Timeline for Infinitesimal generators of stochastic processes
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Jun 18, 2011 at 14:18 | comment | added | Adam Skalski | I have only now noticed John's response - hence the delay. The book of Ma and Rockner deals precisely with the generalization of the symmetric case I described in the last paragraph of my answer above - the standing assumption throughout the book is that the non-symmetric part (of the semigroup/form/generator) is dominated by the symmetric one. | |
| Apr 15, 2011 at 13:06 | comment | added | John Baez | Thanks very much, Adam. I'd really like to avoid the symmetry assumption you mention, because the stochastic processes I'm interested in don't usually satisfy it. Over on my blog, Martin Gisser recommended Introduction to the Theory of (Non-Symmetric) Dirichlet Forms by Z. M. Ma and M. Röckner. I haven't looked at it yet, but the title hints that the usual theory of Dirichlet forms is part of a bigger theory that drops the symmetry assumption. | |
| Apr 11, 2011 at 10:59 | comment | added | Adam Skalski | Indeed, it does - if the measure on the finite set you consider is the uniform one. Symmetry assumption is sometimes viewed as a version of symmetry under the time reversal; this however is not a universal point of view. In the infinite-dimensional framework the symmetry (or at least some aspect of it, as described in the main text) makes it possible to work with semigroups on all $L^p$-spaces, which seems to be a very natural requirement. | |
| Apr 11, 2011 at 10:44 | comment | added | András Bátkai | That is an interesting point what you write. Why is it usual to assume $L^2$-selfadjointness? I am not familiar whith this, but in the matrix case doesn't this mean that you have symmetric matrices? | |
| Apr 11, 2011 at 10:31 | history | answered | Adam Skalski | CC BY-SA 3.0 |