Appendix1.1 Gronwall-Type Inequalities

The first subsection of Appendix is devoted to Gronwall-type inequality for the Fokker–Planck equation. In particular, we show how the expectation of an observable v with respect to \(\rho _,\lambda }\) can be controlled by means of a simple Lyapunov condition on \(\text \, v\), in the spirit of Sect. 7.1 in [49]. A generalisation of Gronwall inequality to systems of differential inequalities is also discussed. These results are extensively used in the next subsections.

Let us start by the following

Proposition 5.1

Let \(v\in C^2(}^d)\cap L^1(\rho _\text )\,\), \(C_0,\,C_1\in }\) such that for all \(x\in }^d\)

$$\begin \text \, v(x) \,\le \, C_0 \,+\, C_1\, v(x). \end$$

(206)

Suppose that for all \(T>0\)

$$\begin & \sup _\,\int _}^d}|v(x)| \,\rho _}(x)\,\mathrm dx \,<\infty , \quad \sup _\,\int _}^d}|\text \, v(x)|\, \rho _}(x)\,\mathrm dx \,<\infty ,\nonumber \\ & \sup _\,\int _}^d}|\nabla v(x)|\, \rho _}(x)\,\mathrm dx \,<\infty . \end$$

(207)

Then for all \(t>0\) we have:

$$\begin \int _}^d} v(x)\,\rho _}(x)\,\mathrm dx \,\le \, e^\int _}^d} v(x)\,\rho _\text (x)\,\mathrm dx \,+\, \left( e^-1\right) \,\frac. \end$$

(208)

Proof

Consider a sequence \(\psi _N\in C^2(}^d)\) such that

$$\begin \psi _N(x)\,=\, 1 & \text \quad |x|\le N\\ 0 & \text \quad |x|\ge N+1 \end\right. } \end$$

(209)

and \(\psi _N\,\), \(\nabla \psi _N\,\), \(\text \,\psi _N\) are uniformly bounded. We have \(\varphi _N\equiv v\,\psi _N\in C^2_c(}^d)\), hence by identity (126)

$$\begin \begin& \int _}^d}\varphi _N(x)\rho _}(x)\,\mathrm dx \,- \int _}^d}\varphi _N(x)\rho _\text (x)\,\mathrm dx \; =\; \int _0^t\!\int _}^d} \text\varphi _N(x)\rho _,\lambda }(x)\,\mathrm dx\,\mathrm ds. \end\nonumber \\ \end$$

(210)

Now,

$$\begin \text \varphi _N \,=\, \text \, v\;\psi _N \,+\, v\,\text \psi _N \,+\, 2\,\nabla v\cdot (\Lambda \beta )^\nabla \psi _N \end$$

(211)

and hence, hypothesis (207) guarantees that there is dominated convergence in (210). Precisely letting \(N\rightarrow \infty \), we find

$$\begin \begin& \int _}^d}v(x)\rho _}(x)\,\mathrm dx \,- \int _}^d}v(x)\rho _\text (x)\,\mathrm dx \,=\, \int _0^t\!\int _}^d} \text\, v(x)\rho _,\lambda }(x)\,\mathrm dx\,\mathrm ds. \end \end$$

(212)

As a consequence, \(t\mapsto \int _}^d}v(x)\rho _}(x)\,\mathrm dx\) is absolutely continuous on compact subsets of \([0,\infty )\) and there exists

$$\begin \frac\int _}^d}v(x)\rho _}(x)\,\mathrm dx \;=\, \int _}^d}\text \, v(x)\rho _}(x)\,\mathrm dx \end$$

(213)

for a.e. \(t>0\,\). Then by hypothesis (206), we have

$$\begin \frac\int _}^d}v(x)\rho _}(x)\,\mathrm dx \;\le \; C_0 + C_1 \int _}^d} v(x)\rho _}(x)\,\mathrm dx. \end$$

(214)

Thesis (208) finally follows applying Gronwall inequality (see Lemma 5.4). \(\square \)

The previous result can be extended to observables v that are not known to be integrable a priori:

Proposition 5.2

Let \(v\in C^2(}^d)\cap L^1(\rho _\text )\,\), \(C_0,\,C_1\in }\) such that inequality (206) holds true for all \(x\in }^d\,\). Suppose \(v\ge 0\) and

$$\begin v(x)\rightarrow \infty \ \ \text |x|\rightarrow \infty . \end$$

(215)

Then, inequality (208) holds true for all \(t>0\,\).

