Gaussian fields, equilibrium potentials and multiplicative chaos for Dirichlet forms

Abstract

For a Dirichlet form $(\mathcal {E},\mathcal {F})$ on L²(E;m), let $\mathbb {G}(\mathcal {E})=\{X_{u};u\in \mathcal {F}_{e}\}$ be the Gaussian field indexed by the extended Dirichlet space $\mathcal {F}_{e}$. We first solve the equilibrium problem for a regular recurrent Dirichlet form $\mathcal {E}$ of finding for a closed set B a probability measure μ^B concentrated on B whose recurrent potential $R\mu ^{B}\in \mathcal {F}_{e}$ is constant q.e. on B (called a Robin constant). We next assume that E is the complex plane $\mathbb {C}$ and $\mathcal {E}$ is a regular recurrent strongly local Dirichlet form. For the closed disk $\bar B(\textbf {x},r)=\{\textbf {z}\in \mathbb {C}:|\textbf {z}-\textbf {x}|\le r\}$, let μ^{x, r} and f(x, r) be its equilibrium measure and Robin constant. Denote the Gaussian random variable $X_{R\mu ^{\textbf {x}.r}}\in \mathbb {G}(\mathcal {E})$ by Y^{x, r} and let, for a given constant γ > 0, $\mu _{r}(A,\omega )={\int \limits }_{A} \exp (\gamma Y^{\textbf {x},r}-(1/2)\gamma ^{2} f(\textbf {x},r))d\textbf {x}.$ Under a certain condition on the growth rate of f(x, r), we prove the convergence in probability of μ_r(A, ω) to a random measure $\overline {\mu }(A,\omega )$ as r ↓ 0. The possible range of γ to admit a non-trivial limit will then be examined in the cases that $(\mathcal {E}.\mathcal {F})$ equals $(\frac 12{\textbf {D}}_{\mathbb {C}},H^{1}(\mathbb {C}))$ and $(\textbf {a},H^{1}(\mathbb {C}))$, where a corresponds to the uniformly elliptic partial differential operator of divergence form.

Appendices

Appendix: Proof of Proposition 4.7

1.1 Proof of Proposition 4.7 (i)

Using the function $\check r_{1}(\textbf {x},\textbf {y}),\textbf {x},\textbf {y}\in F,$ in Eq. 4.13, define

$$\check {r_{1}^{1}}(\textbf{x},\textbf{y})=\check r_{1}(\textbf{x},\textbf{y}),\quad \check {r_{1}^{n}}(\textbf{x},\textbf{y})=\int \check r_{1}(\textbf{x},\textbf{z})\check r_{1}^{n-1}(\textbf{z},\textbf{y})m_{F}(d\textbf{z}),\ n\ge 2.$$

$\check {r_{1}^{n}}(\textbf {x},\textbf {y})$ is the density function of the kernel $\check R^{n}(\textbf {x},d\textbf {y})$ on $(F,{\mathcal B}(E))$ with respect to m_F. $\check {R_{1}^{n}}(\textbf {x},d\textbf {y})$ is m_F-symmetric and $\check {R_{1}^{n}}1_{F}(\textbf {x})=1, \textbf {x}\in F,$ so that $\widetilde m_{F}\check {R_{1}^{n}}=\widetilde m_{F}.$ Consequently,

$$\check {R_{1}^{n}}(\textbf{x},A)-\widetilde m_{F}(A)={\int}_{F} [\check {R_{1}^{n}}(\textbf{x},A)-\check {R_{1}^{n}}(\mathbf{x}^{\prime},A)]\widetilde m_{F}(d\mathbf{x}^{\prime}),\quad A\in {\mathcal B}(F).$$

Denote by ||μ|| the total variation of a signed measure μ on F, We then get from the above identity and an estimate [14, (3.4)]

$$ \sup_{\textbf{x}\in F}||\check {R_{1}^{n}}(\textbf{x},\cdot)-\widetilde m_{F}(\cdot)||\le 2\gamma^{n},\quad \text{for some constant}\ \gamma\in (0,1). $$

(7.1)

Therefore, if we let

$$ \check r^{(\pm)}(\textbf{x},A)={\int}_{A} {\sum}_{n=1}^{\infty} (\check {r_{1}^{n}}(\textbf{x},\textbf{y})-1/m(F))^{\pm} m_{F}(d\textbf{y}),\ \textbf{x}\in F,\ A\in {\mathcal B}(F), $$

(7.2)

then, $\check r^{(+)}(\textbf {x},A),\ \check r^{(-)}(\textbf {x},A)$ are positive kernels on $(F,{\mathcal B}(F))$ satisfying $\sup _{\textbf {x}\in F}\check r^{(\pm )}(\textbf {x},F)<\infty$ and, for any $\varphi \in L^{\infty }(F;m_{F})$,

$$ \check R\varphi(\textbf{x})={\int}_{F}\check r^{(+)}(\textbf{x},d\textbf{y})\varphi(\textbf{y}) - {\int}_{F}\check r^{(-)}(\textbf{x},d\textbf{y})\varphi(\textbf{y})\ \text{for}\ m\text{-a.e.}\ \textbf{x}\in F. $$

(7.3)

on account of Eq. 4.14. This identity can be readily verified to hold also for φ ∈ L²(F; m_F).

Define $r^{(\pm )}(\textbf {x},\textbf {y})={\int \limits }_{F}{\int \limits }_{F\times F} r^{g}(\textbf {x},\textbf {z}) {\check r}^{(\pm )}(\textbf {z}, d\textbf {w}) r^{g}(\textbf {w},\textbf {y}) m_{F}(d\textbf {z})$ for $\textbf {x},\textbf {y}\in \mathbb {C}$, r^(±)(x, y) are symmetric and $\mathbb {M}^{g}$-excessive for each variable x and y. ${\int \limits }_{\mathbb {C}} r^{(+)}(\textbf {x},\textbf {y})h(\textbf {y})m(d\textbf {y})$ is finite for each $\textbf {x}\in \mathbb {C}$ for any non-negative bounded Borel function h on $\mathbb {C}$ vanishing outside a bounded set, because R^gh is bounded on $\mathbb {C}$ by Lemma 3.1 and so $\psi (\textbf {z})={\int \limits }_{F} \check r^{(+)}(\textbf {z},d\textbf {w})R^{g}h(\textbf {w})$ is bounded on F by a constant C > 0, and furthermore ${\int \limits }_{\mathbb {C}} r^{(+)}(\textbf {x},\textbf {y})h(\textbf {y})m(d\textbf {y})= R^{g}(1_{F}\cdot \psi )(\textbf {x})\le C R^{g}g(\textbf {x})=C$ in view of [14, (3.28)]. Therefore r^(±)(x, y) is finite for m-a.e.y and hence q.e.$\textbf {y}\in \mathbb {C}$.

