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I have a technical question on a continuity of green function.

Setting

Let $E$ be a locally compact separable metric space and $m$ a locally finite measure on $E$.

Let $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ be a transient diffusion process. We assume that there exists a continuous function $p_{t}(x,y):(0,\infty) \times E \times E \to [0,\infty)$ such that \begin{align*} E_{x}[f(X_t)]:=\int_{E}p_{t}(x,y)\,dm(y),\quad t>0,\ x,y \in E \end{align*}

My question

Then, under what conditions, can we show that $g(x,y)=\int_{0}^{\infty}p_{t}(x,y)\,dt$ is a continuous function on $E \times E$ off the diagonal.

Aim

I am interested in the case that $X$ is a part process of a recurrent process. For example,

  • $E=\bar{\mathbb{D}} \setminus K$, where $\mathbb{D}$ is the unit disk of $\mathbb{R}^2$ and $K \subset \mathbb{D}$ is a closed ball,
  • $X$ is a part process on $\bar{\mathbb{D}} \setminus K$ of a reflected Brownian motion on $\bar{D}$.

In this situation, $X$ possesses a continuous heat kernel $p_{t}(x,y)$. However, I do not even know whether $x \mapsto g(x,y)$ is locally bounded on $E \setminus \{y\}$.

  • If $K$ is the closed ball centered at the origin with radius $r_0<1$, it is known that there exists $C \in (0,\infty)$ such that \begin{align*} (\ast) \quad g(x,y) \le C-2\log |x-y|,\quad m \otimes m\text{-a.e. on } E \times E \end{align*} However, I do not know whether $(\ast)$ holds for any $(x,y) \in E \times E$. If $(\ast)$ holds for any $(x,y) \in E \times E$, $x \mapsto g(x,y)$ is locally bounded on $E \setminus \{y\}$.
  • There is a function $G(x,y)$ which is continuous on $E \times E$ off the diagonal. It holds that $g(x,y)=G(x,y),\quad m\otimes m\text{-a.e.}$

I think that the continuity of green functions is a fundamental problem, but is there a standard proof method?

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  • $\begingroup$ The Green function $g(\cdot,y)$ is harmonic (with respect to the diffusion) in $E \setminus \{y\}$. In most cases this property alone implies local Hölder continuity (or even $C^\infty$ when the diffusion is the Brownian motion). Does this work for you? $\endgroup$ – Mateusz Kwaśnicki Jun 9 at 19:42
  • $\begingroup$ @MateuszKwaśnicki Thank you for your reply. I know that the green function is harmonic in $E \setminus \{y\}$ for the stochastic process. But does this fact always implies that the green function is harmonic for the Dirichlet form? If the green function is locally bounded, it is harmonic for the Dirichlet form and then we may obtain a continuity. $\endgroup$ – sharpe Jun 9 at 20:04
  • $\begingroup$ @MateuszKwaśnicki In the cases you are talking about, you may already know that harmonic functions are locally bounded. $\endgroup$ – sharpe Jun 9 at 20:29
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(A comment too long for a comment).


I am not an expert, but I guess an answer will depend on how specific you want it to be. If this is about reflecting Brownian motions, then we know a lot, including heat kernel bounds for nice enough domains; see Bass–Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991) and Yang–Zhang, Estimates of Heat Kernels with Neumann Boundary Conditions, Potential Anal. 38 (2013). If I understand correctly, in particular the Green function (away from the diagonal) is smooth in the interior of the domain, and Hölder continuous at the boundary (if the domain is Lipschitz).


For more general diffusions (or differential operators), one often knows "a priori" supremum bounds for and Hölder continuity results for harmonic functions. This should also give the desired continuity of the Green function. To my understanding, this is often parallel to heat kernel bounds, which are either assumed (in the general setting of metric measure spaces) or proved (for, say, certain elliptic operators on $\mathbb{R}^n$, or on smooth manifolds).

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  • $\begingroup$ Let $D$ be a bdd Lipschitz domain and let $K \subset D$ be a closed disk. Then, we can define the Brownian motion which is reflected at $\partial D$ and absorbed on $\partial K$. We can also define the green function from the Brownian motion. The green function $G^K(x,y)$ is defined for any $(\bar{D} \setminus K) \times (\bar{D} \setminus K)$. $G^{K}(x,y)$ possesses a version which is continuous on $(D \setminus K) \times (D \setminus K) $ off the diagonal. The version may be locally bounded. But, I do not know $x \mapsto G^{K}(x,y)$ is locally bounded on $\bar{D} \setminus K \setminus \{y\}$. $\endgroup$ – sharpe Jun 9 at 23:23
  • $\begingroup$ @sharpe: Have you looked into the Bass–Hsu paper? Corollary 3.3 gives you the upper bound for the Green's function. (For simplicity, they consider $d \geqslant 3$, but if your favourite $D$ is two-dimensional, you can consider $D \times \mathbb{R}$ instead, and then simply integrate out the last coordinate. Obviously, the Green function for the reflected process will become infinite then; but everything will be just fine for Green function for the Brownian motion reflected at $\partial D$ and killed on $\partial K$.) $\endgroup$ – Mateusz Kwaśnicki Jun 10 at 7:47
  • $\begingroup$ Sorry. I do not understand what you are saying. How do we obtain an estimate of $G^{K}(x,y)$? Can we obtain an estimate like as $(\ast)$ from the green function on $D \times \mathbb{R}$? $\endgroup$ – sharpe Jun 10 at 9:56
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    $\begingroup$ @sharpe: I think so: all you need to know is a local bound for $G_{D\times \mathbb{R}}$ from Bass–Hsu and a bound for $G_{D\times \mathbb{R}}^{K\times \mathbb{R}}$ at infinity (which is perhaps not immediate, but quite standard). It is perhaps even simpler to integrate the bounds for the heat kernel, though: Thm 3.3 in Bass–Hsu gives you an upper bound in $D \times \mathbb{R}$, integrating it you find the corresponding bound for $p_D(t,x,y)$ in $D$, the argument used in Cor. 2.5 (or Thm 2.4) gives exponential decay of $p^K_D(t,x,y)$, integrate with respect to time, and you are done. $\endgroup$ – Mateusz Kwaśnicki Jun 10 at 10:28
  • $\begingroup$ Ah! I see. It seems to go well. I should have read the paper more properly. Thank you for teaching me. To prove exponential decay of $p_{D}^{K}(t,x,y)$ is important! $\endgroup$ – sharpe Jun 10 at 11:31

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