Random walk in RStan

Modeling
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4

Update
The authors of the paper have kindly provided their rbi code, rbi being the R interface to LibBi. I can now see that rather than using the exponential decay function, derived by integration, the authors use the ordinary differential equation (ODE) and solve it internally with an RK4 ODE solver, which I know is also possible in Stan.

model decay {
  state W, k, K
  noise r
  param sigma_r
  obs W_obs
  sub parameter {
    sigma_r ~ normal(0.1, 0.01)
  }
  const prop_std = 0.5;
  sub proposal_parameter {
    sigma_r ~ normal(sigma_r, 0.01*prop_std)
  }
  sub initial {
    k ~ normal(log(0.003), 0.8)
    K ~ log_normal(log(0.003), 0.8)
    W ~ normal(1.0, 0.1)
    25
  }
  sub transition(delta = 1.0) {
    r ~ normal(0.0, sigma_r)
    k <- k + r
    K <- exp(k)
    ode(h = 0.01, atoler = 1.0e-6, rtoler = 1.0e-9, alg = 'RK4(3)'){
      dW/dt = -K*W
    }
  }
  sub observation {
    W_obs~log_normal(log(W),0.1)
  }
}

Is there someone fluent in LibBi and Stan, who can translate this into Stan?