We see from Eq. 4.14 that, for φ ∈ L²(F; m_F), $\check R\varphi =\check R_{1}\varphi -{\langle }\widetilde m_{F},\varphi \>+ \check R_{1}\check R\varphi$. Consider any $\mu \in {\mathcal S}_{0}^{g,(0)}$ with $\mu (\mathbb {C})<\infty .$ Since $R^{g}\mu \in L^{2}(\mathbb {C},m_{F})$ and ${\langle }\widetilde m_{F},R^{g}\mu \>=\frac {1}{m(F)}{\langle }R^{g}g, \mu \>= \mu (\mathbb {C})/m(F),$ we have

$$ H_{F}\check R(1_{F}R^{g}\mu)(\textbf{x})=R^{g}(1_{F}R^{g}\mu)(\textbf{x})-\mu(\mathbb{C})/m(F) + R^{g}(1_{F}\check R(1_{F}R^{g}\mu))(\textbf{x}),\quad \textbf{x}\in \mathbb{C}, $$

which combined with Eqs. 4.12 and 7.3 implies that Rμ admits an expression Eq. 4.26 by a kernel $\frak r(\textbf {x},\textbf {y})$ defined by

$$ \begin{array}{@{}rcl@{}} \frak r(\textbf{x},\textbf{y})&=&{\int}_{F} r^{g}(\textbf{x},\textbf{z})r^{g}(\textbf{z},\textbf{y})m_{F}(d\textbf{z})+ r^{(+)}(\textbf{x},\textbf{y})-r^{(-)}(\textbf{x},\textbf{y})\\&&+r^{g}(\textbf{x},\textbf{y})-2/m(F),\quad \textbf{x}, \textbf{y}\in \mathbb{C}. \end{array} $$

(7.4)

$\frak r(\textbf {x},\textbf {y})$ is symmetric and, for each $\textbf {x}\in \mathbb {C},$ it is a difference of M^g-excessive functions finite for q.e. $\textbf {y}\in \mathbb {C}.$ This property for the first term of the righthand side can be verified in a similar way to the proof for other terms given previously.

Proof of Proposition 4.7 (ii)

We take x, y ∈ B(S − 1) with |x −y| > η. Since there exists a constant M₂ such that $|\widehat R\mu ^{\textbf {y},r_{2}}|\leq M_{2}$ on B(x, η/8) for any r₂ < η/8 by Lemma 4.6, the stated uniform boundedness of ${\langle }\mu ^{\textbf {x},r_{1}}, R\mu ^{\textbf {y},r_{2}}\> ={\langle }\mu ^{\textbf {x},r_{1}}, \widehat R\mu ^{\textbf {y},r_{2}}\>$ holds true. To prove Eq. 4.27, we first show that

$$ \lim_{r_{1},r_{2}\downarrow 0} {\langle}\mu^{\textbf{x},r_{1}}, R^{g}\mu^{\textbf{y},r_{2}}\>= r^{g}(\textbf{x},\textbf{y}), $$

(7.5)

for m × m-a.e. (x, y) ∈ B(S − 1) × B(S − 1) ∩{(x, y) : |x −y| > η}.

Since $e^{-t} p_{t}(\textbf {z},\textbf {w})\leq {p^{g}_{t}}(\textbf {z},\textbf {w})\leq p_{t}(\textbf {z},\textbf {w})$, we can use Eq. 4.1 to find for any ε > 0 a positive t₀ satisfying

$$ {{\int}_{0}^{t}} {p^{g}_{s}}(\textbf{z},\textbf{w})ds \leq {{\int}_{0}^{t}} {K_{2}\over s} e^{-9 k_{2} \eta^{2}/16s}ds <{\varepsilon} $$

(7.6)

for any t ≤ t₀ and z, w ∈ E such that |z −w| > 3η/4. In particular, ${{\int \limits }_{0}^{t}} {\langle }\mu ^{\textbf {x},r_{1}}, {P^{g}_{s}}\mu ^{\textbf {y},r_{2}}\>ds<{\varepsilon }$.

Let M₁ be a constant satisfying $R^{g} \mu ^{\textbf {y},r_{2}}\leq M_{1}$ on $\mathbb {C}\setminus B(\textbf {y},\eta /2)$ for all r₂ ≤ η/8. Such constant M₁ exists by Lemma 4.6. By Eq. 4.1 and the tail estimate Eq. 4.32, we may assume that, by taking smaller t₀ > 0 if necessary,

$$ {\int}_{\mathbb{C}\setminus B(\textbf{x},\eta/2)}{p^{g}_{t}}(\textbf{z},\textbf{w})m(d\textbf{w})\leq {\int}_{\{|\textbf{w}|>3\eta/8\}} {K_{2}\over t} e^{-k_{2}|\textbf{w}|^{2}/t} d\textbf{w} \leq K_{2}\sqrt{\pi/k_{2}t} e^{-9\eta^{2}k_{2}/64t}< {{\varepsilon}\over M_{1}} $$

for all t ≤ t₀ and z ∈ B(x, η/8). In particular,

$$ \begin{array}{@{}rcl@{}} &&{\langle}\mu^{\textbf{x},r_{1}} {P^{g}_{t}}, 1_{\mathbb{C}\setminus B(S-1/2)} R^{g} \mu^{\textbf{y},r_{2}}\>\\ &&=\int\int {p^{g}_{t}}(\textbf{z},\textbf{w}) 1_{\mathbb{C}\setminus B(S-1/2)}(\textbf{w}) R^{g}\mu^{\textbf{y},r_{2}}(\textbf{w})m(d\textbf{w})\mu^{\textbf{x},r_{1}}(d\textbf{z})\\ &&\leq M_{1} {\int}_{B(\textbf{x},\eta/8)} \mu^{\textbf{x},r_{1}}(d\textbf{z}){\int}_{\mathbb{C}\setminus B(\textbf{x},\eta/2)}{p^{g}_{t}}(\textbf{z},\textbf{w})m(d\textbf{w})<{\varepsilon}. \end{array} $$

