Averaging Principles for Markovian Models of Plasticity

Abstract

In this paper we consider a stochastic system with two connected nodes, whose unidirectional connection is variable and depends on point processes associated to each node. The input node is represented by an homogeneous Poisson process, whereas the output node jumps with an intensity that depends on the jumps of the input nodes and the connection intensity. We study a scaling regime when the rate of both point processes is large compared to the dynamics of the connection. In neuroscience, this system corresponds to a neural network composed by two neurons, connected by a single synapse. The strength of this synapse depends on the past activity of both neurons, the notion of synaptic plasticity refers to the associated mechanism. A general class of such stochastic models has been introduced in Robert and Vignoud (Stochastic models of synaptic plasticity in neural networks, 2020, arxiv: 2010.08195) to describe most of the models of long-term synaptic plasticity investigated in the literature. The scaling regime corresponds to a classical assumption in computational neuroscience that cellular processes evolve much more rapidly than the synaptic strength. The central result of the paper is an averaging principle for the time evolution of the connection intensity. Mathematically, the key variable is the point process, associated to the output node, whose intensity depends on the past activity of the system. The proof of the result involves a detailed analysis of several of its unbounded additive functionals in the slow-fast limit, and technical results on interacting shot-noise processes.

Appendices

Appendix A: Proofs of Technical Results for Occupation Times

1.1 Proof of Lemma 15

Denote, for $t{\ge }0$, a, b, $c{\in }{\mathbb {R}}_+$, and $\varepsilon {>}0$,

$$\begin{aligned} L_{\varepsilon }(t) {\mathop {=}\limits ^{{\mathrm{def.}}}} a\overline{X}_{\varepsilon }^K(t){+}b \overline{Z}_{\varepsilon }^K(t){+}c\overline{X}_{\varepsilon }^K(t)\overline{Z}_{\varepsilon }^K(t). \end{aligned}$$

Let G be a continuous bounded Borelian function. From Proposition 12, we can extract a sub-sequence $(\varepsilon _n)$ such that, for the convergence in distribution

$$\begin{aligned} \lim _{n\rightarrow +\infty } \left( \int _0^tL_{\varepsilon _{n}}(u)G\left( \overline{X}_{\varepsilon _{n}}^K(u),\overline{Z}_{\varepsilon _{n}}^K(u)\right) {\mathrm {d}}u\right) =(L(t)), \end{aligned}$$

where (L(t)) is a continuous càdlàgprocess.

We will now prove that the process (L(t)) is such that,

$$\begin{aligned} (L(t)) = \left( \int _0^t\int _{{\mathbb {R}}^2} (ax{+}bz{+}cxz)G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right) \end{aligned}$$

holds almost-surely for $t{\in }[0,T]$.

For $A{>}0$, the convergence of $(\overline{\nu }^K_{\varepsilon _n})$ to $\overline{\nu }^K$ gives the convergence in distribution,

$$\begin{aligned}&\lim _{n\rightarrow +\infty } \left( \int _0^t A{\wedge } L_{\varepsilon _{n}}(s)G\left( \overline{X}_{\varepsilon _{n}}^K(s),\overline{Z}_{\varepsilon _{k_n}}^K(s)\right) {\mathrm {d}}s\right) \nonumber \\&\quad =\left( \int _0^t\int _{{\mathbb {R}}_+^2} A{\wedge } (ax{+}bz{+}cxz) G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right) . \end{aligned}$$

(38)

Using again the upper-bound, Relation (29), for $(\overline{X}^K_\varepsilon (t))$ with Relation (23), and Proposition 24 for $R_{\varepsilon }$, we obtain that

$$\begin{aligned} C_{L}{\mathop {=}\limits ^{{\mathrm{def.}}}} \sup _{\begin{array}{c} 0{<}\varepsilon {<}1\\ 0{\le }t{\le }T \end{array}} {\mathbb {E}}\left[ L_\varepsilon (s)^2\right] {<}+\infty , \end{aligned}$$

hence, for $\eta {>}0$,

$$\begin{aligned}&{\mathbb {P}}\left( \int _0^T (L_\varepsilon (s){-}A)^+{\mathrm {d}}s{\ge }\eta \right) \le \frac{1}{\eta }\int _0^T{\mathbb {E}}\left[ (L_\varepsilon (s){-}A)^+\right] {\mathrm {d}}s \\&\quad \le \frac{1}{\eta A}\int _0^T{\mathbb {E}}\left[ L_\varepsilon (s)^2\right] {\mathrm {d}}s\le \frac{C_{L}T}{\eta A}. \end{aligned}$$

