Large Deviations for the Dynamic $\Phi ^{2n}_d$ Model

Abstract

We are dealing with the validity of a large deviation principle for a class of reaction–diffusion equations with polynomial non-linearity, perturbed by a Gaussian random forcing. We are here interested in the regime where both the strength of the noise and its correlation are vanishing, on a length scale $\epsilon$ and $\delta (\epsilon )$, respectively, with $0<\epsilon ,\delta (\epsilon )\ll 1$. We prove that, under the assumption that $\epsilon$ and $\delta (\epsilon )$ satisfy a suitable scaling limit, a large deviation principle holds in the space of continuous trajectories with values both in the space of square-integrable functions and in Sobolev spaces of negative exponent. Our result is valid, without any restriction on the degree of the polynomial nor on the space dimension.

Appendix A

For every $\delta >0$ and $\theta \in \,(0,1)$, we denote

$$\begin{aligned} z_{\delta ,\theta }(t)=\int _0^t (t-s)^{-\frac{\theta }{2}} e^{(t-s)A} dw^\delta (s),\ \ \ \ \ t\ge 0. \end{aligned}$$

(A.1)

In case $\theta =0$, we denote $z_{\delta ,0}(t)=z_\delta (t)$.

Lemma A.1

Under Hypotheses 1 and 2, there exists $\bar{\theta } \in \,(0,1)$ such that for any $\kappa , p\ge 1$ and $T>0$ and for any $\delta \in \,(0,1)$ and $\theta \in \,[0,\bar{\theta })$ we have

$$\begin{aligned} \sup _{t \in \,[0,T]}\,\mathbb {E} |z_{\delta ,\theta }(t)|^\kappa _{p}\le c_{\kappa ,p}(T)\,\Lambda _\theta (\delta )^{\frac{\kappa }{2}}, \end{aligned}$$

(A.2)

where

$$\begin{aligned} \Lambda _\theta (\delta )={\left\{ \begin{array}{ll} \displaystyle {\log \delta ^{-1},} &{} \text {if}\ \alpha =\theta =0,\ d=2,\\ \displaystyle {\delta ^{-(d-2(1-\theta )+\alpha )},} &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Proof

According to (2.3), for every $p\ge 2$ we have

$$\begin{aligned} \begin{array}{l} \displaystyle {\mathbb {E} |z_{\delta ,\theta }(t)|^p_{p}=\mathbb {E} \int _D\left| \,\sum _{k=1}^\infty \int _0^t(t-r)^{-\frac{\theta }{2}}e^{-(t-r)\alpha _k}\lambda _k(\delta ) e_k(\xi )\,d\beta _k(r)\right| ^p\,d\xi }\\ \displaystyle {\le c_p\int _D\left( \int _0^t (t-r)^{-\theta }\sum _{k=1}^\infty e^{-2(t-r)\alpha _k}\lambda _k^2(\delta ) |e_k(\xi )|^2\,dr\right) ^{\frac{p}{2}}\,d\xi }\\ \displaystyle {\le c_p |D| \left( \,\sum _{k=1}^\infty \lambda _k^2(\delta ) k^{\frac{\alpha }{d}}\int _0^t r^{-\theta } e^{-2r \alpha _k}\,dr\right) ^{\frac{p}{2}} \le c_p |D| \left( \,\sum _{k=1}^\infty \frac{\lambda _k^2(\delta )\, k^{\frac{\alpha }{d}}}{\alpha _k^{1-\theta }}\right) ^{\frac{p}{2}}.} \end{array} \end{aligned}$$

Hence, thanks to (2.2) and (3.2), we obtain,

$$\begin{aligned} \mathbb {E} |z_{\delta ,\theta }(t)|^p_{p}\le c_p(T)\,|D|\left( \,\sum _{k=1}^\infty \frac{1}{k^{\frac{2(1-\theta )-\alpha }{d}}(1+\delta k^{\frac{1}{d}})^{2\beta }}\right) ^{\frac{p}{2}}. \end{aligned}$$

(A.3)

Notice that, due to (3.3), there exists $\bar{\theta } \in \,(0,1)$ such that the series above is convergent, for every fixed $\delta >0$ and $\theta \in \,[0,\bar{\theta })$.