(7.7)

Put $D(\textbf {y})=B(S-1/2)\setminus \overline {B(\textbf {y},\eta /2)}$. Since ${p^{g}_{t}}(\textbf {z},\textbf {w})\leq (K_{2}/t)e^{-9k_{2}\eta ^{2}/64t}$ for any z ∈ B(x, η/8) and w ∈ B(y, η/2) and R^g1_B(S− 1) ≤ M₄ on $\mathbb {C}$ for some constant M₄ by Lemma 3.1 (i),

$$ {\langle}\mu^{\textbf{x},r_{1}} {P^{g}_{t}}, 1_{B(\textbf{y},\eta/2)} \cdot R^{g}\mu^{\textbf{y},r_{2}}\> \!\leq\! {K_{2}\over t} e^{-9k_{2}\eta^{2}/64 t} {\langle}\mu^{\textbf{y},r_{2}},R^{g} 1_{B(\textbf{y},\eta/2)}\> \!\leq\! {K_{2} M_{4}\over t} e^{-9k_{2}\eta^{2}/64t}\!<\!{\varepsilon} $$

(7.8)

for any t < t₀ by taking smaller t₀ if necessary. Further, since the distance between F and B(x, η/8) exceeds 1/2, we get by putting $A_{t}={{\int \limits }_{0}^{t}} 1_{F}(X_{s})ds$,

$$ \begin{array}{@{}rcl@{}} P_{t_{0}}(\textbf{z}, D(\textbf{y}))-P^{g}_{t_{0}}(\textbf{z},D(\textbf{y}))&=&{\mathbb E}_{\textbf{z}}\left[(1-e^{-A_{t_{0}}})1_{D(\textbf{y})}(X_{t_{0}})\right]\leq {\mathbb E}_{\textbf{z}}\left[{\int}_{0}^{t_{0}} 1_{F}(X_{s}) ds\right]\\ &\leq& {\int}_{0}^{t_{0}} {K_{2} m(F)\over s} e^{-k_{2}/4s}ds \quad\text{for any}\ \textbf{z}\in B(\textbf{x},\eta/8). \end{array} $$

Hence we may also assume that

$$ {\langle}\mu^{\textbf{x},r_{1}}, (P_{t}-{P^{g}_{t}})(1_{D} R^{g}\mu^{\textbf{y},r_{2}})\><{{\varepsilon}} \quad\text{for all}\ t\leq t_{0}, $$

(7.9)

because $R^{g} \mu ^{\textbf {x},r_{2}}\le M_{1}$ on D(y).

Therefore, in the decomposition

$$ \begin{array}{@{}rcl@{}} {\langle}\mu^{\textbf{x},r_{1}}, R^{g} \mu^{\textbf{y},r_{2}}\>&=&{{\int}_{0}^{t}}{\langle}\mu^{\textbf{x},r_{1}}, {P^{g}_{s}}\mu^{\textbf{y},r_{2}}\>ds+{\langle}\mu^{\textbf{x},r_{1}} ,{P^{g}_{t}} R^{g}\mu^{\textbf{y},r_{2}}\>\\ &=&{{\int}_{0}^{t}}{\langle}\mu^{\textbf{x},r_{1}}, {P^{g}_{s}}\mu^{\textbf{y},r_{2}}\>ds+{\langle}\mu^{\textbf{x},r_{1}} {P^{g}_{t}}, 1_{\mathbb{C}\setminus B(S-1/2)} \cdot R^{g}\mu^{\textbf{y},r_{2}}\>\\ &&+{\langle}\mu^{\textbf{x},r_{1}} {P^{g}_{t}}, 1_{B(\textbf{y},\eta/2)} \cdot R^{g}\mu^{\textbf{y},r_{2}}\>+{\langle}\mu^{\textbf{x},r_{1}}, ({P^{g}_{t}}-P_{t})(1_{D(\textbf{y})} R^{g}\mu^{\textbf{y},r_{2}})\>\\ &&+{\langle}\mu^{\textbf{x},r_{1}},P_{t}(1_{D(\textbf{y})} R^{g}\mu^{\textbf{y},r_{2}})\>, \end{array} $$

the sum of the first four terms of the righthand side is smaller than 4ε for any r₁, r₂ ∈ (0, η/8) and t ≤ t₀.

Since p_t(z, w) is uniformly continuous relative to (z, w) on $\overline {B(\textbf {x},\eta /8)}\times \overline {D}(\textbf {y})$, by putting $\delta (t,r_{1})=\sup \{|p_{t}(\textbf {z},\textbf {w})-p_{t}(\textbf {x},\textbf {w})|:\textbf {z}\in \overline {B(\textbf {x},r_{1})},\textbf {w}\in \overline {D}(\textbf {y})\}$, we can see that the difference of the last term of the righthand side and ${\int \limits }_{D(\textbf {y})} p_{t}(\textbf {x},\textbf {w}) R^{g}\mu ^{\textbf {y},r_{2}}(\textbf {w})m(d\textbf {w})$ is smaller than M₁δ(t, r₁) which converges to zero as r₁ ↓ 0 for each t < t₀. Furthermore, for $f^{\textbf {x}}_{t}(\textbf {w})=1_{D(\textbf {y})}(\textbf {w})p_{t}(\textbf {x},\textbf {w})$, $R^{g} f_{t}^{\textbf {x}}$ is $\mathcal {E}$-harmonic on B(y, η/8) by Lemma 3.1 and continuous there as in the proof of Lemma 4.4. Consequently, $\lim _{r_{2}\to 0}{\int \limits }_{D(\textbf {y})} p_{t}(\textbf {x},\textbf {w}) R^{g}\mu ^{\textbf {y},r_{2}}(\textbf {w})m(d\textbf {w})=\lim _{r_{2}\to 0}{\langle }\mu ^{\textbf {y},r_{2}}, R^{g} f_{t}^{\textbf {x}}\>=R^{g} f_{t}^{\textbf {x}}(\textbf {y}).$ Accordingly

$$ \limsup_{r_{1},r_{2}\downarrow 0} |{\langle}\mu^{\textbf{x},r_{1}}, R^{g}\mu^{\textbf{y},r_{2}}\>-R^{g} f_{t}^{\textbf{x}}(\textbf{y})|<4{\varepsilon} $$

(7.10)

for any t ≤ t₀ and any x, y ∈ B(S − 1) with |x −y| > η.