Since G is bounded, with the elementary relation $x{=}x{\wedge }A{+}(x{-}A)^+$, $x{\ge }0$, then, for $n{\ge }1$,

$$\begin{aligned}&{\mathbb {P}}\left( \sup _{0\le t\le T}\left| \int _0^tL_{\varepsilon _{n}}(u)G\left( \overline{X}_{\varepsilon _{n}}^K(u),\overline{Z}_{\varepsilon _{n}}^K(u)\right) {\mathrm {d}}u\right. \right. \nonumber \\&\quad \left. \left. {-}\int _0^t A{\wedge }L_{\varepsilon _{n}}(u)G\left( \overline{X}_{\varepsilon _{n}}^K(u),\overline{Z}_{\varepsilon _{n}}^K(u)\right) {\mathrm {d}}u\right| {\ge }\eta \right) \le \frac{C_{L} T}{\eta A}\Vert G\Vert _{\infty }. \end{aligned}$$

(39)

For any $A{>}0$ and $n{\ge }1$, Cauchy-Schwartz’s inequality gives the relation

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T A{\wedge }L_{\varepsilon _{n}}(u)G\left( \overline{X}_{\varepsilon _{n}}^K(u),\overline{Z}_{\varepsilon _{n}}^K(u)\right) {\mathrm {d}}u\right] \le \sqrt{C_L}T\Vert G\Vert _{\infty }. \end{aligned}$$

With Relation (38) and the fact that the left-hand side of (38) has a bounded second moment, by letting n go to infinity, we get the inequality

$$\begin{aligned} {\mathbb {E}}\left[ \int _0^T\int _{{\mathbb {R}}_+^2} A{\wedge } (ax{+}bz{+}cxz) G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right] \le \sqrt{C_L}T\Vert G\Vert _{\infty }. \end{aligned}$$

By letting A go to infinity, the monotone convergence theorem shows that

$$\begin{aligned}&\lim _{A\rightarrow +\infty } {\mathbb {E}}\left[ \int _0^T\int _{{\mathbb {R}}_+^2} A{\wedge }(ax{+}bz{+}cxz) G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right] \\&\quad = {\mathbb {E}}\left[ \int _0^T\int _{{\mathbb {R}}_+^2} (ax{+}bz{+}cxz) G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right] \le \sqrt{C_L}T\Vert G\Vert _{\infty } . <{+}\infty . \end{aligned}$$

With Relation (39) and the integrability properties proven just above, we obtain that, for $\varepsilon {>}0$, there exists $n_0$ such that if $n{\ge }n_0$, then the relation

$$\begin{aligned}&{\mathbb {P}}\left( \sup _{0\le t\le T}\left| \int _0^tL_{\varepsilon _{n}}(u)G\left( \overline{X}_{\varepsilon _{n}}^K(u),\overline{Z}_{\varepsilon _{n}}^K(u)\right) {\mathrm {d}}u\right. \right. \\&\quad \left. \left. {-}\int _0^T\int _{{\mathbb {R}}_+^2} (ax{+}bz{+}cxz) G(x,z)\overline{\nu }^K({\mathrm {d}}s, {\mathrm {d}}x,{\mathrm {d}}z)\right| {\ge }\eta \right) \le \varepsilon \end{aligned}$$

holds. Lemma 15 is proved.

1.2 Proof of Lemma 16

Following Papanicolalou et al. [24] and Kurtz [21], we first show that there exists an optional process $(\overline{\Gamma }^{K}_s)$, with values in the set of probability distributions on ${\mathbb {R}}_+^2$ such that, almost surely, for any bounded Borelian function G on ${\mathbb {R}}_+{\times }{\mathbb {R}}_+^2$,

$$\begin{aligned} \int _{{\mathbb {R}}_+{\times }{\mathbb {R}}_+^2} G(s,x,z)\overline{\nu }^{K}({\mathrm {d}}s, {\mathrm {d}}x, {\mathrm {d}}z)=\int _{{\mathbb {R}}_+{\times }{\mathbb {R}}_+^2} G(s,x,z)\overline{\Gamma }_s^{K}({\mathrm {d}}x, {\mathrm {d}}z)\,{\mathrm {d}}s. \end{aligned}$$

(40)

Recall that the optional $\sigma$-algebra is the smallest $\sigma$-algebra containing adapted càdlàgprocesses. See Sect. VI.4 of Rogers and Williams [29] for example. This is a simple consequence of Lemma 1.4 of Kurtz [21] and the fact that, due to Relation (19), the measure $\overline{\nu }^{K}({\mathrm {d}}s,{\mathbb {R}}^2)$ is the Lebesgue measure on [0, T].