Proposition 5.2 is essentially a reinterpretation of Theorem 7.1.1 in [49] for the Fokker–Planck operator \(\text \) defined in (122). Note in particular that thanks to the regularity of our setting, we do not need any assumption on the signs of \(C_0,\,C_1\) unlike in [49].

Proof

Let \(\zeta _N\in C^2([0,\infty ))\) such that

$$\begin \zeta _N(r) \,=\, r & \text \quad r\le N-1 \\ N & \text \quad r\ge N+1 \end\right. }, \end$$

(216)

\(0\le \zeta _N'\le 1\) and \(\zeta _N''\le 0\,\). Observe that \(\varphi _N\,\equiv \,\zeta _N\circ v - N\,\in C^2_c(}^d)\) since \(v(x)\rightarrow \infty \) as \(|x|\rightarrow \infty \), hence applying identities (130)–(129) to \(\varphi _N\) we find:

$$\begin \frac\,\int _}^d} \zeta _N(v(x)) \rho _}(x)\,\mathrm dx \,=\, \int _}^d} \text (\zeta _N\circ v)(x) \rho _}(x)\,\mathrm dx \end$$

(217)

for all \(t>0\) and

$$\begin \int _}^d} \zeta _N(v(x)) \rho _}(x)\,\mathrm dx \,\xrightarrow [t\rightarrow 0]\, \int _}^d} \zeta _N(v(x)) \rho _\text (x)\,\mathrm dx. \end$$

(218)

Now observe that since \(\nabla (\zeta _N\circ v) = (\zeta _N'\circ v)\,\nabla v\) and \(\text (\zeta _n\circ v) = (\zeta _N''\circ v)\,\nabla v\,\nabla ^T\!v \,+\, (\zeta _N'\circ v)\text \, v\,\), by concavity of \(\zeta _N\), we have

$$\begin \text (\zeta _N\circ v) \,\le \, (\zeta _N'\circ v)\,\text \, v. \end$$

(219)

Using the properties of \(\zeta _N\), one can check that \(\zeta _N'(r)\,r\le \zeta _N(r)\) for all \(r\ge 0\). Therefore, using hypothesis (206) it follows that

$$\begin \text (\zeta _N\circ v) \,\le \, C_0 \,+\, C_1\, (\zeta _N\circ v). \end$$

(220)

Substituting estimate (220) into (217) we find

$$\begin \frac\,\int _}^d} \zeta _N(v(x)) \rho _}(x)\,\mathrm dx \;\le \; C_0\,+\, C_1\,\int _}^d} \zeta _N(v(x)) \rho _}(x)\,\mathrm dx. \end$$

(221)

Therefore, applying Gronwall inequality (see Lemma 5.4), from (221), (218) we obtain

$$\begin \begin \int _}^d} \zeta _N(v(x)) \rho _}(x)\,\mathrm dx \,\le \, e^\,\int _}^d} \zeta _N(v(x)) \rho _\text (x)\,\mathrm dx \,+\,\left( e^-1\right) \,\frac. \end\qquad \end$$

(222)

Finally, the thesis (208) follows by monotone convergence letting \(N\rightarrow \infty \,\). \(\square \)

Remark 5.3

The hypothesis of Propositions 5.1 and 5.2 can be weakened. Instead of bound (206), it suffices to assume:

$$\begin \text \, v(x) \,\le \, u_0(x) \,+\, C_1\, v(x) \end$$

(223)

where the function \(u_0\in \cap _L^1(\rho _})\,\) is such that

$$\begin \sup _\int _}^d} u(x) \rho _}(x)\,\mathrm dx \,\le \, C_0. \end$$

(224)

For completeness, let us briefly recall the classical Gronwall inequality used in the previous proofs.