Thus, to verify Eq. 7.5, it suffices to show that $\lim _{t\to 0} R^{g} f_{t}^{\textbf {x}}(\textbf {y})=\lim _{t\to 0} P_{t}(1_{D(\textbf {y})}\cdot r^{g}(\cdot ,\textbf {y}))(\textbf {x})=r^{g}(\textbf {x},\textbf {y})$ for m × m-a.e.(x, y) ∈ B(S − 1) × B(S − 1) ∩{|x −y| > η}. For any y ∈ B(S − 1), let $E_{1}(\textbf {y})=\{\textbf {x}:r^{g}(\textbf {x},\textbf {y})<\infty \}.$ As $\mathbb {C}\setminus E_{1}(\textbf {y})$ is polar and

$$ {P^{g}_{t}}(1_{D(\textbf{y})}r^{g}(\cdot,\textbf{y}))(\textbf{x})\leq\ P_{t}(1_{D(\textbf{y})}r^{g}(\cdot,\textbf{y}))(\textbf{x})\leq e^{t} {P^{g}_{t}}(1_{D(\textbf{y})}r^{g}(\cdot,\textbf{y}))(\textbf{x}), $$

it is enough to show that $\lim _{t\to 0} {P^{g}_{t}}(1_{D(\textbf {y})}\cdot r^{g}(\cdot ,\textbf {y}))(\textbf {x})=r^{g}(\textbf {x},\textbf {y})$ for any x ∈ D(y) ∩ E₁(y). Since r^g(⋅, y) is $\mathbb {M}^{g}$-excessive and 1_D(y)(X_t)r^g(X_t, y) is right continuous at t = 0 a.s.ℙ_x for x ∈ D(y) ∩ E₁(y), we have

$$ \begin{array}{@{}rcl@{}} r^{g}(\textbf{x},\textbf{y})\wedge n&=&\lim_{t\to 0} {\mathbb E}_{\textbf{x}}^{g}\left[1_{D(\textbf{y})}(X_{t}) r^{g}(X_{t},\textbf{y})\wedge n\right] \leq \mathop{\underline{\lim}}_{t\to 0} {\mathbb E}_{\textbf{x}}^{g}\left[1_{D(\textbf{y})}(X_{t}) r^{g}(X_{t},\textbf{y})\right]\\ &\leq&\mathop{\overline{\lim}}_{t\to 0} {\mathbb E}_{\textbf{x}}^{g}\left[1_{D(\textbf{y})}(X_{t}) r^{g}(X_{t},\textbf{y})\right] \leq\lim_{t\to 0} {\mathbb E}_{\textbf{x}}^{g}\left[r^{g}(X_{t},\textbf{y})\right]=r^{g}(\textbf{x},\textbf{y}),\ \ n\ge 1. \end{array} $$

By letting $n\to \infty$, we arrive at Eq. 7.5.

We shall next show that, for the kernels r⁺(x, y) and r⁻(x, y) appearing in the proof of Proposition 4.7 (i),

$$ \lim_{r_{1},r_{2}\downarrow 0} {\langle}\mu^{\textbf{x},r_{1}}, R^{(\pm)}\mu^{\textbf{y},r_{2}}\>= r^{(\pm)}(\textbf{x},\textbf{y}), $$

(7.11)

for m × m-a.e. (x, y) ∈ B(S − 1) × B(S − 1). Here we let $R^{(\pm )}\mu (\textbf {x})={\int \limits }_{\mathbb {C}} r^{(\pm )}(\textbf {x},\textbf {z})\mu (d\textbf {z}),\ \textbf {x}\in \mathbb {C}.$ Consider the function on $\mathbb {C}$ defined by $\textbf {x}i^{\textbf {y},r_{2}}_{+}(\textbf {z})=1_{F}(\textbf {z}){\check R}^{(+)}(1_{F}R^{g}\mu ^{\textbf {y},r_{2}})(\textbf {z}),\ \textbf {z}\in \mathbb {C}.$ Since $R^{g}\mu ^{\textbf {y},r_{2}}(\textbf {z})$ is bounded in z ∈ F and r₂ by Lemma 4.6 (i) and $\check {R}^{(+)}$ is a bounded linear operator on $L^{\infty }(F;m_{F})$, there exists a constant M > 0 such that for any $\textbf {z}\in \mathbb {C}, r_{2}\in (0,\eta /8)$,

$$ \textbf{x}i^{\textbf{y},r_{2}}_{+}(\textbf{z})\le M,\ R^{g}\textbf{x}i_{+}^{\textbf{y},r_{2}}(\textbf{z})={\int}_{F} r^{g}(\textbf{z},\textbf{w})\textbf{x}i_{+}^{\textbf{y},r_{2}}(\textbf{w})m(d\textbf{w})=H_{F}(R^{g}\textbf{x}i_{+}^{\textbf{y},r_{2}})(\textbf{z})\le M. $$

(7.12)

