x [ (X_t-x  )^2  ] \asymp t^{2/d_w(x)}.\end{aligned}$$

It remains open in which cases this definition coincides with the Definition \[DefWalkD\].

\[SkipfreeP\] For all $t \geq 0$, we have $\mathbb{P}_x$-almost everywhere, that $X_t \in {\operatorname{supp}}(m)$.

We have $\left((X_t )_{t \geq 0}, (\mathcal{F}_{\hat{\Phi}^{-1}_t})_{t \geq 0}, (\theta_{\hat{\Phi}^{-1}_t} )_{t \geq 0}, ( \mathbb{P}_x )_{x \in {\operatorname{supp}}(m)}\right)$ defines a Feller process. Namely, the tuple defines is a strong Markov process such that, for all $f \in C_b ({\operatorname{supp}}(m) )$, the function $x \mapsto \mathbb{E}_x[f(X_t) ]$ belongs to $C_b({\operatorname{supp}}(m) )$ and $\lim_{ t \downarrow 0} \mathbb{E}_x[f(X_t)] =f(x)$. Here, $C_b ({\operatorname{supp}}(m) )$ denotes the set of bounded continuous function with domain ${\operatorname{supp}}(m)$.

To compute the walk dimension and infinitesimal generator, we need the following lemma.

\[PotentialGapD\] Fix $x_0,x_1 \in {\operatorname{supp}}(m)$ with $x_0<x_1$ and set $\tau_{(x_0,x_1)} \coloneqq \inf\{ t \geq 0 \colon X_t \notin (x_0,x_1)\}$. For $x \in (x_0,x_1) \cap {\operatorname{supp}}(m)$, the following holds.

1.  For all $t \geq 0$ and all bounded measurable functions $f$ on ${\operatorname{supp}}(m)$ we have $$\begin{aligned}
    \qquad\quad\mathbb{E}_x\left[ \int_0^{\tau_{[x_0,x_1]}}f(X_s)\mathrm{d}s \right]=\int_{[x_0,x_1]} G_{x_0,x_1}(x,y)f(y)\;{\mathrm{d}m}(y),
    \quad \text{where} \quad
    G_{x_0,x_1}(x,y) \coloneqq  2\dfrac{(x \wedge y-x_0)(x_1-x \vee y)}{x_1-x_0}
    $$ for $x,y \in (x_0,x_1)$. In particular, we obtain $\mathbb{E}_x [ \tau_{(x_0,x_1)} ] < \infty$.

2.  We have $$\begin{aligned}
    \mathbb{P}_x\left(X_{\tau_{(x_0,x_1)}} =x_0\right)=\dfrac{x_1-x}{x_1-x_0}
    \quad \text{and} \quad
    \mathbb{P}_x\left(X_{\tau_{(x_0,x_1)}} =x_1\right)=\dfrac{x-x_0}{x_1-x_0}.\end{aligned}$$

3.  We have for $x<x_1$ with $x,x_1 \in {\operatorname{supp}}(m)$ $$\begin{aligned}
    \mathbb{E}_x\left[ \int_0^{\tau_{(-\infty,x_1]}}f(X_s)ds\right]=\int_{(-\infty,x_1]}2(x_1-x \vee y)f(y)\;{\mathrm{d}m}(y)\end{aligned}$$ and for $x_2 \in supp(m)$ such that $x_2<x$ $$\begin{aligned}
    \mathbb{E}_x\left[ \int_0^{\tau_{(x_2,\infty)}}f(X_s)ds\right]=\int_{[x_2,\infty)}2(x \wedge y-x_2)f(y)\;{\mathrm{d}m}(y).\end{aligned}$$

The proof for (i) and (ii) one can find in [@Burkhardt1983], p. 42, Lemma 2.4.5. The proof of (iii) follows in a similar way as for (ii) taking into account that the following hold for $a<x,y<b$ $$\mathbb{E}_x[ L_y^{\tau_{(-\infty,a)}} ]=2(y \wedge x-a) 
\quad \text{and} \quad
\mathbb{E}_x[ L_y^{\tau_{(b,\infty)}} ]=2(b-y \vee x).
\qedhere$$

