Stochastic quantization associated with the $\exp (\Phi )_2$-quantum field model driven by space-time white noise on the torus

Abstract

We consider a quantum field model with exponential interactions on the two-dimensional torus, which is called the $\exp (\Phi )_{2}$-quantum field model or Høegh-Krohn’s model. In the present paper, we study the stochastic quantization of this model by singular stochastic partial differential equations, which is recently developed. By the method, we construct a unique time-global solution and the invariant probability measure of the corresponding stochastic quantization equation and identify it with an infinite-dimensional diffusion process, which has been constructed by the Dirichlet form approach.

Appendix

1.1 Besov space

Let $(\chi ,\rho )$ be a dyadic partition of unity, i.e., they are smooth radial functions on ${\mathbb {R}}^2$ such that,

• $0\le \chi \le 1$, $0\le \rho \le 1$,

• $\chi$ is supported in $\{x;|x|\le \frac{4}{3}\}$, $\rho$ is supported in $\{x;\frac{3}{4}\le |x|\le \frac{8}{3}\}$,

• $\chi (\xi )+\sum _{j=0}^\infty \rho (2^{-j}\xi )=1$ for any $\xi \in {\mathbb {R}}^2$.

Denote $\rho _{-1}=\chi$ and $\rho _j=\rho (2^{-j}\cdot )$ for $j\ge 0$. Define

$$\begin{aligned} \Delta _jf=\sum _{k\in {\mathbb {Z}}^2}\rho _j(k)\langle f, \mathbf{e }_k\rangle \mathbf{e }_k. \end{aligned}$$

For $s\in {\mathbb {R}}$ and $p,q\in [1,\infty ]$, we define the inhomogeneous Besov norm

$$\begin{aligned} \Vert f\Vert _{B_{p,q}^s}:=\left\| \{2^{js}\Vert \Delta _jf \Vert _{L^p(\Lambda )}\}_{j\ge -1}\right\| _{\ell ^q}. \end{aligned}$$

Proposition A.1

([12, Page 99]). For any $s\in {\mathbb {R}}$, $H^s=B_{2,2}^s$.

1.2 Schauder estimates

Proposition A.2

([42, Propositions 5 and 6]). Let $s\in {\mathbb {R}}$, $p,q\in [1,\infty ]$ and $\mu >0$.

• (i) For every $\delta \ge 0$, $\Vert e^{\frac{1}{2}(\triangle -1)t}u\Vert _{B_{p,q}^{s+2\delta }}\lesssim t^{-\delta }\Vert u\Vert _{B_{p,q}^s}$ uniformly over $t>0$.

• (ii) For every $\delta \in [0,1]$, $\Vert (e^{\frac{1}{2}(\triangle -1)t}-1)u\Vert _{B_{p,q}^{s-2\delta }}\lesssim t^\delta \Vert u\Vert _{B_{p,q}^s}$ uniformly over $t>0$.

Proposition A.3

Let u solve the equation (in the mild form)

$$\begin{aligned} \left\{ \begin{aligned} \partial _tu(t,x)&=\frac{1}{2}(\triangle -1)u(t,x)+U(t,x),\quad t>0,\quad x\in \Lambda ,\\ u(0,\cdot )&=u_0,\quad x\in \Lambda . \end{aligned} \right. \end{aligned}$$

Let $r\in (1,\infty ]$ and define $r'\in [1,\infty )$ by $1/r+1/r'=1$. Then, for any $p,q\in [1,\infty ]$, $\theta \in {\mathbb {R}}$, $\varepsilon >0$, and $\eta \in (0,2/r')$, one has

$$\begin{aligned}&\Vert u\Vert _{L^r([0,T];B_{p,q}^{\theta +2-\varepsilon })\cap C([0,T];B_{p,q}^{\theta +2/r'-\varepsilon })\cap C^{\eta /2}([0,T];B_{p,q}^{\theta +2/r'-\varepsilon -\eta })}\\&\quad \lesssim \Vert u_0\Vert _{B_{p,q}^{\theta +2-\varepsilon }}+ \Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })}. \end{aligned}$$

In particular, for $\beta \in (0,1)$ and $\delta \in (0,1-\beta )$, setting $r=p=q=2$, $\theta = -\,\beta$, $\varepsilon = 1-\beta -\delta$ and $\eta = \delta$, one has

$$\begin{aligned} \Vert u\Vert _{L^2([0,T];H^{1+\delta })\cap C([0,T];H^{\delta })\cap C^{\delta /2}([0,T];L^2)} \lesssim \Vert u_0\Vert _{H^{1+\delta }}+\Vert U\Vert _{L^2([0,T];H^{-\beta })}. \end{aligned}$$