In view of definition, we have the identity $R^{(+)} \mu ^{\textbf {y},r_{2}}=R^{g} \textbf {x}i_{+}^{\textbf {y},r_{2}}$. Accordingly, as in the previous proof of Eq. 7.5, we can decompose ${\langle }\mu ^{\textbf {x},r_{1}}, R^{(+)}\mu ^{\textbf {y},r_{2}}\>$ as

$$ \begin{array}{@{}rcl@{}} {\langle}\mu^{\textbf{x},r_{1}},R^{(+)}\mu^{\textbf{y},r_{2}}\> &=&{{\int}_{0}^{t}}{\langle}\mu^{\textbf{x},r_{1}}, {P^{g}_{s}}\textbf{x}i_{+}^{\textbf{y},r_{2}}\>ds+{\langle}\mu^{\textbf{x},r_{1}} {P^{g}_{t}}, 1_{\mathbb{C}\setminus B(S-1/2)} \cdot R^{g}\textbf{x}i_{+}^{\textbf{y},r_{2}}\>\\ &&+{\langle}\mu^{\textbf{x},r_{1}}, ({P^{g}_{t}}-P_{t})(1_{B(S-1/2)} R^{g}\textbf{x}i_{+}^{\textbf{y},r_{2}})\> \\&&+{\langle}\mu^{\textbf{x},r_{1}},P_{t}(1_{B(S-1/2)} R^{g}\textbf{x}i_{+}^{\textbf{y},r_{2}})\>. \end{array} $$

For any ε > 0, we can take t₁ such that the first term of the righthand side is less than ε for any t ∈ (0, t₁) as Eq. 7.6 because of dist(F, B(x, r₁)) > 1/2 and the bound Eq. 7.12. Because also of the bound Eq. 7.12, we can take t₁ such that the second term is less than ε for any t ∈ (0, t₁) as Eq. 7.7. Further, as Eq. 7.9, we may suppose that the third term is less than ε for all t ≤ t₁.

Since $\sup \{|p_{t}(\textbf {z},\textbf {w})-p_{t}(\textbf {x},\textbf {w})|:\textbf {z}\in B(\textbf {x},r_{1}), \textbf {w}\in B(S-1/2)\}\to 0$ as r₁ → 0, $\lim _{r_{1}\to 0} {\langle }\mu ^{\textbf {x},r_{1}},P_{t}(1_{B(S-1/2)}R^{g} \textbf {x}i^{\textbf {y},r_{2}}_{+})\> = P_{t}(1_{B(S-1/2)}R^{g} \textbf {x}i^{\textbf {y},r_{2}}_{+})(\textbf {x})$ uniformly in r₂ ≤ η/8. Put $h^{\textbf {x}}_{t}(\textbf {w})=1_{F}(\textbf {w}){\check R}^{(+)}R^{g} (1_{B(S-1/2)} p_{t}(\cdot , \textbf {x}))(\textbf {w})$. Since $h^{\textbf {x}}_{t}$ vanishes outside of F, we can see as before that $R^{g}h^{\textbf {x}}_{t}(\textbf {w})$ is continuous on B(S − 1) and consequently

$$ \lim_{r_{2}\to 0} P_{t}(1_{B(S-1/2)}R^{g} \textbf{x}i^{\textbf{y},r_{2}}_{+})(\textbf{x})=\lim_{r_{2}\to 0}{\langle}\mu^{\textbf{y},r_{2}}, R^{g} h^{\textbf{x}}_{t}\>=R^{g} h^{\textbf{x}}_{t}(\textbf{y}). $$

Therefore, as Eq. 7.10, $\limsup _{r_{1},r_{2}\downarrow 0} |{\langle }\mu ^{\textbf {x},r_{1}},R^{(+)}\mu ^{\textbf {y},r_{2}}\>- R^{g}h_{t}^{\textbf {x}}(\textbf {y})|<3{\varepsilon }$ for any t ≤ t₁.

As $R^{g} h^{\textbf {x}}_{t}(\textbf {y})=R^{(+)}(1_{B(S-1/2)} p_{t}(\cdot ,\textbf {x}))(\textbf {y})=P_{t} (1_{B(S-1/2)}\cdot r^{(+)}(\cdot ,\textbf {y}))(\textbf {x}),$ and r⁽⁺⁾(⋅, y) is $\mathbb {M}^{g}$-excessive and finite q.e., we obtain similarly to the above proof of Eq. 7.5, that $\lim _{t\to 0} R^{g} h^{\textbf {x}}_{t}(\textbf {y})=r^{(+)}(\textbf {x},\textbf {y})$ for q.e.x ∈ B(S − 1) for each y ∈ B(S − 1), and consequently, the validity of Eq. 7.11 for R⁽⁺⁾ and r⁽⁺⁾. In the same way Eq. 7.11 for R⁽⁻⁾ and r⁽⁻⁾ is valid.

It remains to prove

$$ \lim_{r_{1},r_{2}\downarrow 0} {\langle}\mu^{\textbf{x},r_{1}}, Q\mu^{\textbf{y},r_{2}}\>= q(\textbf{x},\textbf{y}), $$

(7.13)

for m × m-a.e. (x, y) ∈ B(S − 1) × B(S − 1). Here q(x, y) is the first term of the righthand side of Eq. 7.4 and $Q\mu (\textbf {x})={\int \limits }_{\mathbb {C}} q(\textbf {x},\textbf {z})\mu (z), \textbf {z}\in \mathbb {C}.$ But this can be shown in exactly the same way as the proof of Eq. 7.11 using $1_{F}(\textbf {z})R^{g}\mu ^{\textbf {y},r_{2}}(\textbf {z})$ in place of $\textbf {x}i_{+}^{\textbf {y},r_{2}}(\textbf {z}).$

Appendix: Proof of Proposition 5.4

Assume that (a_ij(x)) is a family of C¹ functions on $\mathbb {C}$ with Hölder continuous derivative satisfying Eq. 1.3. Let $b_{i}(\textbf {x})={\sum }_{i,j=1}^{2} \partial a_{ij}(\textbf {x})/\partial x_{j}$ and L be the infinitesimal generator corresponding to the form a:

$$Lu(\textbf{x})={\sum}_{i,j=1}^{2} {\partial\over \partial x_{i}}\left( a_{ij}(\textbf{x}){\partial u\over \partial x_{j}}\right) ={\sum}_{i,j=1}^{2} a_{ij}(\textbf{x}){\partial^{2} u\over \partial x_{i}\partial x_{j}}+{\sum}_{i=1}^{2} b_{i}(\textbf{x}){\partial u\over \partial x_{i}}. $$