Now, we will give the connection between generalised Kreĭn-operators $D_{\nu}D_{\mu}$ and the infinitesimal generator of transformed gap diffusions. We consider, Borel measures $\mu,\nu$ with compact support and $
\nu( \{0,1\})=0$ and ${\operatorname{supp}}(\nu) \subseteq {\operatorname{supp}}(\mu) \subset [0,1]$. In the case if $F_{\mu}$ is strictly increasing see [@Burkhardt1983] p. 64-65. The only difference to the calculation in [@Burkhardt1983] is the replacement of the inverse with the pseudo-inverse and ${\operatorname{supp}}(\mu) \neq [0,1]$ if $F_{\mu}$ is not strictly increasing. Now, let $(X_t)_{ t \geq 0}$ a Liouville Brownian motion with speed measure $\nu \circ F_{\mu}^{-1}$. We call $Y_t \coloneqq \check{F}_{\mu}^{-1}(X_t)$ (generalised) $\nu$-$\mu$-Liouville Brownian motion with speed measure $\nu \circ F_{\mu}^{-1}$ which is again strong Markov process thus it is a deterministic transformation of $(X_t)_{t \geq 0}$.

Let $(Z_t)_{t \geq 0}$ denote a Markov process with compact state space $E$ and let $C(E)$ denote the continuous functions on $E$. A function $f \in C(E)$ is said to belong the domain $\mathcal{D}(A)$ of the infinitesimal generator of $(Z_t)_{t \geq 0}$ if the limit $$\begin{aligned}
    Af(x)=\lim_{t \downarrow 0} ( \mathbb{E}_x[f(Z_t) ]-f(x))/t
    \end{aligned}$$ exists in $C(E)$ with respect to $\lVert \cdot \rVert_{\infty}$.

Now, we compute the infinitesimal generator of $(Y_t)_{t \geq 0}$.

\[GapDSec\] Let $(X_t)_{ t \geq 0}$ a $\nu \circ F_{\mu}^{-1}$-Liouville Brownian motion and let $A$ denote the infinitesimal generator of $(Y_t)_{t \geq 0}=(\check{F}^{-1}_{\mu}(X_t))_{t \geq 0}$. For $f \in \mathcal{D}(A)$ there exists a continuous continuation of $f$ in $C_{\nu,\mu}$ (also denoted by $f$) such that $$\begin{aligned}
\label{SecondOrder}
f(x)=f(0)
+\int_{0}^x (F_{\mu}(x)-F_{\mu}(y) )2Af(y)d\nu(y), \; \; x \in \R,  \end{aligned}$$ and the Neumann boundary condition $\nabla_{\mu} f(0)=\nabla_{\mu} f(1)=0$ are satisfied.\

We will present two proofs. The first proof uses and a result of [@Burkhardt1983]. The second proof will make use of Lemma \[PotentialGapD\].

Let $A_Y$ the infintesimale generator of $(Y_t)_{t \geq 0}$ on $C_{\nu,\mu}$ and $A_X$ the infinitesimal generator of $(X_t)_{t \geq 0}$ on $C_{\nu \circ F_{\mu}^{-1},\Lambda}$. At first note that we have for $f \circ \check{F}_{\mu}^{-1} \in \mathcal{D}(A_X)$ and $x \in F_{\mu}( {\operatorname{supp}}(\nu))$ $$\begin{aligned}
\lim_{t \downarrow 0} ( \mathbb{E}_x[ f( \check{F}_{\mu}^{-1}(X_t))-f(\check{F}^{-1}_{\mu}(x)) ] )/t
=\lim_{t \downarrow 0} ( \mathbb{E}_{\check{F}_{\mu}^{-1}(x)}[ f  (Y_t) -f(\check{F}^{-1}_{\mu}(x))] )/t
=\lim_{t \downarrow 0} ( \mathbb{E}_{\check{F}_{\mu}^{-1}(x)}[ f  (Y_t) -f(Y_0)] )/t\end{aligned}$$ From this, it follows that $$\begin{aligned}
f \circ \check{F}_{\mu}^{-1} \in \mathcal{D}(A_X) \rightarrow 
f  \in \mathcal{D}(A_Y)
\quad \text{and} \quad
A_X(f \circ \check{F}_{\mu}^{-1})=A_Y ( f) \circ \check{F}_{\mu}^{-1}.\end{aligned}$$ From [@Burkhardt1983 p.49], it is known that $2A_X(f)=\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}(f), \ f \in \mathcal{D}(A_X)$ with $\nabla_{\mu} f(0)=\nabla_{\mu} f(1)=0$. This in tandem with and the fact that $A_y(f) \in C_{\nu,\mu}$ yields $$\Delta_{\nu,\mu}(f)=
\Delta_{\nu \circ F_{\mu}^{-1}, \Lambda}(f \circ \check{F}_{\mu}^{-1})\circ F_{\mu}=2A_Y(f)
\quad \text{and} \quad
\nabla_{\mu} f \circ  \check{F}_{\mu}^{-1}(0)=\nabla_{\mu} f \circ \check{F}_{\mu}^{-1}(1)=0.
\qedhere$$

