Timeline for Can a diffusion process admit an invariant measure with a non-differentiable density?
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| S Jul 12, 2023 at 16:05 | history | bounty ended | CommunityBot | ||
| S Jul 12, 2023 at 16:05 | history | notice removed | CommunityBot | ||
| S Jul 4, 2023 at 15:01 | history | bounty started | 0xbadf00d | ||
| S Jul 4, 2023 at 15:01 | history | notice added | 0xbadf00d | Canonical answer required | |
| Jul 4, 2023 at 15:01 | history | edited | 0xbadf00d | CC BY-SA 4.0 |
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| Jul 1, 2023 at 20:00 | comment | added | 0xbadf00d | @m7e I've seen papers on the relation between the invariant measures of the diffusion process and its discretization, but I wasn't able to find formulas for the specific form the invariant measures of the discretizations: Asked for that on MSE. Do you know a reference for that? According to this question, the discretization should have an invariant measure (though we should need to rule out some cases as I've pointed out in my other question). | |
| Jul 1, 2023 at 19:58 | comment | added | 0xbadf00d | @m7e Well, okay, I've assumed in the discussion above that the drift and diffusion coefficients are functions satisfying the usual Lipschitz condition. In that case, $\mathcal D(A)\subseteq C^2(H)$ should hold. Sticking to the Ito diffusion case, I think I should consider the Euler-Maruyama discretization of it. | |
| Jul 1, 2023 at 19:42 | comment | added | m7e | No, $\mathcal{D}(A)\subset C^2(H)$ does not necessarily hold. In fact, there are examples (if the drift coefficient of the diffusion is a distribution) such that $\mathcal{D}(A)$ and $C^2(H)$ are disjoint. | |
| Jul 1, 2023 at 13:37 | comment | added | 0xbadf00d | @m7e Sorry, but I cannot follow. We have $\mathcal D(A)\subseteq C^2(H)$. On the other hand, if $\mu$ is invariant, then $p$ must belong to $\mathcal D(A^\ast)$. If, for example, $A$ is self-adjoint, then this means $p\in C^2(H)$. What am I missing? | |
| Jul 1, 2023 at 13:21 | comment | added | m7e | It seems you're mixing up $C^2_c(H)\subset \mathcal{D}(A)$ with $\mathcal{D}(A)\subset C^2_c(H)$. Otherwise, I don't see how you would arrive to the conclusion that $p$ has to be differentiable (which is not true). | |
| Jun 30, 2023 at 20:29 | comment | added | Thomas Kojar | In SPDEs, I've seen the use of Stochastic-quantization where they start from some very singular measure eg. that with distributional density (Euclidean free field), and construct an SPDE converging to it up to renormalizations. | |
| Jun 30, 2023 at 20:27 | comment | added | Thomas Kojar | Maybe I am misunderstanding, but are you starting with some general measure $\mu$ that has poor regularity and trying to construct a diffusion process that converges to it in law? The closest think I've seen in such a generality is using Dirichlet forms eg. see here "Construction of diffusion processes on fractals, d-sets, and general metric measure spaces" projecteuclid.org/journals/kyoto-journal-of-mathematics/… | |
| Jun 30, 2023 at 14:22 | history | asked | 0xbadf00d | CC BY-SA 4.0 |