2024/12/10 by Hu, Kevin, Ramanan, Kavita · 2 citations
#35Q84 #60J60 (Primary) 60J70 #60K35 #82C22 #82C31 (Secondary) #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2412.07710
We study the long-time behavior of the κ-Markov local-field equation (κ-MLFE), which is a conditional McKean-Vlasov equation associated with interacting diffusions on the κ-regular tree. Under suitable assumptions on the coefficients, we prove well-posedness of the κ-MLFE. We also establish an H-theorem by identifying an energy functional, referred to as the sparse free energy, whose derivative along the measure flow of the κ-MLFE is given by a nonnegative functional that can be viewed as a modified Fisher information. Moreover, we show that the zeros of the latter functional coincide with the set of stationary distributions of the κ-MLFE and are also marginals of splitting Gibbs measures on the κ-regular tree. Furthermore, we show that for a natural class of initial conditions, the corresponding measure flow converges to one of the stationary distributions, thus demonstrating that the sparse free energy acts as a global Lyapunov function. Under mild additional conditions, in the case κ= 2 we prove that the sparse free energy arises naturally as the renormalized limit of certain relative entropies. We exploit this characterization to prove a modified logarithmic Sobolev inequality and establish an exponential rate of convergence of the 2-MLFE measure flow to its unique stationary distribution.