Stochastic differential equations with interaction and constant diffusion
Анотація
We analyze the evolution of probability measures associated with stochastic differential equations with interaction (SDEWI) featuring constant diffusion coefficients. Our main results concern the propagation of the logarithmic Sobolev inequality (LSI); we prove that if the initial measure satisfies an LSI, the solution preserves this property almost surely. Additionally, we provide exact growth estimates for the Wasserstein-2 distance between two solutions driven by the same Brownian motion, improving upon general stability bounds by utilizing the constant diffusion structure. Finally, we derive bounds for the expected displacement of the measure from its initial state, characterizing the short-time behavior of the system.
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