Let $f{\in }{\mathcal {C}}_b^1({\mathbb {R}}_+^2)$ be a bounded ${\mathcal {C}}^1$-function on ${\mathbb {R}}_+^2$ with bounded partial derivatives, we have the relation

$$\begin{aligned} \varepsilon f\left( \overline{X}^{K}_\varepsilon (t),\overline{Z}^{K}_\varepsilon (t)\right)= & {} \varepsilon f(\overline{x}_0,\overline{z}_0){+}\varepsilon \overline{M}_\varepsilon ^f(t)\nonumber \\&{+}\int _0^t B^F_{K{\wedge }\overline{W}^{K}_\varepsilon (s)}(f)\left( \overline{X}^{K}_\varepsilon (s),\overline{Z}^{K}_\varepsilon (s)\right) {\mathrm {d}}s, \end{aligned}$$

(41)

where, for $t{\ge }0$, if $(\overline{V}^K_\varepsilon (s)){{\mathop {=}\limits ^{{\mathrm{def.}}}}}(\overline{X}^{K}_\varepsilon (s),\overline{Z}^{K}_\varepsilon (s))$,

$$\begin{aligned} \overline{M}_{\varepsilon }^f(t)&{\mathop {=}\limits ^{{\mathrm{def.}}}} \int _0^t \left( f\left( \overline{V}^{K}_\varepsilon (s-){+}\left( K{\wedge }\overline{W}^{K}_\varepsilon (s),1\right) \right) {-}f\left( \overline{V}^{K}_\varepsilon (s{-})\right) \right) \left[ {\mathcal {N}}_{\lambda /\varepsilon }({\mathrm {d}}s){-}\frac{\lambda }{\varepsilon }{\mathrm {d}}s\right] \\&\quad {+}\int _0^t\left( f\left( \overline{V}^{K}_\varepsilon (s-){+}(0,1)\right) {-}f\left( \overline{V}^{K}_\varepsilon (s-)\right) \right) \left[ {\mathcal {N}}_{\overline{\beta }/\varepsilon ,\overline{X}^{K}_\varepsilon }({\mathrm {d}}s){-}\frac{\overline{\beta }\left( \overline{X}_{\varepsilon }^{K}(s)\right) }{\varepsilon }{\mathrm {d}}s\right] , \end{aligned}$$

Proposition 12 shows that the martingale $(\varepsilon \overline{M}_{\varepsilon }^f(t))$ is converging in distribution to 0 as $\varepsilon$ goes to 0.

Relation (41) gives therefore the convergence in distribution

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0} \left( \int _0^t B^F_{K{\wedge }\overline{W}^{K}_\varepsilon (s)}(f)\left( \overline{X}_\varepsilon ^{K}(s),\overline{Z}_\varepsilon ^{K}(s)\right) {\mathrm {d}}s\right) {=}0. \end{aligned}$$

The convergence in distribution of $(\overline{\Omega }^{K}_{\varepsilon _n}(t), \overline{W}^{K}_{\varepsilon _n}(t), \overline{\nu }_{\varepsilon _n}^{K})$, Proposition 15 and Relation (40) give that, for any f in ${\mathcal {C}}_b^1({\mathbb {R}}_+^2)$, the relation

$$\begin{aligned} \left( \int _0^t \int _{{\mathbb {R}}_+^2} B^F_{K{\wedge }\overline{w}^{K}(s)}(f)(x,z)\overline{\Gamma }_s^{K}({\mathrm {d}}x,{\mathrm {d}}z) {\mathrm {d}}s,\, 0{\le }t{\le }T\right) =(0,\, 0{\le }t{\le }T). \end{aligned}$$

(42)

holds with probability 1.

Let $(f_n)$ be a dense countable sequence in ${\mathcal {C}}_b^1({\mathbb {R}}_+^2)$ and ${\mathcal {E}}_1$ be the event, where Relation (42) holds for all $f{=}f_n$, $n{\ge }1$. Note that that ${\mathbb {P}}\left( {\mathcal {E}}_1\right) {=}1$. On ${\mathcal {E}}_1$, there exists a (random) subset $S_1$ of [0, T] with Lebesgue measure T such that

$$\begin{aligned} \int _{{\mathbb {R}}_+^2} B^F_{K{\wedge }\overline{w}^{K}(s)}(f_n)(x,z)\overline{\Gamma }_s^{K}({\mathrm {d}}x,{\mathrm {d}}z)=0, \forall s{\in }S_1 \text { and }\forall n{\ge }1, \end{aligned}$$

and, consequently,

$$\begin{aligned} \int _{{\mathbb {R}}_+^2} B^F_{K{\wedge }\overline{w}^{K}(s)}(f)(x,z)\overline{\Gamma }_s^{K}({\mathrm {d}}x,{\mathrm {d}}z)=0, \forall s{\in }S_1 \text { and }\forall f{\in } {\mathcal {C}}_b^1({\mathbb {R}}_+^2). \end{aligned}$$