Lemma 5.4

(Gronwall inequality). Let \(f\!:[0,\infty )\rightarrow }\) be absolutely continuous on compact sets. Let \(a,b\!:[0,\infty )\rightarrow }\) continuous such that

$$\begin f'(t) \,\le \, a(t)\,f(t) \,+\, b(t) \end$$

(225)

for almost every \(t>0\). Then setting

$$\begin e(t) \,\equiv \, \exp \int _0^t a(s)\,\mathrm ds, \end$$

(226)

we have for all \(t\ge 0\)

$$\begin f(t) \,\le \, e(t)\,f(0) \,+\, e(t) \int _0^t \frac\,\mathrm ds. \end$$

(227)

In particular, if a, b are constant, inequality (227) becomes:

$$\begin f(t) \,\le \, e^\,f(0) \,+\, \big (e^-1\big )\,\frac. \end$$

(228)

The next lemma deals with a system of differential inequalities, extending the classical Gronwall inequality. This result was essentially due to [51]. A proof can be obtained also by adapting Lemma E.4 in [52] to the two-dimensional case with two absolutely continuous functions. The symbol \(\preceq \) will denote a componentwise inequality; namely, we write \((u_1,u_2)\preceq (v_1,v_2)\) for “\(\,u_1\le v_1\) and \(u_2\le v_2\)”.

Lemma 5.5

(Comparison lemma for systems of differential inequalities). Let \(f_1,\,f_2\!:[0,\infty )\rightarrow }\) be absolutely continuous functions on compact sets; let \(f\equiv (f_1,f_2)\,\). Let \(A\equiv (A_1,A_2):}\times }^2\rightarrow }^2\) be a Lipschitz continuous function such that

$$\begin f'(t) \,\preceq \, A\big (t,f(t)\big ) \end$$

(229)

for almost every \(t>0\). Suppose that:

i.

the map \(\xi _2\mapsto A_1(t,\xi _1,\xi _2)\) is non-decreasing on \(}\), for every \((t,\xi _1)\in [0,\infty )\times }\,\);

ii.

the map \(\xi _1\mapsto A_2(t,\xi _1,\xi _2)\) is non-decreasing on \(}\), for every \((t,\xi _2)\in [0,\infty )\times }\,\).

Then, we have

$$\begin f(t) \,\preceq \, g(t) \end$$

(230)

for all \(t\ge 0\,\), where g is the unique solution of the Cauchy problem

$$\begin \,g'(t) = A(t,g(t)) \\ \,g(0) = f(0) \end\right. }. \end$$

(231)

In particular, if A is linear and time-independent, namely \(A(t,\xi )=A\,\xi +B\) for a suitable \(2\times 2\) real matrix \(A\equiv \beginA_ & A_ \\ A_ & A_ \end\) with \(A_,A_\ge 0\) and a suitable vector \(B\equiv \beginB_1 \\ B_2 \end\in }^2\), then inequality (230) becomes:

$$\begin f(t) \,\preceq \, e^\,f(0) \,+\, \big (e^-I_2\big )\, A^ B. \end$$

(232)

1.2 Expectations of Polynomials with Respect to \(\rho _,\lambda }\,\): Proof of Theorem 4.3

Assumption (A2) provides polynomial bounds on the derivatives of V. Theorem 4.3 will follow if we show that any positive power of the variables \(|x_1|\,\), \(|x_2|\) has uniformly bounded expectations in the measure \(\rho _,\lambda }\) for all \(t>0\,\), \(\lambda \ge 1\,\). Because of the different time scales in the evolution of the variables \(x_1,\,x_2\), we have to start by the following auxiliary

Proposition 5.6

For \(x=(x_1,x_2)\in }^d\), let

$$\begin v_(x) \;\equiv \; \frac\,|x_1|^ +\, |x_2|^. \end$$

(233)

In the hypothesis of Theorem 4.1, for every \(r\in [1,\infty )\) there exists a finite nonnegative constant \(M_=M_(\beta _1,\beta _2,\rho _\text ,V)\) such that for all \(t>0\), \(\lambda \ge 1\)

$$\begin \int _}^d} v_(x)^r\rho _,\lambda }(x)\;\mathrm dx \;\le \, M_. \end$$

(234)

In particular, by assumption (A3) there exist \(a\in (0,\infty )\), \(a_0\in [0,\infty )\) such that for every \(x\in }^d\)

$$\begin x\cdot \nabla V(x) \,\ge \, a\,|x|^2 - a_0, \end$$

(235)

and then, a suitable choice for \(M_\) is given by

$$\begin M_\equiv & \int _}^d}|x|^\rho _\text (x)\,\mathrm dx \,+\,\frac}\nonumber \\ & \bigg (a_0\,+\,\frac+\frac+\frac\bigg )^. \end$$

