Convergence to the uniform distribution of moderately self-interacting diffusions on compact Riemannian manifolds

2023/07/04 by Holbach, Simon, Raimond, Olivier
#60K35 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2307.01538

Abstract

We consider a self-interacting diffusion X on a smooth compact Riemannian manifold \mathbb M, described by the stochastic differential equation dX_t = √(2) dW_t(X_t)- β(t) ∇ V_t(X_t)dt, where β is suitably lower-bounded and grows at most logarithmically, and V_t(x)=(1)/(t)∫₀^t V(x,X_s)ds for a suitable smooth function V\colon \mathbb M²→\mathbb R that makes the term -∇ V_t(X_t) self-repelling. We prove that almost surely the normalized occupation measure μ_t of X converges weakly to the uniform distribution \mathcal U, and we provide a polynomial rate of convergence for smooth test functions. The key to this result is showing that if f\colon\mathbb M→\mathbb R is smooth, then μ_e^t(f) shadows the flow generated by the ordinary differential equation ν_t(f)=-ν_t(f)+\mathcal U(f).

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