# How to prove Feller property without using heat kernel estimates

I have a question about Markov processes.

Let $\mathbb{M}=(X_t,P_x)$ be a Markov process on a locally compact separable metric measure space $(E,\mu)$.

$\mathbb{M}$ is called Feller process if its semigroup $\{p_{t}\}_{t>0}$ satisfies the following: for all $t>0$, \begin{align*} (0)\quad p_{t}(C_{\infty}(E)) \subset C_{\infty}(E), \end{align*} where $C_{\infty}(E)$ is the set of continuous functions which vanish at infinity.

If we know $\mathbb{M}$ has the following property: \begin{align*} (1)\quad\lim_{r \to \infty}\sup_{x \in E}P_{x}(X_{t} \in E \setminus B(x,r))=0, \end{align*} we can prove $p_{t}f$ vanishes at infinity for all $f \in C_{c}(E)$. Indeed, for all $f \in C_{c}(E)$, we have

\begin{align*} &|p_{t}f(x)| \le E_{x}[|f(X_t)|] \\ &=\int_{E \cap B(x,r)}p_{t}(x,y)|f(y)|\,\mu(dy)+\int_{E \setminus B(x,r)}p_{t}(x,y)|f(y)|\,\mu(dy) \\ &\le \sup_{y \in E \cap B(x,r)}|f(y)|+\|f\|_{\infty} \sup_{x \in E}P_{x}(X_{t} \in E \setminus B(x,r))\end{align*} Then, letting $x \to \infty$ and then $r \to \infty$, we obtain the assertion.

A sufficient condition for (1)

Let us consider the case $\mathbb{M}$ is a diffusion process on a Euclidean domain $D$. If the transition density $p_{t}(x,y)$ of $\mathbb{M}$ has the following estimate: \begin{align*} (2) \quad p_{t}(x,y) \le a_{1}e^{t} t^{-d/2} \exp(-|x-y|^2/a_{2}t)\quad (a_1,a_2 \text{ are some constants indep of $t,x,y$}), \end{align*} we can prove (1).

My question

I am interested in the property (1) of reflecting Brownian motions on smooth domains. These processes are generated by the following classical Dirichlet form: \begin{align*} (3)\quad\mathcal{E}(f,g)=\frac{1}{2}\int_{D}(\nabla f, \nabla g)\,dx,\quad f,g \in H^{1}(D). \end{align*}

When the boundary of $D$ is sufficiently smooth, it is known that (3) is regular on $\bar{D}$ and we can construct a processes $(\{X_t\},\{P_x\})$ whose Dirichlet form is (3). Furthermore, $(\{X_t\},\{P_x\})$ solves the following Skorohod SDE: \begin{align*} X_{t}=x+B_{t}+\int_{0}^{t}n(X_s)dL_s, \end{align*} where $B_t$ is the $d$-dim B.M. and $n$ is the inward unit normal on $\partial D$ and $\{L_t\}$ is boundaly local time.

If transition density of $X$ has a estimate like (2) and $D$ is bounded, we can compute expectation of $L_t$.

In the case of a diffusion, (1) is implied for example by having bounded coefficients. This follows immediately from applying BDG to $X_t-x$ and doesn't require (2) which is much harder to get. Note by the way that (1) itself is quite a bit overkill since it rules out the OU process, which is the prototypical example of a Feller process, even in your "strong" sense.

I also want to point out that the definition of "Feller property" isn't consistent across the literature. For many authors, mapping $C_b$ (continuous bounded functions) into itself is sufficient for a Markov process to be called "Feller". Mapping $C_\infty$ to itself rules out simple examples like $dx = -x^3\,dt + dW$.

• Thank you for your comment. As you say this condition (1) is it too strong to prove Feller property in my strong sense. By the way, are there other good ways to prove Feller property in my strong sense? – sharpe Sep 5 '17 at 8:20
• Regarding the name: I believe Feller originally considered processes with compact state space; certainly Dynkin in his book on Markov processes requires a Feller process to have compact state space. Extension to locally compact spaces is then immediate via one-point compactification: this is the (now-standard, as far as I can tell) definition of the Feller property with $C_\infty$ (often denoted $C_0$). A definition that uses $C_b$ instead is sometimes referred to as the $C_b$-Feller property (for example in Niels Jacob's book). – Mateusz Kwaśnicki Sep 5 '17 at 9:00
• @Mateusz Kwaśnicki I think that what is "standard" really depends on the community you're talking to. For example, Meyn-Tweedie, Da Prato-Zabczyk and Oksendal all refer to the definition with $C_b$ when defining what a "Feller process" is and I'm sure that you have in mind just as many books going the other way around. Note though that if you take the definition with $C_0$, then "strong Feller" does not imply "Feller", which is why I personally prefer the "$C_b$" definition... – Martin Hairer Sep 5 '17 at 13:30
• @MartinHairer: Certainly, you are right. This is why I tried to emphasize that "now-standard" is only my impression, but apparently failed. My point was that the origin of the name is related with the $C_0$ definition. – Mateusz Kwaśnicki Sep 5 '17 at 18:51

The answer depends on what you are working with.

1. If you have a pseudo-differential operator and ask whether it generates a Feller semigroup, the best answer known to date is contained in Walter Hoh's works; see a three-volume book Pseudo-Differential Operators and Markov Processes by Niels Jacob for rigorous statements.

2. If you have ask what differential operators generate Feller semigroups, or what SDEs determine Feller diffusions, then this is closely related to a question about the possibility of starting a process "at infinity", and in some sense dual to the "no explosion in finite time" problem. This can be a delicate question. Feller property is known if the coefficients of the generator grow slowly enough (quadratic growth for the coefficients at the second order derivative and linear growth for the linear term). For a detailed discussion and references, see Properties at infinity of diffusion semigroups and stochastic flows via weak uniform covers by Xue-Mei Li.

3. For general conditions, you may see a Primer on Feller semigroups and Feller processes. For example, Theorem 1.10 gives an equivalent condition expressed in terms of transition kernels.

• Thank you for your comment. What I particularly care about is Feller property (in my sense) of a reflecting Brownian motions on smooth domains. I edited the article to explain the details. – sharpe Sep 5 '17 at 9:57
• Oh, now this is a completely different question! I would expect that if your domain has sufficiently narrow (infinite) "horns", you cannot hope for the Feller property. Indeed, the first coordinate of a reflecting BM in a horn $\{(x,y):x>0,|y|<H(x)\}$ behaves roughly as the one-dimensional BM with drift $H'(x)/(2 H(x))$, so choosing $H(x)=\exp(-x^4)$ should produce a counterexample. For a more detailed discussion, see p. 5 of Pinsky's article. – Mateusz Kwaśnicki Sep 5 '17 at 10:23