The paper constructs a Lyapunov function (sparse free energy) for the Markovian local-field equation on regular trees, proves convergence to stationary distributions, and characterizes those distributions as tree Gibbs marginals.
Long time behavior of one-dimensional McKean-Vlasov SDEs with common noise
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abstract
In this paper, by introducing a new type asymptotic coupling by reflection, we explore the long time behavior of random probability measure flows associated with a large class of one-dimensional McKean-Vlasov SDEs with common noise. Concerning the McKean-Vlasov SDEs with common noise under consideration in the present work, in contrast to the existing literature, the drift terms are much more general rather than of the convolution form, and, in particular, can be of polynomial growth with respect to the spatial variables, and moreover idiosyncratic noises are allowed to be of multiplicative type. Most importantly, our main result indicates that both the common noise and the idiosyncratic noise facilitate the exponential contractivity of the associated measure-valued processes.
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An H-theorem for a conditional McKean-Vlasov process related to interacting diffusions on regular trees
The paper constructs a Lyapunov function (sparse free energy) for the Markovian local-field equation on regular trees, proves convergence to stationary distributions, and characterizes those distributions as tree Gibbs marginals.