Let us fix an open disk G containing $\overline {B(S+1)}.$ A function Γ(x, y) is said to be a fundamental solution of L on G if it satisfies − LΓ(x, y) = δ(x −y) weakly, that is, for all $u\in {C^{1}_{c}}(G)$,

$$ {\int}_{G}{\sum}_{i,j=1}^{2} a_{ij}(\textbf{x}) {\partial u(\textbf{x})\over \partial x_{i}}{\partial {\Gamma}(\textbf{x},\textbf{y})\over \partial x_{j}} d\textbf{x}=u(\textbf{y}), \quad \forall \textbf{y}\in G. $$

(7.14)

For any fixed y ∈ G, let $L_{0}u(\textbf {x})={\sum }_{i,j=1}^{2} a_{ij}(\textbf {y}){\partial ^{2} u\over \partial x_{i}\partial x_{j}}.$ Then Γ₀(x, y) defined by Eq. 5.19 is a fundamental solution of L₀ on G. We shall briefly describe a construction of a fundamental solution of L from the parametrix Γ₀(x, y) as is stated in [11, §5.6] under the condition that the coefficients of L are Hölder continuous.

Since $a_{ij}\in {C^{1}_{b}}(G)$, the function k₀(x, y) = (L − L₀)Γ₀(x, y) satisfies, for some constant K₁ > 0, |k₀(x, y)|≤ K₁/|x −y|, ∀x, y ∈ G. Define $k^{(n)}_{0}(\textbf {x},\textbf {y})$ by $k^{(1)}_{0}(\textbf {x},\textbf {y})=k_{0}(\textbf {x},\textbf {y})$ and $k^{(n)}_{0}(\textbf {x},\textbf {y})={\int \limits }_{G}k_{0}(\textbf {x},\textbf {z}) k^{(n-1)}_{0}(\textbf {z},\textbf {y})d\textbf {z}$. Then $|k^{(2)}_{0}(\textbf {x},\textbf {y})|\leq K_{2} \log (1/|\textbf {x}-\textbf {y}|)+K_{3}$ and $|k^{(3)}_{0}(\textbf {x},\textbf {y})|\leq K_{4}$ for some constants K₂, K₃ and K₄. Put $K^{(n)}_{0}f(\textbf {x})={\int \limits }_{G} k^{(n)}_{0}(\textbf {x},\textbf {y})f(\textbf {y})d\textbf {y}$.

A fundamental solution Γ(x, y) of L on G can be constructed by

$$ {\Gamma}(\textbf{x},\textbf{y})={\Gamma}_{0}(\textbf{x},\textbf{y})+{\int}_{G} {\Gamma}_{0}(\textbf{x},\textbf{z}) {\Phi}(\textbf{z},\textbf{y})d\textbf{z}+\sum \alpha_{i}(\textbf{x}){\upbeta}_{i}(\textbf{y}) $$

(7.15)

for suitable continuous functions Φ(x, y), α_i(x) and β_i(y). In order to make Γ to satisfy − LΓ(x, y) = δ(x −y), Φ(x, y) needs to be a solution of the following Fredholm integral equation.

$$ {\Phi}(\textbf{x},\textbf{y})=k_{0}(\textbf{x},\textbf{y})+{\int}_{G} k_{0}(\textbf{x},\textbf{z}){\Phi}(\textbf{z},\textbf{y})d\textbf{z}+\sum L\alpha_{i}(\textbf{x}){\upbeta}_{i}(\textbf{y}). $$

(7.16)

Note that $k^{(n)}_{0}(\textbf {x},\textbf {y})$ is continuous on G for any n ≥ 3. Let us take a continuous function

$$ g(\textbf{x},\textbf{y})= k_{0}^{(4)}(\textbf{x},\textbf{y})+k_{0}^{(5)}(\textbf{x},\textbf{y})+k_{0}^{(6)}(\textbf{x},\textbf{y}) + \sum (K_{0}^{(3)}+K_{0}^{(4)}+K_{0}^{(5)})(L\alpha_{i})(\textbf{x}){\upbeta}_{i}(\textbf{y}). $$

Here α_i = β_i = 0 for all i if λ = 1 is not an eigenvalue of the dual operator $(K^{*}_{0})^{(3)}$ on C_b(G) of $K^{(3)}_{0}$ defined by $(K_{0}^{*})^{(3)}f(\textbf {x})={\int \limits } (k_{0}^{*})^{(3)}(\textbf {x},\textbf {y})f(\textbf {y})d\textbf {y}$ with $k_{0}^{*}(\textbf {x},\textbf {y})=k_{0}(\textbf {y},\textbf {x})$, while, if λ = 1 is an eigenvalue, then α_i,β_i are chosen to satisfy (g(⋅, y), ψ_j) = 0 for all eigenfunctions {ψ_j} corresponding to the eigenvalue λ = 1 of $(K^{*}_{0})^{(3)}$. Then the Fredholm equation $w(\textbf {x},\textbf {y})=K^{(3)}_{0} w(\textbf {x},\textbf {y})+g(\textbf {x},\textbf {y})$ has a unique continuous solution w(x, y) for any y ∈ G. Using this solution, the unique solution of Eq. 7.16 is given by ${\Phi }(\textbf {x},\textbf {y})=k_{0}(\textbf {x},\textbf {y})+k^{(2)}_{0}(\textbf {x},\textbf {y})+k^{(3)}_{0}(\textbf {x},\textbf {y})+w(\textbf {x},\textbf {y}).$ We notice that, according to the construction of Γ from Γ₀ by Eq. 7.15,

$$ {\Gamma}(\textbf{y},\textbf{z})-{\Gamma}_{0}(\textbf{y},\textbf{z})\text{is bounded in}\ (\textbf{y},\textbf{z})\in G\times G. $$

(7.17)

We now proceed to a proof of Eq. 4.19 with κ = Λ/λ. For x ∈ B(S − 1) and 0 < 5r ≤ t ≤ 1/3, let μ^{x, r} be the equilibrium measure for $\overline {B(\textbf {x},r)}$ relative to the admissible set $F=\overline {B(S+1)}\setminus B(S)$ for the Dirichlet form a on $H^{1}(\mathbb {C})$. We first show that the logarithmic potential

$$U\mu^{\textbf{x},r}(\textbf{y})={1\over \pi}\int \log{1\over |\textbf{y}-\textbf{z}|}\mu^{\textbf{x},r}(d\textbf{z}),\ \textbf{y} \in \mathbb{C},$$

of μ^{x, r} has the properties

$$ {\langle}\mu^{\textbf{x},r},U\mu^{\textbf{x},r}\> <\infty\quad \text{and}\quad U\mu^{\textbf{x},r}\in L^{2}_{\text{loc}}(\mathbb{C}). $$

(7.18)

Since μ^{x, r} is a measure of 0-order finite energy for the perturbed form a^g of a by g = 1_F, so it is for the perturbed Dirichlet integral (1/2)D(u, u) + (u, u)_g.

