5
$\begingroup$

This question is continuous for the one asked here: Local solutions of renormalized stochastic PDE but it was better to ask it separetely.

Again, we are interested in the local behavior of the $\Phi_2^4$ model after applying Debussche Daprato $\phi=v+Y$: $$(\partial_r-\Delta)v=-(Y^{:3:}+3Y^{:2:}v+3Yv^2+v^3) \\\\ (1),$$ $Y^{:k:}$ are the Wick powers. We also recall that the approximate equation refers to the renormalized equation (by a mollification) of $(1)$ depending on $\epsilon.$

We denote by $T$ the maximal stopping time of $(1)$ (corresponding to $v$), can we find a family of finite stopping times $(T_\epsilon)_\epsilon$ such that $T_\epsilon$ increases to $T$ when $\epsilon$ decreases to 0, $(v_\epsilon,T_\epsilon)$ is a solution for the approximate equation, $\sup_{r\in[0,T_\epsilon]}\Vert v_\epsilon(r)-v(r)\Vert_{\mathscr{C}^\alpha}$ converges in probability to 0 ? How is this related to the continuity dependence?

Proofs or references are appreciated

Improve this question
$\endgroup$

1 Answer 1

Reset to default
5
+100
$\begingroup$

You can take $T_\epsilon = L_\epsilon \wedge \inf\{t > 0\,:\, \|v_\epsilon\| \ge L_\epsilon\}$ for a sequence $L_\epsilon$ going to infinity sufficiently slowly. How slow exactly depends on details of the problem you're looking at, the norms involved, etc. In virtually all cases of interest (in particular your example and other examples of that type), $L_\epsilon = |\log\log \epsilon|$ works.

Improve this answer
$\endgroup$
6
  • $\begingroup$ We still need to ensure thaf $u$ is also a solution, so probably we need: $T_\epsilon:=L_\epsilon \wedge \inf\{r\geq 0,, \max(\Vert v\Vert,\Vert v_\epsilon\Vert) \geq L_\epsilon\}$? $\endgroup$
    – mathex
    Commented Jun 17 at 18:51
  • $\begingroup$ @mathex That depends on how you interpret the statement if $v$ blows up. My interpretation is that there may be a non-zero (but small) probability that $\|v_\epsilon - v\|_{\mathcal{C}^\alpha}$ is infinite, but that doesn't prevent it from converging to zero in probability... $\endgroup$
    – Martin Hairer
    Commented Jun 17 at 20:03
  • $\begingroup$ Last question: denoting $\tau,\tau_\epsilon$ the explosion times of $v,v_\epsilon$ respectively, in the definition of $T_\epsilon,$ should we include the condition $0<t<\tau_\epsilon$ (unless we still have convergence even when it's infinite) $\endgroup$
    – mathex
    Commented Jun 18 at 17:13
  • $\begingroup$ @mathex In all these kind of problems it's always the case that the norm diverges at the explosion time, so I'm not sure I understand what the question is or rather what difference this would make. $\endgroup$
    – Martin Hairer
    Commented Jun 18 at 17:40
  • $\begingroup$ One last inquiry before closing this topic: how do we define the solution of the initial equation in $\phi$? $\endgroup$
    – mathex
    Commented Jun 23 at 21:53

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .