Timeline for On the existence and uniqueness of solution to SPDE with nonlinear growth coefficients
Current License: CC BY-SA 3.0
6 events
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| Jan 23, 2014 at 17:59 | comment | added | epsilon | Joris, thanks again for the comments. That's exactly the thought I had before. In fact, the form of this particular SPDE was motivated by some of Mao's recent work on SDEs. | |
| Jan 23, 2014 at 10:35 | history | edited | Joris Bierkens | CC BY-SA 3.0 |
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| Jan 23, 2014 at 10:27 | comment | added | Joris Bierkens | @MartinHairer: Thanks for pointing this out, I overlooked the effect of that nonlinearity. I proposed the substitution mostly to enforce nonnegativity. I have now two questions that might still help towards an answer: i) is it true that solutions of the equation without nonlinear part are a.e. nonnegative (as I would expect), and ii) can the formulation of X. Mao be extended to Hilbert space setting (in Da Prato & Zabczyk I encounter a weaker variant). This could work since $F[u](x) := -u^2(x) \mathbb 1_{u(x) \geq 0}$ still has this 'dissipativity' property. I'll look into this. | |
| Jan 22, 2014 at 22:27 | comment | added | epsilon | Thank you all for the comments. In finite dimensional case, I know a way to derive global existence. Thanks to local Lip condition, a local solution exists. Then one argues that the explosion time is $\infty$ a.s. I am not sure whether similar ideas work in infinite dimensions. | |
| Jan 22, 2014 at 18:04 | comment | added | Martin Hairer | The problem is that when you write $({\partial \over \partial x})^2 \Psi_t$ you really mean $({\partial \Psi_t \over \partial x})^2$. But this makes no sense because the solution is not differentiable... The transformation you are performing is the inverse of the Cole-Hopf transformation, so you end up with a KPZ-type equation which is certainly not easier to analyse than the original equation by any stretch. (See my recent paper on "Solving the KPZ equation"...) | |
| Jan 22, 2014 at 11:08 | history | answered | Joris Bierkens | CC BY-SA 3.0 |