Asymptotically unbiased approximation of the QSD of diffusion processes with a decreasing time step Euler scheme

TL;DR

The paper develops a decreasing-step Euler scheme with a redistribution mechanism to approximate the quasistationary distribution $\mu^*$ of a diffusion killed at the boundary of a bounded domain. It proves almost-sure convergence of the occupation measure to $\mu^*$, and establishes convergence in distribution of the discretized process $\overline{X}_{\Gamma_n}$ to $\mu^*$ via finite-horizon weak-error bounds between the diffusion with renewal and its discretization. The contributions include a flexible framework allowing general redistribution measures $\mathfrak{p}_n$, a thorough tightness analysis using a one-dimensional reflected Brownian bound process, and a robust discretization analysis (including Wasserstein estimates) that yields an unbiased QSD approximation as $n\to\infty$ with $\gamma_n\to0$. The results have practical impact for simulating QSDs in applications such as population dynamics and molecular dynamics, providing unbiased QSD estimation and a pathway to estimating the survival rate $\lambda^*$ without discretization bias. Overall, the work extends stochastic approximation techniques to diffusion QSDs with renewal, offering both theoretical guarantees and implementable strategies for accurate, memory-conscious simulations.

Abstract

We build and study a recursive algorithm based on the occupation measure of an Euler scheme with decreasing step for the numerical approximation of the quasistationary distribution (QSD) of an elliptic diffusion in a bounded domain. We prove the almost sure convergence of the procedure for a family of redistributions and show that we can also recover the approximation of the rate of survival and the convergence in distribution of the algorithm. This last point follows from some new bounds on the weak error related to diffusion dynamics with renewal.

Figures (2)

• Figure 1: Construction of the process $(Z_r)_{r \geq 0}$. Vertical dotted blue lines indicate the times $\Delta_n$. The solid black curve is $\psi_D(\overline{X}_{\tau(r)})$, the dashed black curve is $\widetilde{\psi}_D(\overline{X}_{\tau(r)})$. Black points represent the values of $\xi_n = \psi_D(\overline{X}_{\tau(\Delta_n)})$. The red curve is $Z_r$, the horizontal red line has coordinate $\eta_0$, and red points represent the values of $\zeta_n$.
• Figure 2: The process $Z_r$ on $[S_q,S_{q+1})$. On the interval $[S_q,T_q)$, it is bounded from above by $Z^+_{q,r}$, which is reflected at $0$ and therefore does not depend on the grid $(\Delta_n)_{n \geq 0}$. The statement of Proposition \ref{['prop:ZZpq']} is that the distance between $Z_r$ and $Z^+_{q,r}$ does not exceed $2\epsilon_q$. For $\epsilon>0$, the hitting times of the respective levels $2\eta_0/3$ and $2\eta_0/3+2\epsilon$ for $Z^+_{q,r}$ are denoted by $T^{+;0}_q$ and $T^{+;\epsilon}_q$.

Theorems & Definitions (89)

• Theorem 1.1: Convergence to the QSD for $\mathfrak{p}_n={\mu}_n$
• Theorem 1.2: Convergence to the QSD
• Corollary 1.3: Almost sure weak convergence of $(\mathfrak{p}_n)_{n \geq 1}$
• Remark 1.4
• Remark 1.5
• Theorem 1.6: Convergence in distribution of $(\overline{X}_{\Gamma_{n}})_{n\ge0}$
• Definition 1.7: Euler schemes with redistribution
• Remark 1.8
• Theorem 1.9: Weak consistency of Euler schemes with redistribution
• Proposition 2.1: Almost sure tightness of $({\mu}_n)_{n \geq 1}$