We have

$$\begin{aligned} \sum _{k=1}^\infty \frac{1}{k^{\frac{2(1-\theta )-\alpha }{d}} (1+\delta k^{\frac{1}{d}})^{2\beta }}\sim \int _1^\infty \frac{1}{x^{\frac{2(1-\theta )-\alpha }{d}}(1+\delta x^{\frac{1}{d}})^{2\beta }}\,dx, \end{aligned}$$

and then, with a change of variable, we obtain

$$\begin{aligned} \sum _{k=1}^\infty \frac{1}{k^{\frac{2(1-\theta )-\alpha }{d}} (1+\delta k^{\frac{1}{d}})^{2\beta }}\sim d\, \delta ^{-(d-2(1-\theta )+\alpha )} \int _\delta ^\infty \frac{1}{x^{1-(d-2(1-\theta )+\alpha )}(1+x)^{2\beta }}\,dx. \end{aligned}$$

Therefore, if $\alpha =\theta =0$ and $d=2$, since $\beta >0$ we have

$$\begin{aligned} \sum _{k=1}^\infty \frac{1}{k^{\frac{2(1-\theta )-\alpha }{d}} (1+\delta k^{\frac{1}{d}})^{2\beta }}\sim c\,\log \frac{1}{\delta }. \end{aligned}$$

Otherwise, according to Hypothesis 2, there exists $\bar{\theta }>0$ such that

$$\begin{aligned} 2\beta -(d-2(1-\theta )+\alpha )>0, \end{aligned}$$

for every $\theta \in \,[0,\bar{\theta })$. Hence, as $d-2(1-\theta )+\alpha >0$, we get

$$\begin{aligned} \begin{array}{l} \displaystyle {\sum _{k=1}^\infty \frac{1}{k^{\frac{2(1-\theta ) -\alpha }{d}}(1+\delta k^{\frac{1}{d}})^{2\beta }}}\\ \displaystyle {\sim d\, \delta ^{-(d-2(1-\theta )+\alpha )}\int _0^\infty \frac{1}{x^{1-(d-2(1-\theta )+\alpha )}(1+x)^{2\beta }}\,dx\le c\, \delta ^{-(d-2(1-\theta )+\alpha )}.} \end{array} \end{aligned}$$

This implies (A.2), in case $\kappa =p$. The general case follows from the Hölder inequality. $\square$

Next, for every $s>0$, we have

$$\begin{aligned} \mathbb {E}\,|z_{\delta ,\theta }(t)|^2_{H^{-s}(D)}= \sum _{k=1}^\infty \int _0^t (t-r)^{-\theta } e^{-2(t-r)\alpha _k} \lambda _k^2(\delta )\alpha _k^{-s}\,dr. \end{aligned}$$

Therefore, by proceeding as in the proof of Lemma A.1 we conclude

Lemma A.2

Under Hypotheses 1 and 2, there exists $\bar{\theta } \in \,(0,1)$ such that for any $s, T>0$ and for any $\delta \in \,(0,1)$ and $\theta \in \,[0,\bar{\theta })$ we have

$$\begin{aligned} \sup _{t \in \,[0,T]}\,\mathbb {E}\, |z_{\delta ,\theta }(t)|^2_{H^{-s}(D)}\le c(T)\,\Gamma _{\theta ,s}(\delta ), \end{aligned}$$

(A.4)

where

$$\begin{aligned} \Gamma _{\theta ,s}(\delta )={\left\{ \begin{array}{ll} \displaystyle {\log \delta ^{-1},} &{} \text {if}\ \theta =s,\ d=2,\\ \displaystyle {\delta ^{-(d-2(1-\theta )-2s)},} &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

Now, let us consider the linear problem

$$\begin{aligned} dz(t)=Az(t)\,dt+dw^\delta (t),\ \ \ \ z(0)=0. \end{aligned}$$

(A.5)

