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Phase transitions arising in stochastic ergodic control associated with viscous Hamilton–Jacobi equations with bounded inward drift by Naoyuki Ichihara is a Mathematics article available to read on EtoBox.

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This paper is concerned with certain phase transition phenomena arising in a family of stochastic ergodic control problems having real parameter b. We show that the large time behavior of the optimal diffusion changes drastically in the vicinity of some critical value b 1⁄4 b c . Specifically, the optimal diffusion is recurrent for b\b c , while it is transient for b [ b c . We also investigate the large time behavior of the optimal diffusion for b 1⁄4 b 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.

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Author

Naoyuki Ichihara

Publisher

Springer Science and Business Media LLC

Published

2021

Language

EN

Field

Mathematics (Physical Sciences)