First, note that ${\operatorname{supp}}(\nu \circ F_{\mu}^{-1})=F_{\mu}({\operatorname{supp}}(\nu))$. Therefore, we have for all $t \geq 0$ $  \check{F}^{-1}_{\mu}(X_t) \in  \check{F}^{-1}_{\mu}(F_{\mu}({\operatorname{supp}}(\nu)) ) $ almost everywhere by . We will show that holds on $ \check{F}^{-1}_{\mu}(F_{\mu}({\operatorname{supp}}(\nu)) )$ which determines $f$ uniquely. To see this note that by we have $\nu( \check{F}^{-1}_{\mu}(F_{\mu}({\operatorname{supp}}(\nu)) )\cap {\operatorname{supp}}(\nu))=1$ proving that the set $\check{F}^{-1}_{\mu}(F_{\mu}({\operatorname{supp}}(\nu)) )$ intersects ${\operatorname{supp}}(\nu)$ densely. Let $x_1 \in \check{F}^{-1}_{\mu}(F_{\mu}({\operatorname{supp}}(\nu)) )\cap {\operatorname{supp}}(\nu)$. Clearly, $\check{F}^{-1}_{\mu}(F_{\mu}(x_{1}))=x_{1}$. Applying (iii) and yields $$\begin{aligned}
\mathbb{E}_{F_{\mu}(0^+)}\left[ \int_{0}^{\tau_{(-\infty,F_{\mu}(x_1))}} 2Af\circ \check{F}_{\mu}^{-1}(X_s)\;{\mathrm{d}}s  \right]
&=\int_{[F_{\mu}(0^+),F_{\mu}(x_1)]}(F_{\mu}({x_1})-y ) 2Af \circ \check{F}_{\mu}^{-1}(y)\;{\mathrm{d}}\nu \circ F_{\mu}^{-1}(y) \\
&=\int_{[0,x_1]}(F_{\mu}(x_1)-F_{\mu}(y)) 2Af(y)\;{\mathrm{d}}\nu (y).\end{aligned}$$ On other site applying Dynkin’s formula ([@Revuz2013], p. 284 Proposition 1.5) and using the fact that $Af$ is bounded and $\tau_{(-\infty,F_{\mu}(x_1))}$ is integrable yields $$\begin{aligned}
    \mathbb{E}_{F_{\mu}(0^+)}\left[ \int_{0}^{\tau_{(-\infty,F_{\mu}(x_2))}} 2Af\circ \check{F}_{\mu}^{-1}(X_s) \;{\mathrm{d}}s  \right]
    =\mathbb{E}_{F_{\mu}(x_0)}[ f (\check{F}_{\mu}^{-1}(X_{\tau_{(-\infty,F_{\mu}(x_1))}}) )]-f( \check{F}_{\mu}^{-1}(F_{\mu}(0^+)))
    =f(x_1)-f( \check{F}_{\mu}^{-1}(F_{\mu}(0^+))).\end{aligned}$$ Therefore, we have $$\begin{aligned}
\label{Neumann}
f(x_1)=f( \check{F}_{\mu}^{-1}(F_{\mu}(0^+)))+\int_{[0,x_1]}(F_{\mu}(x_1)-F_{\mu}(y)) 2Af(y)\;{\mathrm{d}}\nu (y).\end{aligned}$$ This combined with $f(0) = f (\check{F}^{-1}_{\mu} (F_{\mu}(0^+ )))$ shows for all $x\in  {\operatorname{supp}}(\nu)$ and implies $\nabla_{\mu}f(0)=0$. In the same way applying the second part of Lemma \[PotentialGapD\] (iii) for $x=1^-$ and $x_2<1$ gives $$f(x_1)-f(1)=\int_{[x_2,1]} 2 ( F_{\mu}(1)-F_{\mu}(y))Af(y) d \nu(y).$$ Therefore, it follows $\nabla_{\mu}f(1)=0$.

