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If a sequence of Dirichlet forms convergence to 0, then what about the diffusion processes associated with these Dirichlet forms? Do the finite dimensional distributions of them converges weakly? and what are the limits?

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What do you mean exactly by "finite dimensional distribution of the diffusion processes" associated with the Dirichlet forms? Perhaps the limit (for $t\to \infty$) of the associated semigroups? –  Delio Mugnolo Jan 27 '13 at 19:36
The sequence of Dirichlet forms depend on a parameter, So the convergence of the associated diffusion processes are about this parameter not time $t\rightarrow\infty$. –  John Young Jan 27 '13 at 22:00
Yes, this is clear to me. Say, if the forms are $(a_n)_{n\in \mathbb N}$, then I am talking about convergence of the associated semigroups $((e^{-ta_n})_{t\ge 0})_{n\in \mathbb N}$. My question was - and is -: what is the "finite dimensional distribution" of the diffusion process (governed by $(e^{-ta_n})_{t\ge 0}$, say)? –  Delio Mugnolo Jan 27 '13 at 23:10
yes, That is what I mean. –  John Young Jan 28 '13 at 20:38
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Kato shows in §VI.3 of his book "Perturbation theory for linear operators" that in particular if a sequence of Dirichlet forms (or, more generally, of bounded closed sesquilinear forms) converges to a limiting form $a$ (in a certain quite natural sense), then the associated operators converge to the operator associated with $a$ in the norm resolvent sense. This in turn implies norm convergence of the associated semigroups, e.g. because of the representation of the semigroups via the backward Euler scheme applied to the resolvents. I guess this answers your question about the diffusion processes associated with the forms.

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