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.

对于L ²（E ; m）上的Dirichlet形式\（（\ mathcal {E}，\ mathcal {F}）\），令\（\ mathbb {G}（\ mathcal {E}）= \ {X_ { \ mathcal {F} _ {e} \} \）中的u}; u \是由扩展Dirichlet空间\（\ mathcal {F} _ {e} \）索引的高斯字段。我们首先解决常规复发狄利克雷形式的平衡问题\（\ mathcal {E} \）找到一个闭集的乙一个概率测度μ^乙集中在乙其复发潜在\（R \亩^ {B} \在\ mathcal {F} _ {e} \）在B上为常数qe（称为Robin常数）。接下来，我们假设E是复平面\（\ mathbb {C} \），而\（\ mathcal {E} \）是规则的递归强局部Dirichlet形式。对于封闭的磁盘\（\ bar B（\ textbf {x}，r）= \ {\ textbf {z} \ in \ mathbb {C}：| \ textbf {z}-\ textbf {x} | \ le r \} \） ，让μ ^{X，- [R}和˚F（X，- [R ）是它的平衡的措施和Robin恒定。表示高斯随机变量\（X_ {R \亩^ {\ textbf {X} .R}} \在\ mathbb {G}（\ mathcal {E}）\）由Ý ^{X， - [R}和让，对于给定常数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}。\）根据上的生长速度在一定条件˚F（X，- [R ）中，我们证明了收敛的概率μ _{- [R} （甲，ω），以一个随机测度\（\划线{\亩}（A，\欧米加）\）作为- [R ↓ 0的可能范围的γ承认非平凡限制将然后在箱子待检查\（ （\ mathcal {E}。\ mathcal {F}）\）等于\（（\ frac 12 {\ textbf {D}} _ {\ mathbb {C}}，H ^ {1}（\ mathbb {C}） ）\）和\（（\ textbf {a}，H ^ {1}（\ mathbb {C}））\），其中a对应于散度形式的均匀椭圆偏微分算子。