(236)

Proof

We are going to apply Proposition 5.2 to \(v_\lambda ^r\). Computing first- and second-order derivatives, one finds

$$\begin \begin \text_\lambda \left( v_\lambda (x)^r\right) =& -\,\frac\;v_\lambda (x)^\, x \cdot \nabla V(x) \,+\, \frac\;v_\lambda (x)^\, \bigg (\frac+\frac\bigg ) \,+ \\& + \frac\; v_\lambda (x)^\;2(r-1)\, \bigg (\frac+\frac\bigg ) \\\le& -\frac\; v_\lambda (x)^\left( x\cdot \nabla V(x) \,-\, k_r\right) , \end \end$$

(237)

where we set \(k_r\equiv \frac+\frac+\frac\,\). Using assumption (A3) and taking \(\lambda \ge 1\), we have for all \(x\in }^d\)

$$\begin x\cdot \nabla V(x) \;\ge \; a\,|x|^2 - a_0 \;\ge \; a\,v_\lambda (x) - a_0. \end$$

(238)

Hence:

$$\begin \text _\lambda \left( v_\lambda (x)^r\right)\le & -\,\frac\;v_\lambda (x)^\,\big (a\, v_\lambda (x) -a_0 -k_r\big ) \nonumber \\\le & -\frac\, \bigg (\frac\; v_\lambda (x)^r - m_r\bigg ) \end$$

(239)

where we set \(-m_r\equiv \min _\left( \frac\,\xi ^r-(a_0+k_r)\,\xi ^\right) = -\big (\frac\big )^\big (\frac\big )^r\). Finally, from inequality (239) and Proposition 5.2 it follows that for all \(t>0\,\), \(\lambda \ge 1\)

$$\begin \begin \int _}^d} v_\lambda (x)^r\rho _,\lambda }(x)\,\mathrm dx \,\le \;&e^}\int _}^d} v_\lambda (x)^r\rho _\text (x)\,\mathrm dx \;+\, \big (1-e^}\big )\,\frac \\\le& \int _}^d} |x|^ \rho _\text (x)\,\mathrm dx \;+\, \frac. \end \end$$

(240)

\(\square \)

Corollary 5.7

Let \(r\in [0,\infty )\,\). In the hypothesis of Theorem 4.1, we have:

$$\begin \sup _\,\int _}^d} |x_2|^r \rho _,\lambda }(x)\;\mathrm dx \;\le \; M_r, \end$$

(241)

where for \(r\ge 2\,\) \(M_\) is defined by Proposition 5.6, while for \(r<2\) we set \(M_r\equiv 1-\frac+\frac\,M_2\,\).

Proof

It follows by Proposition 5.6 since

$$\begin |x_2|^r \le \, v_\lambda (x_1,x_2)^\frac \end$$

(242)

for all \((x_1,x_2)\in }^d\). In addition, if \(r\in [0,2)\), one has \(\xi ^r\le 1-\frac+\frac\,\xi ^2\) for all \(\xi \ge 0\,\). \(\square \)

Corollary 5.8

Let \(r\in [0,\infty )\,\). In the hypothesis of Theorem 4.1, we have for all \(\lambda \ge 1\)

$$\begin \sup _\;\int _}^d} |x_1|^r\rho _,\lambda }(x)\;\mathrm dx \;\le \; \lambda \;M_r, \end$$

(243)

where \(M_\) is defined as in Corollary 5.7.

Proof

It follows by Proposition 5.6 since \( |x_1|^r \le \lambda \,v_\lambda (x_1,x_2)^\frac \,\). \(\square \)

Proposition 5.9

In the hypothesis of Theorem 4.1 plus (A4), for every \(r\in [0,\infty )\) there exists a finite nonnegative constant \(M_r'=M_r'(\beta _1,\beta _2,\rho _\text ,V)\) such that

$$\begin \sup _\,\int _}^d} |x_1|^r\rho _,\lambda }(x)\;\mathrm dx \;\le \; M_r'. \end$$

(244)

In particular by assumption (A4), there exist \(a_1\in (0,\infty )\), \(a_0,a_2,p\in [0,\infty )\) such that for every \((x_1,x_2)\in }^d\)

$$\begin x_1\cdot \nabla _V(x_1,x_2) \,\ge \, a_1\,|x_1|^2 - a_2\,|x_2|^p -a_0, \end$$

