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Birth–death processes form a natural class where ideas and results on large deviations can be tested. We derive a large-deviation principle under an assumption that the rate of jump down (death) grows asymptotically linearly with the population size, while the rate of jump up (birth) grows sublinearly. We establish a large-deviation principle under various forms of scaling of the underlying process and the corresponding normalization of the logarithm of the large-deviation probabilities. The results show interesting features of dependence of the rate functional upon the parameters of the process and the forms of scaling and normalization.

Let X be a d-dimensional continuous super-Brownian motion with branching rate ε, which might be described symbolically by the "stochastic equation" a space-time white noise. A Schilder type theorem is established concerning large deviation probabilities of X on path space as ε → 0, with a representation of the rate functional via an L2 -functional on a generalized "Cameron-Martin space" of measure-valued paths.

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A large-deviation principle for birth–death processes with a linear rate of downward jumps

A Schilder Type Theorem for Super-Brownian Motion

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2 results

A large-deviation principle for birth–death processes with a linear rate of downward jumps

A Schilder Type Theorem for Super-Brownian Motion