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Denote $E = C([0, 1])$. I am consider a 1-dimensional stochastic heat equation on $h$:

$$\partial_tu(t, x) = \partial_x^2u(t, x) - V'(u(t, x)) + \dot{W}(t, x), \quad\text{ for all } (t, x) \in (0, \infty)\times(0, 1),$$

$$u(t, 0) = 0 \quad\text{for all }t > 0,\qquad u(0) = u \in E.$$

Here $W$ is the space-time white noise, $V$ is a bounded differentiable potential on $\mathbb{R}$ and $V'$ is its derivative.

We know that the mild solution $u(t) \in E$ forms an $E$-valued Markov process. If we denote its semigroup by $P_t$, then it should have a infinitesimal generator $L$ due to the Hille-Yoshida Theorem. We can write out the explicit form of $L$ on functions such as $F(u) = f(\langle u, \phi_1\rangle,..., \langle u, \phi_m\rangle)$ for $f$ and $\phi_j$ with good regularities. But such functions is not enough to form a core of $L$, as I guess.

I am wondering if there is any results on the core of generator of SPDE.

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  • $\begingroup$ A good reference for this, if you haven't already found it, may be Da Prato & Zabczyk, 2nd Order PDEs in Hilbert Spaces, Chapter 12 (Gradient systems), although it considers the case where the state space is Hilbert $H$ and not Banach as in your case. If I understand correctly, the Markov generator is then defined on a space $\mathcal E_A(H)$ of functions of the form $e^{i \langle h, x \rangle}$ which may give a suggestion for your case. Also the book lists many references. A selection of this material can also be found in Da Prato, An introduction to infinite-dimensional analysis (2006). $\endgroup$
    – Joris Bierkens
    Commented Jan 7, 2014 at 8:38
  • $\begingroup$ Only problem is that your nonlinear term is in fact not of gradient form from an infinite-dimensional point of view. Still, the reference I mentioned may be useful. $\endgroup$
    – Joris Bierkens
    Commented Jan 7, 2014 at 9:42
  • $\begingroup$ Thank you very much for helping me. I have read the book about infinite-dimensional analysis by Da Prato. As you said, they built the theories on a Hilbert space but I've proved that it does not matter, so we do not need to mind it. As there result, on $\mathcal{E}_A(E)$ we can give the definition of $L$, and the "real" generator $\mathcal{L}$ (the one appeared in Hille-Yoshida) is actually the Friedrichs extension of $(L, \mathcal{E}_A(E))$. Now I want to find a subspace larger than $\mathcal{E}_A(E)$, such that $\mathcal{L}$ is the closure of its restriction on it. $\endgroup$
    – gregarki khayal
    Commented Jan 7, 2014 at 10:18
  • $\begingroup$ We can find an invariant and ergodic measure $\pi$ for $u(t)$. Roughly, if we define a bilinear form $(\mathcal{E}, \mathcal{E}_A(E))$ such that $\mathcal{E}(F, G)=<-LF, G>_\pi$, then the closure of $\mathcal{E}$ over $L^2(\pi)$ just coincides with the Dirichlet form of $u(t)$; but it does not mean that the closure of $L$ is just $\mathcal{L}$. This is the key problem, I believe. $\endgroup$
    – gregarki khayal
    Commented Jan 7, 2014 at 10:26

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