(245)

then a suitable choice for \(M_r'\) is given by:

$$\begin \begin M_r' \,\equiv \, \int _}^d}|x_1|^r\rho _\text (x)\,\mathrm dx + \frac\,\big (2\,a_2\,M_+r\,m_r\big ) & \text\quad r>2 \\ \int _}^d}|x_1|^2\rho _\text (x)\,\mathrm dx + \frac\,\big (a_2\,M_+a_0+\frac\big ) & \text\quad r=2 \\ 1-\frac+\frac\,M'_2 & \text\quad r<2 \end\right. }, \end\qquad \end$$

(246)

where \(M_\) is defined by Corollary 5.7 and \(m_r\equiv -\min _\big (\frac\,\xi ^r - a_2\,\frac\,\xi ^ - (a_0+\frac)\,\xi ^\big )\,\).

Proof

Let \(r>2\). We want to apply Proposition 5.1 to

$$\begin v(x)\,\equiv \, |x_1|^r. \end$$

(247)

First of all observe that v satisfies the integrability hypothesis (207) thanks to Corollary 5.8 and assumption (A2) for \(\nabla V\). Now, computing first and second derivatives we have

$$\begin \text \, v(x) \,=\, -r\,|x_1|^ \left( x_1\cdot \nabla _V(x) - k_r\right) , \end$$

(248)

setting \(k_r\equiv \frac\,\). By assumption (A4)

$$\begin x_1\cdot \nabla _V(x) \,\ge \, a_1\,|x_1|^2 - a_2\,|x_2|^p - a_0, \end$$

(249)

hence

$$\begin \text \, v(x) \,\le -r\, \big (a_1\,|x_1|^r - a_2\,|x_1|^\,|x_2|^p - (a_0+k_r)\,|x_1|^ \big ). \end$$

(250)

Young’s inequality guarantees that

$$\begin |x_1|^\,|x_2|^p \,\le \, \frac\,|x_1|^ + \frac\,|x_2|^ \end$$

(251)

for every \(\sigma ,\tau >1\,\), \(\sigma ^+\tau ^=1\,\). For example, \(\sigma \equiv \frac\) ensures \((r-2)\,\sigma <r\) and enforces \(\tau \equiv r-1\,\). For this choice of \(\sigma ,\tau \), we obtain:

$$\begin \text \, v(x)\le & -r\, \bigg (a_1\,|x_1|^r - a_2\,\frac\,|x_1|^ - \frac\,|x_2|^ - (a_0+k_r)\,|x_1|^ \bigg )\nonumber \\\le & -r \, \bigg (\frac\,|x_1|^r - \frac\,|x_2|^ - m_r \bigg ) \end$$

(252)

where we set \(-m_r\equiv \min _\big (\frac\,\xi ^r - a_2\,\frac\,\xi ^ - (a_0+k_r)\,\xi ^\big )\,\). By Corollary 5.7, we know that for all \(t>0\,\), \(\lambda \ge 1\)

$$\begin \int _}^d}|x_2|^\rho _,\lambda }(x_1,x_2)\,\mathrm dx_1\,\mathrm dx_2 \;\le \, M_. \end$$

(253)

Therefore by Proposition 5.1 and Remark 5.3, we have for all \(t>0\,\), \(\lambda \ge 1\)

$$\begin \int _}^d}|x_1|^r\rho _,\lambda }(x)\,\mathrm dx \,&\le \;&e^t} \int _}^d}|x_1|^r\rho _\text (x)\,\mathrm dx +\big (1-e^t}\big )\,\Big (\frac}+\frac\Big ) \nonumber \\ \le & \int _}^d}|x_1|^r\rho _\text (x)\,\mathrm dx \,+\, \frac}+\frac \end$$

(254)

which concludes the proof in the case \(r>2\). If \(r=2\), Young inequality is not needed and inequality (250) suffices to apply Proposition 5.1 and Remark 5.3, obtaining for all \(t>0\,\), \(\lambda \ge 1\)

$$\begin \begin \int _}^d}|x_1|^2\rho _,\lambda }(x)\,\mathrm dx \;\le \;&e^ \int _}^d}|x_1|^2\rho _\text (x)\,\mathrm dx \,+\, \big (1-e^\big )\,\frac+k_2'} \\\le& \int _}^d}|x_1|^2\rho _\text (x)\,\mathrm dx \,+\, \frac+k_2'}. \end \end$$

