Feller Property

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Feller Property

Let $X$ and $Y$ be Borel spaces.

Definition (Weak Feller)

The stochastic kernel $P \in\mathcal{P}(X\mid Y)$ is said to be weakly continuous, or satisfying the Feller property (or most commonly: Weak Feller) if the function $y\mapsto \int\limits f(x) \, P(dx\mid y) =:Pf(y)$ is Continuous and bounded in $Y$ whenever $f$ is continuous and bounded in $X$ i.e. $\text{for }f\in C_{b}(X): Pf(y)\in C_{b}(Y)\implies P\text{ is Weak Feller}$

Remark

In simple terms, we can think of the function $g(y)$ as a probability $p(y)$ and what we’re saying is that our stochastic kernel is Weak Feller if it maps bounded continuous functions to bounded continuous functions. This basically means that the transition probabilities “respect” continuity: if you start at nearby states, your expected future values don’t jump around wildly.

Definition (Weak Feller)

The Stochastic Kernel $P(\cdot\mid y) \in \mathcal{P}(X)$ is Weak Feller if $\forall(y_{n})_{n\in\mathbb{N}}\subseteq Y$ such that $y_{n}\to y$ then $P(\cdot\mid y_{n})\to P(\cdot\mid y)\text{ weakly.}$

Remark

So this definition implicitly is equivalent to our initial definition of given that for a weakly continuous $P(\cdot\mid y)$ then we get that $\int\limits_{\mathbb{X}} f(x) \, P(dx\mid y_{n})\to \int\limits _{\mathbb{X}}f(x) \, P(dx\mid y)$ for every Continuous and bounded $f:\mathbb{X}\to \mathbb{R}$ .

Definition (Strong Feller)

The stochastic kernel $P \in\mathcal{P}(X\mid Y)$ is said to be strongly continuous or Strong Feller if the function $y\mapsto \int\limits f(x) \, P(dx\mid y) =:Pf(y)$ is Continuous and bounded in $Y$ whenever $f$ is measurable and bounded in $X$ i.e. $\text{for }f\in L_{\infty}(X;\mathbb{R}): Pf(y)\in C_{b}(Y)\implies P\text{ is Strong Feller}$

Remark

This property relaxes the Weak Feller property in that it maps bounded measurable functions to bounded continuous functions. So essentially, our kernel “regularizes” any initial function we throw at it.

Linked from

Measurable Selection Conditions

Belief MDP

Prokhorov's Theorem

Invariant probability measure