Its unique mild solution $z_\delta (t)$ coincides with the process $z_{\delta ,0}(t)$ defined in (A.1), for $\theta =0$. Notice that, due to (A.2), we have

$$\begin{aligned} \sup _{t \in \,[0,T]}\,{\mathbb {E}} |z_{\delta }(t)|^\kappa _{L^p(D)}\le c_{\kappa ,p}(T)|D|{\left\{ \begin{array}{ll} \left( \log \delta ^{-1} \right) ^{\frac{\kappa }{2}}, &{} \text {if } \alpha =0 \text { and } d=2,\\ &{}\\ \delta ^{-\frac{\kappa }{2}(d-2+\alpha )}, &{} \text {otherwise.} \end{array}\right. } \end{aligned}$$

(A.6)

By using a stochastic factorization argument, for every $\theta \in \,(0,1)$, we have

$$\begin{aligned} z_\delta (t)=\frac{\sin (\theta \pi )}{2\pi }\int _0^t (t-\sigma )^{\frac{\theta }{2}-1}e^{(t-\sigma )A}z_{\delta ,\theta }(\sigma )\,d\sigma . \end{aligned}$$

If we take $\kappa >2/\theta$, we have

$$\begin{aligned} |z_\delta (t)|^\kappa _H\le c_{\kappa ,\theta }\left( \int _0^T \sigma ^{\left( \frac{\theta }{2}-1\right) \frac{\kappa }{\kappa -1}}\, d\sigma \right) ^{\kappa -1}\int _0^t|z_{\delta ,\theta }(\sigma )|_H^\kappa \, d\sigma \le c_{\kappa ,\theta }(T)\int _0^t|z_{\delta ,\theta }(\sigma ) |_H^\kappa \,d\sigma . \end{aligned}$$

Therefore, if we fix $\gamma >d-2+\alpha$ and we pick $\theta _\gamma \in \,(0,\bar{\theta })$ such that

$$\begin{aligned} d-2(1-\theta _\gamma )+\alpha <\gamma , \end{aligned}$$

thanks to (A.2), we get

$$\begin{aligned} \mathbb {E}\,\sup _{t \in \,[0,T]} |z_\delta (t)|^\kappa _H\le c_{\kappa ,\sigma }(T)\,\delta ^{-\frac{\gamma \kappa }{2}}. \end{aligned}$$

(A.7)

Thus, we have proven the following result.

Lemma A.3

Under Hypotheses 1 and 2, for every $\kappa \ge 2$ and $\delta >0$ we have that for every

$$\begin{aligned} \gamma >d-2+\alpha , \end{aligned}$$

it holds

$$\begin{aligned} \mathbb {E}\sup _{t \in \,[0,T]} |z_\delta (t)|^\kappa _H\le c_{\kappa ,\gamma }(T)\, \delta ^{- \frac{\gamma \kappa }{2}},\ \ \ \ \delta \in \,(0,1). \end{aligned}$$

(A.8)

Finally, by using again a factorization argument, for every $s>0$ and $\kappa >\frac{2}{s}\vee 1$ we have

$$\begin{aligned}&|z_\delta (t)|^\kappa _{H^{-s}(D)}\le c\,\left( \int _0^T \sigma ^{-\left( \frac{s}{2}-1\right) \frac{\kappa }{\kappa -1}}\, d\sigma \right) ^{\kappa -1} \int _0^t |z_{\delta ,s}(\sigma )|_{H^{-s}(D)}^\kappa \,d\sigma \\&\quad \le c(T)\int _0^t |z_{\delta ,s}(\sigma )|_{H^{-s}(D)}^\kappa \,d\sigma . \end{aligned}$$

Therefore, due to (A.4) we can conclude that the following result is true.

Lemma A.4

Under Hypotheses 1 and 2, for every $s>0$, $\delta \in \,(0,1)$ and $\kappa \ge 1$we have that

$$\begin{aligned} \mathbb {E}\sup _{t \in \,[0,T]} |z_\delta (t)|^\kappa _{H^{-s}(D)}\le c_{\rho }(T){\left\{ \begin{array}{ll} \log \delta ^{-1},\ \ \ \ \text {if}\ d=2,\\ \delta ^{-(d-2)},\ \ \ \ \text {if}\ d\ge 3. \end{array}\right. }. \end{aligned}$$

(A.9)