(255)

Finally, if \(r<2\), we can just use the bound \(|x_1|^r\le 1-\frac+\frac|x_1|^2\) and come back to the previous case. \(\square \)

Proof of Theorem 4.3

It follows by combining assumption (A2) with Corollary 5.7 and Proposition 5.9. \(\square \)

Proof of Proposition 4.6

We rely on Corollary 7.3.8 in [49] that ensures

$$\begin \Vert \rho \Vert _}^d\!\times (0,T))} \,<\infty , \end$$

(256)

provided \(\rho _\text \) is bounded on \(}^d\) (assumption (B1)) and

$$\begin \int _0^T \!\!\int _}^d} |\nabla V(x)|^\rho _}(x)\,\mathrm dx\,\mathrm dt \,<\infty \,, \end$$

(257)

which in turn follows from Corollaries 5.7, 5.8 together with assumption (A2). We also make use of the inequality \(|\!\log \xi |^2\,\xi \le \root 4 \of \,\) for \(\xi \in [0,1]\,\). Therefore:

$$\begin & \int _} |\log \rho _}(x)|^2\,\rho _}(x)\,\mathrm dx \le \left( \,\log \Vert \rho \Vert _}^d\!\times (0,T))}\right) ^2;\end$$

(258)

$$\begin & \int _\}} |\log \rho _}(x)|^2\,\rho _}(x)\,\mathrm dx \le \int _}^d} e^}\,\mathrm dx;\end$$

(259)

$$\begin & \int _<\,\rho _t<1\}} |\log \rho _}(x)|^2\,\rho _}(x)\,\mathrm dx \le \int _}^d} |x|^2 \rho _}(x)\,\mathrm dx. \end$$

(260)

The r.h.s. of (258) is finite by (256). The r.h.s. of (260) has finite supremum over \(t\in (0,T)\) by Corollaries 5.7, 5.8. This concludes the proof. \(\square \)

Proof of Theorem 4.7

We refer to Theorem 7.4.1 in [49]. It ensures that (136) is a consequence of two integrability conditions:

$$\begin&\int _0^T \!\!\int _}^d} |\nabla V(x)|^2\,\rho _}(x)\,\mathrm dx\,\mathrm dt <\infty , \end$$

(261)

$$\begin&\int _0^T \!\!\int _}^d} \log ^2\max (1,|x|)\,\rho _}(x)\,\mathrm dx\,\mathrm dt <\infty \end$$

(262)

which in turn follow from Corollaries 5.7 and 5.8. \(\square \)

Proof of Proposition 4.11

Since \(\rho ^}\) is solution of a suitable Fokker–Planck equation with drift \(\frac\,\langle \nabla _V\rangle _t\) (Theorem 4.8), Corollary 7.3.8 in [49] ensures that

$$\begin \Vert \rho ^}\Vert _}^\!\times (0,T))} \,<\infty , \end$$

(263)

provided \(\rho _\text ^}\) is bounded on \(}^\) (assumption (B2)) and

$$\begin \int _0^T \!\!\int _}^} \big |\langle \nabla _V\rangle _t(x_2)\big |^\rho _}^}(x_2)\;\mathrm dx_2\,\mathrm dt \,<\infty , \end$$

(264)

which is true by Corollaries 5.7 and 5.8 and Jensen inequality. The proof is then concluded by miming the proof of Proposition 4.6. \(\square \)

Proof of Theorem 4.12

Being \(\rho ^}\) solution of a suitable Fokker–Planck equation, we may refer to Theorem 7.4.1 in [49]. The latter ensures that (149) follows from two integrability conditions:

$$\begin&\int _0^T \!\!\int _}^}\! \big |\langle \nabla _V\rangle _t(x_2)\big |^2\,\rho _}^}(x_2)\,\mathrm dx_2\,\mathrm dt<\infty , \end$$

(265)

$$\begin&\int _0^T \!\!\int _}^}\! \log ^2\max (1,|x_2|)\,\rho _}^}(x_2)\,\mathrm dx_2\,\mathrm dt <\infty \end$$

(266)

which in turn follow from Corollaries 5.7 and 5.8. \(\square \)

