Summary
Interacting diffusion systems are a class of diffusion processes taking values in an infinite product space of an interval, which models the variety of phenomena in physics and biology. This paper surveys the subject of interacting diffusion systems and related problems. Topics include:
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1.
Introduction.
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2.
Stationary distributions and ergodic theorems in transient case.
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3.
Zd-shift invariance of stationary distributions.
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4.
Local extinction in transient case.
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5.
Uniformity and local extinction in recurrent case.
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6.
Parabolic Anderson model and sample Lyapunov exponent.
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7.
Finite systems of interacting diffusions.
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8.
Approximation of infinite systems via finite systems.
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9.
Methods and some technicalities.
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9.1
Duality.
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9.2
Comparison.
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9.3
Coupling.
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9.4
Liouville property.
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9.5
Random walk estimates.
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9.6
Moments estimates.
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9.7
Idea of the proof of the approximation result.
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Shiga, T. (1996). Interacting diffusion systems over Zd . In: Ikeda, N., Watanabe, S., Fukushima, M., Kunita, H. (eds) Itô’s Stochastic Calculus and Probability Theory. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68532-6_20
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DOI: https://doi.org/10.1007/978-4-431-68532-6_20
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