  We consider an one-dimensional conservation law with random space-time
forcing and calculate using large deviations the exponentially small
probabilities of anomalous shock profile displacements. Under suitable
hypotheses on the spatial support and structure of random forces, we analyze
the scaling behavior of the rate function, which is the exponential decay rate
of the displacement probabilities. For small displacements we show that the
rate function is bounded above and below by the square of the displacement
divided by time. For large displacements the corresponding bounds for the rate
function are proportional to the displacement. We calculate numerically the
rate function under different conditions and show that the theoretical analysis
of scaling behavior is confirmed. We also apply a large-deviation-based
importance sampling Monte Carlo strategy to estimate the displacement
probabilities. We use a biased distribution centered on the forcing that gives
the most probable transition path for the anomalous shock profile, which is the
minimizer of the rate function. The numerical simulations indicate that this
strategy is much more effective and robust than basic Monte Carlo.
