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 μB 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 μx, 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 Yx, 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 μr(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.
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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
\(\check {r_{1}^{n}}(\textbf {x},\textbf {y})\) is the density function of the kernel \(\check R^{n}(\textbf {x},d\textbf {y})\) on \((F,{\mathcal B}(E))\) with respect to mF. \(\check {R_{1}^{n}}(\textbf {x},d\textbf {y})\) is mF-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,
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)]
Therefore, if we let
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 \(\varphi \in L^{\infty }(F;m_{F})\),
on account of Eq. 4.14. This identity can be readily verified to hold also for φ ∈ L2(F; mF).
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 Rgh 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; mF), \(\check R\varphi =\check R_{1}\varphi -{\langle }\widetilde m_{F},\varphi \>+ \check R_{1}\check R\varphi \). 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
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
\(\frak r(\textbf {x},\textbf {y})\) is symmetric and, for each \(\textbf {x}\in \mathbb {C},\) it is a difference of Mg-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
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
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,
for all t ≤ t0 and z ∈ B(x, η/8). In particular,
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 Rg1B(S− 1) ≤ M4 on \(\mathbb {C}\) for some constant M4 by Lemma 3.1 (i),
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 \(A_{t}={{\int \limits }_{0}^{t}} 1_{F}(X_{s})ds\),
Hence we may also assume that
because \(R^{g} \mu ^{\textbf {x},r_{2}}\le M_{1}\) on D(y).
Therefore, in the decomposition
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 pt(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
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
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 rg(⋅, y) is \(\mathbb {M}^{g}\)-excessive and 1D(y)(Xt)rg(Xt, y) is right continuous at t = 0 a.s.ℙx for x ∈ D(y) ∩ E1(y), we have
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),
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 \(\textbf {z}\in \mathbb {C}, r_{2}\in (0,\eta /8)\),
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
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
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
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 (aij(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:
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 \(u\in {C^{1}_{c}}(G)\),
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
for suitable continuous functions Φ(x, y), αi(x) and βi(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.
Note that \(k^{(n)}_{0}(\textbf {x},\textbf {y})\) is continuous on G for any n ≥ 3. Let us take a continuous function
Here αi = βi = 0 for all i if λ = 1 is not an eigenvalue of the dual operator \((K^{*}_{0})^{(3)}\) on Cb(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 αi,βi are chosen to satisfy (g(⋅, y), ψj) = 0 for all eigenfunctions {ψj} 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,
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 μx, 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
of μx, r has the properties
Since μx, r is a measure of 0-order finite energy for the perturbed form ag of a by g = 1F, 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
According to the fundamental identity of the logarithmic potential (cf. [13, (2.13)]),
which readily implies Eq. 7.18.
Define \({\Gamma }\mu ^{\textbf {x},r}(\textbf {y})={\int \limits } {\Gamma }(\textbf {y},\textbf {z})\mu ^{\textbf {x},r}(d\textbf {z}), y\in \mathbb {C}\). Γ0μx, 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) = (aij(y)). Since the weak derivative ∇Γ0μx, r is given by
we get
which is finite by Eq. 7.18. Consequently Γ0μx, r ∈BL(G). By Eq. 7.15, Γμx, 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 = Γμx, r.
In what follows, we let T = S − 1/4. By virtue of Lemma 3.8, it holds that
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
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 Γμx, 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
where \(\widehat R\mu ^{\textbf {x},r}\) is a version of Rμx, r introduced in Lemma 4.4.
By Lemma 4.6,
By Eq. 7.15, Γ(y, z) is jointly continuous on G × G off the diagonal set, and consequently
Therefore, it follows from Eq. 7.19 and Lemma 4.4 that
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,
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
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.
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Fukushima, M., Oshima, Y. Gaussian fields, equilibrium potentials and multiplicative chaos for Dirichlet forms. Potential Anal 55, 285–337 (2021). https://doi.org/10.1007/s11118-020-09858-0
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DOI: https://doi.org/10.1007/s11118-020-09858-0