Construction of $\varPhi ^4_3$ Diagrams for Pedestrians

Abstract

We aim to give a pedagogic and essentially self-contained presentation of the construction of various stochastic objects appearing in the dynamical $\varPhi ^4_3$ model. The construction presented here is based on the use of paraproducts. The emphasis is on describing the stochastic objects themselves rather than introducing a solution theory for the equation.

Appendix

We now give a second proof of Proposition 4, based on the following logarithmic Sobolev inequality (see [4, Sect. 1.6] and [23, Sects. 1.1 and 1.5]).

Lemma 7

(log-Sobolev inequality) Let $\mu$ be a Gaussian measure and $X \in W^{1,2}(\mu )$. We have

$$\begin{aligned} \mathbb {E}(|X|^{2} \log |X|) \le \mathbb {E}|\mathbf {D}X|^{2} + \frac{1}{2} \mathbb {E}|X|^{2} \log (\mathbb {E}|X|^{2}), \end{aligned}$$

(89)

where $\mathbf {D}$ is the Malliavin derivative, and $\mathbb {E}$ is the expectation taken with respect to $\mu$.

Now, let $T_{t}$ be the Ornstein-Uhlenbeck semigroup defined by

$$\begin{aligned} T_{t} X = \sum _{n=0}^{+\infty } e^{-nt} X_{n}, \end{aligned}$$

(90)

where $X_{n}$ is the component of X in $\mathcal H_{n}$. The Ornstein-Uhlenbeck semigroup is closely related to the Malliavin derivative, because it determines the quadratic form associated with the infinitesimal generator L of $T_t$. More precisely, for sufficiently nice random variables X, Y, we have

$$\begin{aligned} \partial _t \mathbb {E}[(T_t X) Y] = \mathbb {E}[(LX) Y] = - \mathbb {E}[ \langle \mathbf {D}X, \mathbf {D}Y \rangle ]. \end{aligned}$$

(91)

See [23, Sect. 1.4] for a more detailed discussion of these objects. The main use of the logarithmic Sobolev inequality will be to show the following hypercontractivity estimate.

Proposition 6

(Hypercontractivity) Let $T_{t}$ be the Ornstein-Uhlenbeck semigroup. We have

$$\begin{aligned} \big ( \mathbb {E}|T_{t} X|^{q} \big )^{\frac{1}{q}} \le \big ( \mathbb {E}|X|^{p} \big )^{\frac{1}{p}}, \end{aligned}$$

(92)

for all $p \ge 2$ and $q = 1 + (p-1) e^{2t}$.

Proof

(Second proof of Proposition  4) If $X \in \mathcal H_{n}$, then $T_{t} X = e^{-nt} X$, and we can see that Proposition 4 is an immediate consequence of Proposition 6. It then remains to prove Proposition 6. We can assume $X \ge 0$ without loss of generality.

Fix $p \ge 2$. Let $q(t) = 1 + (p-1) e^{2t}$, and let

$$\begin{aligned} F(t) = \mathbb {E}|T_{t} X|^{q(t)}, \qquad G(t) = F(t)^{\frac{1}{q(t)}}. \end{aligned}$$

Our aim is to show that $G'(t) \le 0$ for all $t > 0$, and (92) will follow. In fact, we have

$$\begin{aligned} G'(t) = G(t) \bigg [ - \frac{q'(t)}{q^{2}(t)} \log F(t) + \frac{F'(t)}{q(t) F(t)} \bigg ]. \end{aligned}$$

Since $q'(t) \ge 0$, it suffices to show that

$$\begin{aligned} -\frac{1}{q(t)} F(t) \log F(t) + \frac{F'(t)}{q'(t)} \le 0. \end{aligned}$$

(93)

Noting that

$$\begin{aligned} F'(t) = \mathbb {E}\bigg [ (T_{t}X)^{q(t)} \bigg ( q'(t) \log (T_{t}X) + q(t) \frac{LT_{t}X}{T_{t}X} \bigg ) \bigg ], \end{aligned}$$

we see that (93) is equivalent to

$$\begin{aligned} -\frac{1}{q(t)} F(t) \log F(t) + \mathbb {E}\bigg [ (T_{t}X)^{q(t)} \log (T_{t}X) \bigg ] + \frac{q(t)}{q'(t)} \mathbb {E}\bigg [ (T_{t}X)^{q(t)-1} (LT_{t}X) \bigg ]. \end{aligned}$$

(94)

Applying the log-Sobolev inequality to the random variable $(T_{t}X)^{\frac{q(t)}{2}}$ and using the integration by parts formula (91) which in the current context becomes

$$\begin{aligned} \mathbb {E}\langle \mathbf {D}Y, \mathbf {D}Z \rangle = - \mathbb {E}\big (Y (LZ) \big ), \end{aligned}$$

we see that (94) follows immediately.