Random walk in RStan

Modeling
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5

Update
I thought about my first attempt in Stan and now understand why I get a linear decline in k rather than the expected exponential decay. It is because r is not coded as a dynamic but a fixed parameter, meaning that k[i] always has the same difference to k[i - 1], resulting in a linear trend. Inspired by this answer I rewrote my Stan code so that the random walk is specified in the model block:

stan_code <- "
data {
  int n; // Number of observations
  vector[n] y;    // Response variable
  vector[n] x;    // Predictor variable
}

parameters {
  real<lower=0> sigma; // Standard deviation
  real<lower=0> a;  // Intercept
  real<lower=0> ks;  // Initial decay rate
  real<lower=0> tau; // Precision of random step
}

model {
  // Priors
  sigma ~ exponential( 1 );
  a ~ lognormal( log(2) , 0.05 );
  ks ~ lognormal( log(0.08) , 0.1 );
  tau ~ lognormal( log(0.01) , 0.1 );
  
  // Random walk
  vector[n] k; // Decay rate
  k[1] = ks;  // Set starting value of random walk
  for ( i in 2:n ) {
    k[i] ~ normal(k[i - 1], tau);
  }
  
  // Prediction function
  vector[n] mu; // Mean prediction for y
  for( i in 1:n ) {
    mu[i] = a * exp(-k[i] * x[i]); // Calculate mu using updated k
  }
  
  // Likelihood
  y ~ normal(mu, sigma);
}
"

Unfortunately, the model does not compile this time and throws errors Error evaluating the log probability at the initial value., normal_lpdf: Random variable is nan, but must be not nan! and Initialization failed.. The suggestion: Try specifying initial values, reducing ranges of constrained values, or reparameterizing the model. Any help with such reparameterisation would be greatly appreciated!