Strong and weak quantitative estimates in slow-fast diffusions using filtering techniques

2025/05/11 by Anugu Sumith Reddy, Anugu, Sumith Reddy, Vivek S. Borkar +1
Economics, Econometrics and Finance · Mathematics · #Stochastic processes and financial applications #Fractional Differential Equations Solutions #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2505.07093

Abstract

The behavior of slow-fast diffusions as the separation of scale diverges is a well-studied problem in the literature. In this short paper, we revisit this problem and obtain a new proof of existing strong quantitative convergence estimates (in particular, L² estimates) and weak convergence estimates in terms of n (the parameter associated with the separation of scales). In particular, we obtain the rate of n^-(1)/(2) in the strong convergence estimates and the rate of n^-1 for weak convergence estimate which are already known to be optimal in the literature. We achieve this using nonlinear filtering theory where we represent the evolution of fast diffusion in terms of its conditional distribution given the slow diffusion. We then use the well-known Kushner-Stratanovich equation which gives the evolution of the conditional distribution of the fast diffusion given the slow diffusion and establish that this conditional distribution approaches the invariant measure of the ``frozen" diffusion (obtained by freezing the slow variable in the evolution equation of the fast diffusion). At the heart of the analysis lies a key estimate of a weighted Lipschitz distance like function between a generic one-parameter family of measures and the family of unique invariant measures (of the ``frozen" diffusion parametrized by a path). This estimate is in terms of the operator norm of the dual of the infinitesimal generator of the ``frozen" diffusion.

Citations

• A limit theorem of nonlinear filtering for multiscale McKean-Vlasov stochastic systems
• A viscous ergodic problem with unbounded and measurable ingredients. Part 1: HJB Equation
• Deep Relaxation of Controlled Stochastic Gradient Descent via Singular Perturbations
• Singular perturbations in stochastic optimal control with unbounded data
• Strong and weak convergence for averaging principle of DDSDE with singular drift
• Strong Convergence Rates in Averaging Principle for Slow-Fast McKean-Vlasov SPDEs
• The averaging principle for stochastic differential equations driven by a Wiener process revisited
• Optimal convergence rates in the averaging principle for slow-fast SPDEs driven by multiplicative noise
• Strong averaging principle for a class of slow-fast singular SPDEs driven by α-stable process
• Approximation of the Filter Equation for Multiple Timescale, Correlated, Nonlinear Systems
• Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels
• Diffusion approximation for fully coupled stochastic differential equations
• Density and gradient estimates for non degenerate Brownian SDEs with unbounded measurable drift
• On Sub-Geometric Ergodicity of Diffusion Processes
• Convergence of nonlinear filterings for multiscale systems with correlated Lévy noises
• Strong convergence order for slow-fast McKean-Vlasov stochastic differential equations
• Strong and weak convergence in the averaging principle for SDEs with Hölder coefficients
• Weak order in averaging principle for stochastic differential equations with jumps
• Strong solutions to stochastic differential equations with rough coefficients
• Multiple Timescales, Mixed Mode Oscillations and Canards in Models of Intracellular Calcium Dynamics
• On Averaging Principles: An Asymptotic Expansion Approach

Related

• Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
• Pointwise gradient estimates in multi-dimensional slow diffusion\n equations with a singular quenching term
• New Lipschitz estimates and long-time asymptotic behavior for porous medium and fast diffusion equations
• Asymptotic behavior for the fast diffusion equation with absorption and singularity
• Large Deviations for Small Noise Diffusions Over Long Time