Let $X$ be a strongly symmetric recurrent Markov process with state
space $S$ and let $L_t^x$ denote the local time of $X$ at $X \in S$. For a
fixed element 0 in the state space S, let
$$ \tau(t) := \inf \{s: L^0_s > t \}. $$
¶ The 0-potential density, $u_{0}(x, y)$, of the process $X$ killed at
$T_0 = \inf \{s:X_s =0\}$ is symmetric and positive definite. Let $\eta =
\{\eta_x; x \in S \}$ be a mean-zero Gaussian process with covariance
¶ $$ E_\eta (\eta_x \eta_y ) = u_{\{0\}}(x, y). $$
The main result of this paper is the following
generalization of the classical second Ray–Knight theorem: for any $b
\in R$ and $t > 0$
$$ \{L_{\tau(t)} + 1/2 (\eta_x + b)^2; x \in S \} =
\{ 1/2 ( \eta_x + \sqrt{2+ b^2})^2 ; x \in S \} \text{ in law}. $$
¶ A version of this theorem is also given when $X$ is transient.