Denote by $\acute {\mathbb {M}}$ the planar Brownian motion. $\acute {R}^{g}(\textbf {x},\textbf {y})$ and $\acute {R}^{\mathbb {C}\setminus F}(\textbf {x},\textbf {y})$ denote the 0-order resolvent density of the subproces of $\acute {\mathbb {M}}$ by $\exp [-{{\int \limits }_{0}^{t}} I_{F}(X_{s})ds]$ and that of the part of $\acute {\mathbb {M}}$ on the set $\mathbb {C}\setminus F$, respectively. Then $\acute {R}^{\mathbb {C}\setminus F}(\textbf {x},\textbf {y})\le \acute {R}^{g}(\textbf {x},\textbf {y})$ so that

$${\langle}\mu^{\textbf{x},r}, \acute{R}^{\mathbb{C}\setminus F}\mu^{\textbf{x},r}\>\!\le\! {\langle}\mu^{\textbf{x},r}, \acute{R}^{g}\mu^{\textbf{x},r}\>\!<\!\infty,\ \text{and}\ \acute R^{\mathbb{C}\setminus F}\mu^{\textbf{x},r}\!\in\! H_{0,e}^{1}(\mathbb{C}\setminus F)\!\subset\! \text{BL}(\mathbb{C})\!\subset\! L_{\text{loc}}^{2}(\mathbb{C}).$$

According to the fundamental identity of the logarithmic potential (cf. [13, (2.13)]),

$$U\mu^{\textbf{x},r}(\textbf{y})=\acute R^{\mathbb{C}\setminus F}\mu^{\textbf{x},r}(\textbf{y})+\acute H_{F} U\mu^{\textbf{x},r}(\textbf{y})-W_{F}(\textbf{y}),\quad \textbf{y}\in \mathbb{C},$$

which readily implies Eq. 7.18.

Define ${\Gamma }\mu ^{\textbf {x},r}(\textbf {y})={\int \limits } {\Gamma }(\textbf {y},\textbf {z})\mu ^{\textbf {x},r}(d\textbf {z}), y\in \mathbb {C}$. Γ₀μ^{x, r} is defined similarly. Since Γ₀(x, y) is bounded by $K_{5} \log (1/|\textbf {x}-\textbf {y}|)+K_{6}$ for some constants K₅ and K₆, we have ${\Gamma }_{0}\mu ^{\textbf {x},r}\in L^{2}_{\text {loc}}(\mathbb {C})$ by Eq. 7.18. By Eq. 7.17, this also holds for Γ in place of Γ₀.

Put A^− 1(y) = (a^ij(y)). Since the weak derivative ∇Γ₀μ^{x, r} is given by

$$ \nabla {\Gamma}_{0} \mu^{\textbf{x},r}(\textbf{w})={\int}_{G}{1\over \pi ({\det}(A^{-1}(\textbf{y})))^{1/2}} {A^{-1}(\textbf{y})(\textbf{w}-\textbf{y})\over {}^{t}(\textbf{w}-\textbf{y})A^{-1}(\textbf{y})(\textbf{w}-\textbf{y})}\mu^{\textbf{x},r}(d\textbf{y}), $$

we get

$$ \begin{array}{@{}rcl@{}} {\int}_{G} |\nabla {\Gamma}_{0}\mu^{\textbf{x},r}(\textbf{w})|^{2}d\textbf{w}&\leq& K_{7} \int\int\int{1\over |\textbf{w}-\textbf{y}||\textbf{w}-\textbf{z}|}d\textbf{w} \mu^{\textbf{x},r}(d\textbf{y})\mu^{\textbf{x},r}(d\textbf{z})\\ &\leq& K_{8} \int\int \log{1\over |\textbf{y}-\textbf{z}|} \mu^{\textbf{x},r}(d\textbf{y})\mu^{\textbf{x},r}(d\textbf{z})+K_{9}. \end{array} $$

which is finite by Eq. 7.18. Consequently Γ₀μ^{x, r} ∈BL(G). By Eq. 7.15, Γμ^{x, r} also belongs to the space BL(G). Since the disk G is an extendable domain for BL-functions ([18]), there exists ${\Psi }\in \text {BL}(\mathbb {C})$ such that Ψ|_G = Γμ^{x, r}.

In what follows, we let T = S − 1/4. By virtue of Lemma 3.8, it holds that

$$\widehat R\mu^{\textbf{x},r}-H_{\mathbb{C}\setminus B(T)}{\widehat R}\mu^{\textbf{x},r}=R^{B(T)}\mu^{\textbf{x},r}\quad \text{q.e.}$$

Further, if we let $\mathcal {F}_{e,B(T)}=\{u\in \text {BL}(\mathbb {C}): \widetilde u =0\text {q.e. on}\ \mathbb {C}\setminus B(T)\},$ then

$$ R^{B(T)}\mu^{\textbf{x},r}\in \mathcal{F}_{e,B(T)},\ \text{and}\ \textbf{a}(R^{B(T)}\mu^{\textbf{x},r},v)={\langle}\mu^{\textbf{x},r}, \widetilde v\>,\ \forall v\in \mathcal{F}_{e,B(T)}.$$

