I have a question about Brownian motions and its heat kernel.

Using Dirichlet form theory, we can construct Brownian motions on manifolds, domains of Euclidean space under mild assumptions. For example, let $D$ be a domain of $\mathbb{R}^d$. Then, we can define the following bilinear form: \begin{equation*} \mathcal{E}(f,g)=\frac{1}{2}\int_{D}(\nabla f, \nabla g)\,dx,\quad f,g \in H^{1}(D), \end{equation*} where $H^{1}(D)$ is the Sobolev space of first order (with Neumann boundary condition). Furthermore, if this form is regular on $L^{2}(\bar{D},dx)$, we can construct reflecting Brownian motion on $\bar{D}$. I think Regularity assumption is very mild assumption. For example, when $D$ has continuous boundary, this form is regular.

My question

I am interested in estimate of heat kernel $p(t,x,y)$ of reflecting Brownian motion as above. In my research, the existence is assured by this paper and references therein. Furthermore, if the Sobolev inequality holds, we can obtain nice (full time) estimate like as \begin{equation} p(t,x,y) \le a_{1}t^{-d/2} \exp(-|x-y|^{2}/a_{2}t)\cdots(1), \end{equation} where $a_1, a_2$ are constants independent of $x,y,t$.

But I don't know nice heat kernel (upper) estimate when the Sobolev inequality is dropped. It is known that if $D$ is expressed as \begin{align*} D_{1}=\{(x_1,x_2)\mid x_{1}>1, 0<x_{2}<e^{-x_1} \}, \end{align*} we can prove Sobolev inequality does not hold. This is heat kernel of reflecting Brownian motion $\left\{X^{1}_{t} \right\}$ on $\bar{D}_{1}$ does not have nice estimate as (1).

On the other hand, if $0<t<1$, the sample path of $\left\{X^{1}_{t} \right\}$ is continuous, its heat kernel has nice estimate as (1). That is, I expected that heat kernel of $\left\{X^{1}_{t} \right\}$ has the following estimate: \begin{equation} p(t,x,y) \le a_{1}t^{-d/2} \exp(-|x-y|^{2}/a_{2}t), \end{equation} for all $x,y \in \bar{D}_{1}$ and $t\in (0,1)$, where $a_1, a_2$ are constants independent of $x,y,t$.

If you know related works on this question, please let me know.


First of all, I do not think Sobolev inequality also holds under the same set of regularity assumptions under which the bilinear form is regular (as a bilinear function over the domain $\bar{D}$). For example, consider a weighted Sobolev space which puts exponential weight on the component with exponential growth rate.

The only/authoritative reference I know about the kind of reflecting( I think what you mean is symmetric in standard terminologies) Brownian motions is [Fukushima-Oshima] Sec 2.4, which is devoted to discussion of Sobolev-type inequality. You can reduce the problem to a estimate problem as a Laplacian but I think a necessary condition to give a better estimate is modulus of continuity on the path of ${X_t^1}$ which allows you to integrate along the path w.r.t. the Dirichlet-form.

The paper you cited requires local structure on the space so it is not completely discarding manifold structure, and that is the reason why it has nicer results than your $D_1$ estimate. But on the other hand I think it is possible to obtain local bound and use a compactness argument to extend to the whole space, which will be interesting to know.

[Fukushima-Oshima]Fukushima, Masatoshi, Yoichi Oshima, and Masayoshi Takeda. Dirichlet forms and symmetric Markov processes. Vol. 19. Walter de Gruyter, 2010.

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