In following we consider the case $\mu=\nu$. The occupation formula of the local time yields for $ t \geq 0$ $$\begin{aligned}
\Phi_t=\int L^{t}_x\;{\mathrm{d}}\nu \circ F_{\mu}^{-1}(x)=
\int L^{t}_x \;{\mathrm{d}}\Lambda(x)
=\int_{0}^t \textbf{1}_{[0,1]}(B_s) \;{\mathrm{d}}\Lambda(s).\end{aligned}$$ Now, for the case that $\mu$ is the $1/3$-Cantor, we demonstrate in Fig. \[fig\] a simulation of a $\mu$-$\mu$-Liouville Brownian path. First we need a simulation of a standard Brownian path $B_{t}$ as depicted in Fig. \[fig\](), the corresponding time-change function $\Phi_t$ with respect to $\Lambda$, see Fig. \[fig\](), and the associated $\Lambda$-Liouville Brownian path $B_{\check{\Phi}^{-1}_{t}}$ as shown in Fig. \[fig\](). Finally, its image under $\check{F}_{\mu}^{-1}$ then shows a realisation of a $\mu$-$\mu$-Liouville Brownian path as shown in Fig. \[fig\]().

[0.49]{}

[0.49]{}

\

[0.49]{}

[0.49]{}

The above theorem allows us to consider the set of (continuous) elements $f$ from $\mathcal{D}(2A)$. With this at hand we observe the following.

$\mathcal{D}(A)=\mathcal{D}^s_{\pi/2, \pi/2}( \Delta_{\nu,\mu})$

We have $\mathcal{D}(A) \subseteq \mathcal{D}^s_{\pi/2, \pi/2 }( \Delta_{\nu,\mu} )$ and $2Af= \Delta_{\nu,\mu}f$ for all $f \in \mathcal{D}(A)$ by . The operator $\Delta_{\nu,\mu}$ restricted to $ \mathcal{D}^s_{\pi/2, \pi/2 }( \Delta_{\nu,\mu} )$ is an infinitesimal generator of a Feller process (). Therefore, [@Revuz2013 Exercise 1.18] shows that the transition functions of both processes are equal.

With we are able to give easy condition to calculate the walk dimension (under some additional assumptions).

\[ThmWalkDim\] Let $(X_s)_{s \geq 0}$ a $m$-Liouville Brownian motion starting in $x \in {\operatorname{supp}}(m)$. If, for $R(r)\coloneqq \inf\{s\geq r\colon s\in {\operatorname{supp}}(m)\}$ and $L(r)\coloneqq \inf\{s\geq r\colon -s\in {\operatorname{supp}}(m)\}$, we have $$\begin{aligned}
\lim_{r \to \infty}\frac{\log R(r)}{\log r} =\lim_{r \to \infty}\frac{\log L(r)}{\log r}=1,\end{aligned}$$ then, for $t\geq 0$, $$\begin{aligned}
\label{Condition1WD}
d_W(x)=1+t\iff \lim_{r \to \infty} \frac{\log{m((-r,r))} }{\log r}=t.\end{aligned}$$ In particular, if the walk dimension exists, then it is independent of the starting point in ${\operatorname{supp}}(m)$.

Set $S(r)\coloneqq R(r+x)\wedge L(r-x) $ and $T(r)\coloneqq R(r+x)\vee L(r-x)$. Using and estimating the Green kernel gives $$\begin{aligned}
(S(r)/2) m((-S(r)/2,S(r)/2)) \leq 
\mathbb{E}_x [\tau_{(x-r,x+r)} ]
\leq {T(r)} m((-T(r),T(r))).\end{aligned}$$ Taking logarithms, using our hypothesis, and dividing by $\log(r)$ then proves the equivalence.

These ideas would also carry over to calculate the local walk dimension.

Assume $m$ is a finite and non-zero measure on $[0,1)$, let ${\widetilde}{m} = m \star \delta_{\mathbb{Z}}$ and let $(X_t)_{t \geq 0}$ denote a Liouville Brownian motion with speed measure ${\widetilde}{m}$. For all $x \in {\operatorname{supp}}( {\widetilde}{m})$, we have $d_W(x)=2$.