1.3 Expectations of Polynomials with Respect to \(\rho _}^}\rho _,\lambda }^}\,\): Proof of Theorem 4.4

Assumption (A2) ensures polynomial bounds for the derivatives of V. We show that the conditional expectation of a polynomial in \(|x_1|\) with respect to the measure \(\rho _}^}(x_1|x_2)\) is bounded by a polynomial in \(|x_2|\). Then from Sect. 5.2 we already know that this quantity has uniformly bounded expectations with respect to the measure \(\rho _,\lambda }^}\) for all \(t>0\,\), \(\lambda \ge 1\,\), and hence, we can prove Theorem 4.4. In particular, the assertion about derivatives of the effective potential \(F\) follows as they can be expressed in terms of conditional expectation of products of derivatives of V.

Proposition 5.10

Suppose that V verifies the bounds (26) for \(\nu =0\) and (29). Then for every \(r\in [0,\infty )\) there exist \(s_r=s_r(V)\in [0,\infty )\) and two finite nonnegative constants \(C_=C_(\beta _1,V)\,\), \(C_=C_(V)\) such that for every \(x_2\in }^\)

$$\begin \int _}^} |x_1|^r\,\rho _}^}(x_1|x_2)\;\mathrm dx_1 \;\le \; C_ + C_ \,|x_2|^. \end$$

(267)

In particular, by hypothesis there exist \(a_1,a_2\in (0,\infty )\), \(a_0\in [0,\infty )\), \(m_1,m_2\in [2,\infty )\), \(\gamma _0,\gamma _1,\gamma _2\in [0,\infty )\) such that for every \((x_1,x_2)\in }^d\)

$$\begin a_1\,|x_1|^2 + a_2\,|x_2|^2 - a_0 \,\le \, V(x_1,x_2) \,\le \, \gamma _1\,|x_1|^ + \gamma _2\,|x_2|^ + \gamma _0, \end$$

(268)

then a suitable choice for \(s_r,\,C_,\,C_\) is given by:

$$\begin \begin s_r\equiv \, r\,\frac,\quad C_ \,\equiv \, \bigg (\frac\bigg )^},\quad C_\equiv \, \frac}\,\Gamma (\frac)}\big )^}\,\Gamma (1+\frac)}\,e^. \end\nonumber \\ \end$$

(269)

Proof

Let \(\Delta >0\) and consider the set

$$\begin A_ \,\equiv \, \left\}^\Big |\; |x_1|^2 \ge \Delta \,|x_2|^ \right\} . \end$$

(270)

for every \(x_2\in }^\,\). Integrating over its complementary set, we have

$$\begin \int _}^\setminus A_}\!\! |x_1|^r\,\rho _}^}(x_1|x_2)\;\mathrm dx_1 \;\le \; \Delta ^\frac\, |x_2|^\,r}. \end$$

(271)

Now, we evaluate the contribution of the integral over \(A_\,\). By (268)

$$\begin \begin \int _} \!|x_1|^r\,e^\,\mathrm dx_1 \,&\le \, \int _}\! |x_1|^r\,e^ \,\mathrm dx_1 \\ &\le \, I_r\ e^\,\Delta \,|x_2|^\,+\,a_2\,|x_2|^2\right) } \end \end$$

(272)

where we set

$$\begin I_r \,\equiv \, \int _}^} |x_1|^r\,e^\,|x_1|^2 \,-\, a_0\right) }\,\mathrm dx_1 \,=\, \frac|}\;\frac\big )}\big )^}}\;e^. \end$$

(273)

On the other hand, by (268)

$$\begin \begin \int _}^} e^\,\mathrm dx_1 \,\ge& \int _}^} e^\,+\,\gamma _2\,|x_2|^\,+\,\gamma _0\right) }\,\mathrm dx_1 \\=& \,J_r\ e^} \end \end$$

(274)

where we set

$$\begin J_r \,\equiv \, \int _}^} e^\,+\,\gamma _0\right) }\,\mathrm dx_1 \,=\, |S_|\; \frac\big )}}}\, e^. \end$$

(275)

By inequalities (272), (274), it follows that:

$$\begin \begin \int _}|x_1|^r\,\rho _}^}(x_1|x_2)\,\mathrm dx_1=&\, \frac}|x_1|^r\,e^\,\mathrm dx_1}}^}e^\,\mathrm dx_1} \\\le&\, \frac\;e^\,\Delta \,|x_2|^\,+\,a_2\,|x_2|^2 \,-\,\gamma _2\,|x_2|^\right) } \\\le&\, \frac \end \end$$

