# Weak Feller Property of Non-linear Filters

## TL;DR

This paper investigates the weak Feller property of non-linear filters in controlled Markov chains, providing new sufficient conditions and alternative proofs that enhance understanding of their stability and approximation capabilities.

## Contribution

The paper establishes new conditions for the weak Feller property of non-linear filters, including cases not previously covered, and offers a simplified proof for existing results.

## Key findings

- Proves weak Feller property under weak continuity and total variation continuity of transition probabilities.
- Provides an alternative, concise proof for a known result on weak Feller property.
- Identifies new conditions under which the weak Feller property holds for non-linear filters.

## Abstract

Weak Feller property of controlled and control-free Markov chains lead to many desirable properties. In control-free setups this leads to the existence of invariant probability measures for compact spaces and applicability of numerical approximation methods. For controlled setups, this leads to existence and approximation results for optimal control policies. We know from stochastic control theory that partially observed systems can be converted to fully observed systems by replacing the original state space with a probability measure-valued state space, with the corresponding kernel acting on probability measures known as the non-linear filter (belief) process. Establishing sufficient conditions for the weak Feller property for such processes is a significant problem, studied under various assumptions and setups in the literature. In this paper, we prove the weak Feller property of the non-linear filter process (i) first under weak continuity of the transition probability of controlled Markov chain and total variation continuity of its observation channel, and then, (ii) under total variation continuity of the transition probability of controlled Markov chain. The former result (i) has first appeared in Feinberg et. al. [Math. Oper. Res. 41(2) (2016) 656-681]. Here, we present a concise and easy to follow alternative proof for this existing result. The latter result (ii) establishes weak Feller property of non-linear filter process under conditions, which have not been previously reported in the literature.