Define ${\Psi }_{B(T)}(\textbf {y})={\Psi }(\textbf {y})-H_{\mathbb {C}\setminus B(T)}{\Psi }(\textbf {y}),\textbf {y}\in \mathbb {C}$. As ${\Psi }\in \text {BL}(\mathbb {C})$, ${\Psi }_{B(T)}\in \mathcal {F}_{e,B(T)}$ and $H_{\mathbb {C}\setminus B(T)}{\Psi }$ is a-harmonic on B(T), namely, $\textbf {a}(H_{\mathbb {C}\setminus B(T)}{\Psi },v)=0,\forall v\in \mathcal {F}_{e,B(T)}$. Since Ψ equals Γμ^{x, r} on G and Γ is a fundamental solution of L on G, we have $\textbf {a}({\Psi }_{B(T)},v)={\langle }\mu ^{\textbf {x},r}, v\>, \forall v\in \mathcal {F}_{e,B(T)}\cap C_{c}(B(T)).$ Therefore $\textbf {a}(R^{B(T)}\mu ^{\textbf {x},r}-{\Psi }_{B(T)}, R^{B(T)}\mu ^{\textbf {x},r}-{\Psi }_{B(T)})=0,$ which in turn implies

$$ \widehat R\mu^{\textbf{x},r}(\textbf{y})-{\Gamma}\mu^{\textbf{x},r}(\textbf{y})=H_{\partial B(T)}\widehat R\mu^{\textbf{x},r}(\textbf{y})-H_{\partial B(T)}{\Gamma}\mu^{\textbf{x},r}(\textbf{y}),\text{for a.e.} \textbf{y}\in B(T)\setminus \overline B(\textbf{x},r), $$

(7.19)

where $\widehat R\mu ^{\textbf {x},r}$ is a version of Rμ^{x, r} introduced in Lemma 4.4.

By Lemma 4.6,

$$ \begin{array}{@{}rcl@{}} \sup_{\textbf{y}\in B(T)}\sup_{\textbf{x}\in B(S-1),0<r<1/8} |H_{\partial B(T)} \widehat R\mu^{\textbf{x},r}(\textbf{y})|&\le& \sup_{\textbf{z}\in \partial B(T)}\sup_{\textbf{x}\in B(S-1),0<r<1/8} |\widehat R\mu^{\textbf{x},r}(\textbf{z})| \\&=:&\ell_{1}<\infty. \end{array} $$

By Eq. 7.15, Γ(y, z) is jointly continuous on G × G off the diagonal set, and consequently

$$\sup_{\textbf{y}\in B(T)} \sup_{\textbf{x}\in B(S-1), 0<r<1/8}|H_{\partial B(T)}{\Gamma}\mu^{\textbf{x},r}(\textbf{y})|\le \sup_{\textbf{y}\in \partial B(T), z\in B(S-1/2)} |{\Gamma}(\textbf{y},\textbf{z})|=:\ell_{2}<\infty. $$

Therefore, it follows from Eq. 7.19 and Lemma 4.4 that

$$\sup_{\textbf{x}\in B(S-1), 0<r<1/8} \sup_{\textbf{y}\in B(T)\setminus B(\textbf{x},r)}|\widehat R\mu^{\textbf{x},r}(\textbf{y})-{\Gamma}\mu^{\textbf{x},r}(\textbf{y})| \le \ell_{1}+\ell_{2} <\infty.$$

By taking Eq. 7.17 into account, it holds further that ${\Gamma }\mu ^{\textbf {x},r}(\textbf {y})-{\Gamma }_{0}\mu ^{\textbf {x},r}(\textbf {y})$ is bounded uniformly in x ∈ B(S − 1) and 0 < r < 1/2.

Hence, there exists a constant K₉ such that, for x ∈ B(S − 1) and 0 < 4r ≤ t < 1/2,

$$ \begin{array}{@{}rcl@{}} &&\max\left\{\widehat R\mu^{\textbf{x},r}(\textbf{y}):\textbf{y}\in \partial B(\textbf{x},t)\right\}\leq \max\left\{{\Gamma}_{0}\mu^{\textbf{x},r}(\textbf{y}): \textbf{y}\in \partial B(\textbf{x},t)\right\}+K_{9}\\ &&\leq {\Lambda\over \pi}\max\left\{{\int}_{B(\textbf{x},r)}\log{\Lambda\over |\textbf{y}-\textbf{z}|^{2}}\mu^{\textbf{x},r}(d\textbf{z}): \textbf{y}\in \partial B(\textbf{x},t)\right\}+K_{9}. \end{array} $$

Since (3/4)t ≤|y −z|≤ (5/4)t for any y ∈ ∂B(x, t) and z ∈ ∂B(x, r), the last expression in the above display is dominated by

$$ \begin{array}{@{}rcl@{}} &&{\Lambda\over \pi}\min\left\{{\int}_{B(\textbf{x},r)}\log{25{\Lambda}\over 9|\textbf{y}-\textbf{z}|^{2}}\mu^{\textbf{x},r}(d\textbf{z}): \textbf{y}\in \partial B(\textbf{x},t)\right\}+K_{9}. \\ &&\leq {\Lambda\over \lambda}\min\left\{{\lambda\over \pi}{\int}_{B(\textbf{x},r)}\log {\lambda\over |\textbf{y}-\textbf{z}|^{2}}\mu^{\textbf{x},r}(d\textbf{z}): \textbf{y}\in \partial B(\textbf{x},t)\right\}\\&&\quad+ {\Lambda\over \pi}(\log (25{\Lambda}/9)-\log\lambda)+K_{9}\\ &&\leq {\Lambda\over \lambda} \min\{{\Gamma}_{0}\mu^{\textbf{x},r}(\textbf{y}): \textbf{y}\in \partial B(\textbf{x},t)\}+K_{10}\\ &&\leq {\Lambda\over \lambda} \min\{\widehat R\mu^{\textbf{x},r}(\textbf{y}): \textbf{y}\in \partial B(\textbf{x},t)\}+K_{9}+K_{10} \end{array} $$

for $K_{10}=K_{9}+({\Lambda }/\pi )(\log (25{\Lambda }/9)-\log \lambda )$. Therefore Eq. 4.19 holds for κ = Λ/λ and C₂ = K₉ + K₁₀.