(276)

where the last inequality holds true for any \(\Delta >\frac\,\). Finally, summing inequalities (271), (276) concludes the proof. \(\square \)

Remark 5.11

If the potential \(V(x_1,x_2)\) grows faster than quadratically in \(x_1\), the exponent \(s_r\) in Proposition 5.10 can be improved. Precisely if there are \(n_1\in [2,\infty )\,\), \(b_2,\,b_1,\,b_0\in [0,\infty )\) such that

$$\begin V(x_1,x_2) \,\ge \, b_1\,|x_1|^ - b_2\,|x_2|^ - b_0 \end$$

(277)

for all \((x_1,x_2)\in }^d\,\), then inequality (267) holds true with

$$\begin s_r \,\equiv \, \frac\,r \end$$

(278)

changing also the constants \(C_\), \(C_\) accordingly. Indeed, the proof of Proposition 5.10 can be suitably modified, starting by considering the set

$$\begin }_ \,\equiv \, \left\}^\Big |\; |x_1|^ \ge \Delta \,|x_2|^ \right\} . \end$$

(279)

Proof of Theorem 4.4

Equation (133) follows combining assumption (A2) with Proposition 5.10 and Corollary 5.7.

Equation (134) for \(|\nu |=0\) follows from assumptions (A2), (A3) which guarantee:

$$\begin a_1\,|x_1|^2 + a_2\,|x_2|^2 - a_0 \,\le \, V(x_1,x_2) \,\le \, \gamma _1 |x_1|^ + \gamma _2 |x_2|^ + \gamma _0. \end$$

(280)

Therefore:

$$\begin F(x_2) \,=\, -\frac\log \int _}^}\!e^\,\mathrm dx_1\, \,\le \, C_0 + \gamma _2\,|x_2|^ \\ \,\ge \, C_1 + a_2\,|x_2|^2 \end\right. } \end$$

(281)

where \(C_0\equiv \gamma _0-\frac\int _}^}e^}\mathrm dx_1\,\), \(C_1\equiv -a_0-\frac\int _}^}e^\mathrm dx_1\) are finite constants. Inequalities (281) combined with Corollary 5.7 ensure that

$$\begin \sup _\int _}^}|F(x_2)|^s\,\rho _,\lambda }^}(x_2)\,\mathrm dx_2 \,<\infty . \end$$

(282)

Finally, Eq. (134) for \(|\nu |\ge 1\) is an application of the multivariate Faà Di Bruno formula (see [53] and references therein):

$$\begin \begin D_^\nu F(x_2)=& \sum _}([\nu ])}\! \frac(|\pi |-1)!}\,\prod _\\ &\sum _}(A)} (-\beta _1)^ \int _}^}\prod _D_^V(x_1,x_2)\,\rho _}^}(x_1|x_2)\;\mathrm dx_1 \end \end$$

(283)

where, for a multi-index \(\nu =(\nu _1,\ldots ,\nu _)\in }^\\,\), \([\nu ]\) denotes the set where each number from 1 to \(d_2\) is repeated in a distinguishable way according to its multiplicity encoded in \(\nu \), i.e., \(\big \,\,\ldots ,\,d_2',\ldots ,d_2^)}\big \}\,\); conversely for a set \(B\subseteq [\nu ]\,\), [B] denotes the multi-index \(\big (|i:1^ \in B|,\,\ldots ,\,|i:d_2^ \in B|\big )\in }^\,\); \(}(A)\) denotes the set of partitions of a set A. Assuming without loss of generality \(s\ge 1\), Eq. (134) for \(|\nu |\ge 1\) then follows from expressions (283) using Jensen inequality and Eq. (133). \(\square \)

1.4 Regularity of \(\rho \): Proof of the Regularity Part of Theorem 4.1

In this and all the following subsections, we will denote

$$\begin \begin A \,\equiv \, (\Lambda \beta )^ \,=\, \begin \frac\,I_ &\quad 0 \\ 0 &\quad \frac\,I_ \end,\quad b(x)\,\equiv \, \Lambda ^\,\nabla V(x) \,=\, \begin\na