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 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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Correspondence to Yoichi Oshima.

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This work was supported by Research Institute for Mathematical Science, A joint Usage/Research Center located in Kyoto University

Appendices

Appendix: Proof of Proposition 4.7

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 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,

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

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 φL2(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 φL2(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

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

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 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, yB(S − 1) with |xy| > 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) : |xy| > 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 tt0 and z, wE such that |zw| > 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 r2η/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 tt0 and zB(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 zB(x, η/8) and wB(y, η/2) and Rg1B(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 tt0.

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

$$ \limsup_{r_{1},r_{2}\downarrow 0} |{\langle}\mu^{\textbf{x},r_{1}}, R^{g}\mu^{\textbf{y},r_{2}}\>-R^{g} f_{t}^{\textbf{x}}(\textbf{y})|<4{\varepsilon} $$
(7.10)

for any tt0 and any x, yB(S − 1) with |xy| > 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) ∩{|xy| > η}. For any yB(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 xD(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 xD(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 zF 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 tt1.

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 r2η/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 tt1.

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.xB(S − 1) for each yB(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 (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:

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

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(xy) 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 yG, 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) = (LL00(x, y) satisfies, for some constant K1 > 0, |k0(x, y)|≤ K1/|xy|, ∀x, yG. 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), αi(x) and βi(y). In order to make Γ to satisfy − LΓ(x, y) = δ(xy), 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 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 αii 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 yG. 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 xB(S − 1) and 0 < 5rt ≤ 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

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

of μx, 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 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

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

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 \({\Gamma }\mu ^{\textbf {x},r}(\textbf {y})={\int \limits } {\Gamma }(\textbf {y},\textbf {z})\mu ^{\textbf {x},r}(d\textbf {z}), y\in \mathbb {C}\). 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

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

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 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

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

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 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

$$ \widehat R\mu^{\textbf{x},r}(\textbf{y})-{\Gamma}\mu^{\textbf{x},r}(\textbf{y})=H_{\partial B(T)}\widehat R\mu^{\textbf{x},r}(\textbf{y})-H_{\partial B(T)}{\Gamma}\mu^{\textbf{x},r}(\textbf{y}),\text{for a.e.} \textbf{y}\in B(T)\setminus \overline B(\textbf{x},r), $$
(7.19)

where \(\widehat R\mu ^{\textbf {x},r}\) is a version of Rμx, 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 xB(S − 1) and 0 < r < 1/2.

Hence, there exists a constant K9 such that, for xB(S − 1) and 0 < 4rt < 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 ≤|yz|≤ (5/4)t for any yB(x, t) and zB(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.

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Fukushima, M., Oshima, Y. Gaussian fields, equilibrium potentials and multiplicative chaos for Dirichlet forms. Potential Anal (2020). https://doi.org/10.1007/s11118-020-09858-0

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Keywords

  • Gaussian field
  • Dirichlet form
  • Equilibrium potential
  • Gaussian multiplicative chaos.

Mathematics Subject Classification (2010)

  • Primary 31C25
  • 60G60
  • Secondary 60J45
  • 60G57