Local solutions of renormalized stochastic PDE

Question

To illustrate the problem consider the mild formulation of the $\Phi^4_2$ model on $[0,T]\times \mathbb{T}^d$: $$\phi=P_r\phi_0+\int_0^rP_{r-q}(-\phi^3(q))dq+Y_r \ \ \ \ \ \ (1)$$ where $(P_r)_{r \geq 0}$ is the semi-group generated by $\Delta,Y$ solves the stochastic heat equation and $\phi_0 \in\mathscr{C}^\alpha (\text{ Holder-Besov space}),\alpha \in ]0,2[.$

The equation isn't deterministic for $d=2,$ this is why we rely on Debussche DaPrato trick to solve it, we need to renormalize, by first introducing $Y_\epsilon:=\mathscr{F}^{-1}\phi_\epsilon*Y,$ where $\phi_\epsilon(x)=\phi(\epsilon x),\phi \in C_{c}^{\infty},\phi(0)=1$ and considering $\phi_\epsilon:=f_\epsilon+Y_\epsilon,$ and letting $Y_\epsilon^{:k:}:=H_{k}(Y_\epsilon,E[Y_\epsilon^2]),H_k$ are Hermite polynomials $k=1,2,3,$ converging to Wick powers $Y^{:k:}$ in $C_T\mathscr{C}^{\alpha-2}.$

We can find several definitions-formulation of the local solution:

1. Debussche Daprato defined the (local) solution of $(1)$ by $\phi=f+Y,$ where $f$ solves $$(\partial_r-\Delta)f=-(Y^{:3:}+3Y^{:2:}f+3Yf^2+f^3)$$ on $[0,T]\times \mathbb{T}^d,T<T^*,T^*$ is a maximal stopping time (actually $T^*=\infty,$ but let's do not worry about this).

2. We can show that a local maximal solution $(v,T^*)$ exists for the equation $$v_r=P_r\phi_0-\int_0^rP_{r-q}(Y^{:3:}(q)+3Y^{:2:}(q)v(q)+3Y(q)v^2(q)+v^3(q))dq$$ such that $v$ depends continuously on $(\phi_0,Y,Y^{:2:},Y^{:3:})$ and a local maximal solution $(v_\delta,T^*_\delta)$ for the approximate equation (that depends on $\delta$) exists such that $\lim_{\delta \to 0}v_\delta=v$ in probability and the convergence doesn't depend on the mollification used.

I am curious about the second part concerning the convergence: in what space does the convergence hold ? Does it mean to prove $$\forall T>0,\lambda>0,\lim_{\delta\to0}P(\sup_{r \in [0,T\wedge T^*\wedge T^*_\delta[}\Vert v_\delta(r)-v(r)\Vert_{\alpha}>\lambda)=0$$

Answer 1

For $L>0$, write $T_\epsilon^L = L \wedge \inf\{t>0\,:\, \|v_\epsilon(t)\|_\alpha \ge L\}$ and similarly for $T^L$ and $v$. Then, one has $v_\epsilon \to v$ in the sense that, for every $\lambda > 0$ and $L>0$ one has $$ \lim_{\epsilon \to 0} \mathbb{P}\Bigl(\sup_{t \le T_\epsilon^L \wedge T^L} \|v_\epsilon(t) - v(t)\|_\alpha > \lambda\Bigr) = 0\;. $$ In fact, given any complete metric space $X$, one can define a space $X^{\mathrm{Sol}}$ of continuous functions with values in $X$ that are allowed to blow up at some finite time. This can be equipped with metric that makes $X^{\mathrm{Sol}}$ a complete metric space and such that the above convergence is just convergence in probability in $(C^\alpha)^{\mathrm{Sol}}$. Furthermore, the solution map is continuous in all parameters (in particular its initial condition). See Section 1.5.1 and in particular Lemma 1.2 of this paper. The construction is quite subtle to guarantee completeness of $X^{\mathrm{Sol}}$ and that elements of that space are forced to blow up at the end of their existence time. Note that all this has nothing to do with the fact that this particular example requires renormalisation. This solution space is useful already for finite-dimensional ODEs with smooth coefficients.