Proof

We decompose

$$\begin{aligned}&u_t=e^{\frac{1}{2}t(\triangle -1)}u_0+\int _0^te^{\frac{1}{2}(t-s)(\triangle -1)}U_s\mathrm{d}s =:u_t^0+u_t^1,\\&u_t-u_s=(e^{\frac{1}{2}(t-s)(\triangle -1)}-1)u_s+\int _s^te^{\frac{1}{2}(t-v)(\triangle -1)}U_v\mathrm{d}v =:u_{ts}^0+u_{ts}^1. \end{aligned}$$

(1) Bound in $L^r([0,T];B_{p,q}^{\theta +2-\varepsilon })$. By Proposition A.2-(i),

$$\begin{aligned} \Vert u_t^0\Vert _{B_{p,q}^{\theta +2-\varepsilon }} \lesssim \Vert u_0\Vert _{B_{p,q}^{\theta +2-\varepsilon }},\quad \Vert u_t^1\Vert _{B_{p,q}^{\theta +2-\varepsilon }} \lesssim \int _0^t(t-s)^{-\frac{2-\varepsilon }{2}}\Vert U_s\Vert _{B_{p,q}^{\theta }}\mathrm{d}s. \end{aligned}$$

By Young’s inequality,

$$\begin{aligned} \Vert u^1\Vert _{L^r([0,T];B_{p,q}^{\theta +2-\varepsilon })} \lesssim \Vert t\mapsto t^{-\frac{2-\varepsilon }{2}}\Vert _{L^1([0,T])} \Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })} \lesssim \Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })}. \end{aligned}$$

(2) Bound in $L^\infty ([0,T];B_{p,q}^{\theta +2/r'-\varepsilon })$. By Proposition A.2-(i),

$$\begin{aligned} \Vert u_t^1\Vert _{B_{p,q}^{\theta +2/r'-\varepsilon }} \lesssim \int _0^t (t-s)^{-\left( \frac{1}{r'}-\frac{\varepsilon }{2}\right) }\Vert U_s\Vert _{B_{p,q}^{\theta }}\mathrm{d}s. \end{aligned}$$

By Young’s inequality,

$$\begin{aligned} \Vert u^1\Vert _{L^\infty ([0,T];H^{\theta +2-2/r-\varepsilon })}&\lesssim \Vert t\mapsto t^{-\left( \frac{1}{r'}-\frac{\varepsilon }{2}\right) }\Vert _{L^{r'}([0,T])} \Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })} \\&\lesssim \Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })}. \end{aligned}$$

(3) Bound in $C^{\eta /2}([0,T];B_{p,q}^{\theta +2/r'-\varepsilon -\eta })$. By Proposition A.2-(ii) and the bound in $L^\infty ([0,T];B_{p,q}^{\theta +2/r'-\varepsilon })$,

$$\begin{aligned} \Vert u_{ts}^0\Vert _{B_{p,q}^{\theta +2/r'-\varepsilon -\eta }} \lesssim (t-s)^{\eta /2}\Vert u_s\Vert _{B_{p,q}^{\theta +2/r'-\varepsilon }} \lesssim (t-s)^{\eta /2}\Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })}, \end{aligned}$$

and by Proposition A.2-(i),

$$\begin{aligned} \Vert u_{ts}^1\Vert _{B_{p,q}^{\theta +2/r'-\varepsilon -\eta }}&\lesssim \int _s^t (t-v)^{-\left( \frac{1}{r'}-\frac{\varepsilon }{2}-\frac{\eta }{2}\right) }\Vert U_v\Vert _{B_{p,q}^{\theta }}\mathrm{d}v\\&\lesssim \left( \int _s^t(t-v)^{-1+(\varepsilon +\eta )\frac{r'}{2}} \mathrm{d}v\right) ^{\frac{1}{r'}}\left( \int _s^t\Vert U_v\Vert _{B_{p,q}^{\theta }}^r\mathrm{d}v\right) ^{\frac{1}{r}}\\&\lesssim (t-s)^{(\varepsilon +\eta )/2}\Vert U\Vert _{L^r([0,T];B_{p,q}^{\theta })}. \end{aligned}$$

Then $u\in C^{\eta /2}([0,T];B_{p,q}^{\theta +2/r'-\varepsilon -\eta })$ implies $u\in C([0,T];B_{p,q}^{\theta +2/r'-\varepsilon -\eta })$. Since $\eta ,\varepsilon >0$ are arbitrary small, one has $u\in C([0,T];B_{p,q}^{\theta +2/r'-\varepsilon })$ for any $\varepsilon >0$. $\square$