I have a question about heat kernel.
Definition
Let $(X,\mu)$ be a $\sigma$-finite measure space and $L$ be a densely defined closed linear operator on real Hilber space $L^{2}(X)$ such that
- $(Lu,u)_{L^{2}}\leq 0$ for all $u \in D(L)$
- $(\alpha -L)(D(L))=L^{2}(X)$ for some $\alpha>0$
Then there exists strongly continuous contraction semigroup (denoted by $e^{tL})$ on $L^{2}(X)$ corresponding to $L$ (Hille-Yosida's theorem) If $e^{tL}$ is given by a kernel $p(t,x,y)$, that is, a measurable function on $X \times X$ such that
$e^{tL}f(x)=\int_{X}p(t,x,y)f(y)\,\mu(dy)$ a.e. $x \in X$ for all $t>0$, $f \in L^{2}(X)$
and such that $|p(t,x,y)| \leq Ct^{-d/2}$ for all $t>0$, where $C,d$ are positive constants (independent of $t,x,y$). We call the kernel $p(t,x,y)$ the heat kernel of $L$.
My question
Are there any sufficient conditions for the existence of heat kernel?
When $X$ is a infinite dimensional space, is it difficult for heat kernel to exist?
If you know, please tell me. Thank you in advance?