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Phase transitions arising in stochastic ergodic control associated with viscous Hamilton–Jacobi equations with bounded inward drift

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  • Naoyuki Ichihara

    (Aoyama Gakuin University)

Abstract

This paper is concerned with certain phase transition phenomena arising in a family of stochastic ergodic control problems having real parameter $$\beta $$ β . We show that the large time behavior of the optimal diffusion changes drastically in the vicinity of some critical value $$\beta ={\beta _{c}}$$ β = β c . Specifically, the optimal diffusion is recurrent for $$\beta {\beta _{c}}$$ β > β c . We also investigate the large time behavior of the optimal diffusion for $$\beta ={\beta _{c}}$$ β = β c which turns out to be different from the previous two cases and more subtle. Our proof is based on the Lyapunov method giving analytical criteria for recurrence and transience of diffusions. The key lies in the analysis of solutions to the associated viscous Hamilton–Jacobi equation with bounded inward drift. In particular, a refined version of the gradient estimate for solutions to viscous Hamilton–Jacobi equations plays a substantial role.

Suggested Citation

  • Naoyuki Ichihara, 2021. "Phase transitions arising in stochastic ergodic control associated with viscous Hamilton–Jacobi equations with bounded inward drift," Partial Differential Equations and Applications, Springer, vol. 2(1), pages 1-28, February.
    • Handle: RePEc:spr:pardea:v:2:y:2021:i:1:d:10.1007_s42985-021-00072-0
    DOI: 10.1007/s42985-021-00072-0
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    References listed on IDEAS

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    1. Ichihara, Naoyuki, 2012. "Large time asymptotic problems for optimal stochastic control with superlinear cost," Stochastic Processes and their Applications, Elsevier, vol. 122(4), pages 1248-1275.
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