Since $ {\widetilde}{m}((-r+x,r+x)) =2r\cdot m([0,1))+O(1)$ and $L(r-x)-r,R(r+x)-r\in [0, 1]$, the claim is a consequence of with $t=1$.

Let $\nu$ be a Borel measure on $\R$ and $ F_{\mu}$ is a non-decreasing and continuous function on $\R$ with associated Lebesgue-Stieltjes measure $\mu$ such that ${\operatorname{supp}}(\nu) \subset {\operatorname{supp}}(\mu)$. Denote by $(X_t)_{t \geq 0}$ the $m$-Liouville Brownian motion with $m \coloneqq \nu \circ F_{\mu}^{-1}$ . Let $d_W$ denote the walk dimension of $(X_t)_{t \geq 0}$ and $\check{d}_W$ denote the walk dimension of $(\check{F}_{\mu}^{-1}(X_t))_{t \geq 0}$. For $x \in  {\operatorname{supp}}(m)$, if $$\begin{aligned}
 \forall r \in \mathbb{R} :
F_{\mu}(-r)=-F_{\mu}(r), \
a \coloneqq \lim_{r \rightarrow \infty} \dfrac{\log F_{\mu}(r) }{\log r}>0 \quad \mbox{ and }\quad
\lim_{r \rightarrow \infty} \dfrac{\log R(r) }{\log r}=\lim_{r \rightarrow \infty} \dfrac{\log L(r) }{\log r}=1,\end{aligned}$$ then $$\begin{aligned}
 \lim_{r \rightarrow \infty} \frac{\log{m(\left(-r,r\right))} }{\log r}=t 
\iff
 1+t=d_W\left(x\right)=
a^{-1} \cdot  \check{d}_W(\check{F}_{\mu}^{-1}(x)).\end{aligned}$$

Let $x \in \check{F}_{\mu}^{-1}\left({\operatorname{supp}}(m)\right)$ and, for $r>0$, define $$R^{\check{F}_{\mu}^{-1}}(r)
    \coloneqq \inf\{ s \geq r \colon s \in \check{F}_{\mu}^{-1}({\operatorname{supp}}(m))\}
    \quad \text{and} \quad
    L^{\check{F}_{\mu}^{-1}}(r)\coloneqq \inf\{ s \geq r \colon -s \in \check{F}_{\mu}^{-1}({\operatorname{supp}}(m)) \}.$$ By assumption $(F_{\mu}(-r)=-F_{\mu}(r))$ and so $$\begin{aligned}
    F_{\mu}(R^{\check{F}_{\mu}^{-1}}(r+x))
= R(F_{\mu}(x+r)), \ F_{\mu}(L^{\check{F}_{\mu}^{-1}}(r-x))
= L(F_{\mu}(r-x)),
        \end{aligned}$$ therefore we have $$\begin{aligned}
\inf\{ t \geq 0 \colon \check{F}_{\mu}^{-1}\left(X_t \right) \in \{ R^{\check{F}^{-1}}(r+x),L^{\check{F}^{-1}}(r-x)\}
&= \inf\{ t \geq 0 \colon X_t \in \{ R(F_{\mu}(r+x)),L(F_{\mu}(r-x))\}\end{aligned}$$ Define for $r>0$ large $$m(r,x):=\min \{F_{\mu}(r+x)-F_{\mu}(x),F_{\mu}(r-x)-F_{\mu}(x)\}, \ M(r,x):=\max \{F_{\mu}(r+x)-F_{\mu}(x),F_{\mu}(r-x)-F_{\mu}(x)\},$$ then $$\begin{aligned}
 \mathbb{E}_{F(x)}[ 
\tau_{(F_{\mu}(x)-m(r,x),F_{\mu}(x)+m(r,x) )} ] \leq \mathbb{E}_{F(x)}[ \tau_{(F_{\mu}(-r+x),F_{\mu}(r+x))}] \leq \mathbb{E}_{F_{\mu}(x)}[  \tau_{(F_{\mu}(x)-M(r,x),F_{\mu}(x)+M(r,x) )} ].\end{aligned}$$ The statement follows now with help of Theorem \[ThmWalkDim\] and the assumption on $F_{\mu}$.

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