By Proposition 25, for $s{\in }S_1$, the probability distribution $\overline{\Gamma }_s^{K}$ is the invariant distribution $\Pi _{K{\wedge }\overline{w}^{K}(s)}$. Lemma 16 is proved.

Appendix B: Shot-Noise Processes

This section presents several technical results on shot-noise processes which are crucial for the proof of Theorem 4. See Schottky [30], Rice [26] and Gilbert and Pollak [9] for an introduction.

1.1 A Scaled Shot-Noise Process

Recall that $(S^x_{\varepsilon }(t))$, with initial point $x{\ge }0$, has been introduced by Definition 6. We will have the following conventions,

$$\begin{aligned} \left( S_{\varepsilon }(t)\right) {\mathop {=}\limits ^{{\mathrm{def.}}}}\left( S^0_{\varepsilon }(t)\right) , \left( S^x(t)\right) {\mathop {=}\limits ^{{\mathrm{def.}}}}\left( S^x_{1}(t)\right) \text { and } \left( S(t)\right) {\mathop {=}\limits ^{{\mathrm{def.}}}}\left( S^0_{1}(t)\right) . \end{aligned}$$

The process (S(t)) is in fact the standard shot-noise process of Lemma 5 associated to the Poisson process ${\mathcal {N}}_\lambda$, for $t{\ge }0$,

$$\begin{aligned} S(t)=\int _0^t e^{-(t-s)}{\mathcal {N}}_\lambda ({\mathrm {d}}s){\mathop {=}\limits ^{{\mathrm{dist.}}}}\int _0^t e^{-s}{\mathcal {N}}_\lambda ({\mathrm {d}}s). \end{aligned}$$

(43)

In particular (S(t)) is a stochastically non-decreasing process, i.e. for $y{\ge }0$ and $s{\le }t$,

$$\begin{aligned} {\mathbb {P}}\left( S(s){\ge }y\right) \le {\mathbb {P}}\left( S(t){\ge }y\right) . \end{aligned}$$

(44)

A classical formula for Poisson processes, see Proposition 1.5 of Robert [27] for example, gives the relation, for $\xi {\in }{\mathbb {R}}$,

$$\begin{aligned} {\mathbb {E}}\left[ e^{\xi S(t)}\right] =\exp \left( {-}\lambda \int _0^t \left( 1{-}\exp \left( \xi e^{-s}\right) \right) {\mathrm {d}}s\right) , \end{aligned}$$

(45)

in particular ${\mathbb {E}}\left[ S^x(t)\right] {=}x\exp (-t){+}\lambda \left( 1{-}\exp ({-} t)\right)$. It also shows that $(S^x(t))$ is converging in distribution to $S(\infty )$ such that,

$$\begin{aligned} {\mathbb {E}}\left[ e^{\xi S(\infty )}\right] =\exp \left( {-}\lambda \int _0^{+\infty } \left( 1{-}\exp \left( \xi e^{-s}\right) \right) {\mathrm {d}}s\right) {<}{+}\infty . \end{aligned}$$

It is easily seen that $(S^x_{\varepsilon }(t)){{\mathop {=}\limits ^{{\mathrm{dist.}}}}} (S^x(t/\varepsilon ))$and thus with Relation (22), $(S^x_{\varepsilon }(t))$ can be represented as, for $t{\ge }0$,

$$\begin{aligned} S^x_{\varepsilon }(t){\mathop {=}\limits ^{{\mathrm{def.}}}} xe^{-t/\varepsilon }{+} \int _0^t e^{-(t-s)/\varepsilon }{\mathcal {N}}_{\lambda /\varepsilon }({\mathrm {d}}s) = xe^{-t/\varepsilon }{+} S_{\varepsilon }(t). \end{aligned}$$

(46)

We remind here the results of Proposition 7, that will be proved in the following paragraph. For $\xi {\in }{\mathbb {R}}$ and $x{\ge }0$, the convergence in distribution of the processes

$$\begin{aligned} \lim _{\varepsilon \searrow 0} \left( \int _0^t e^{\xi S^x_{\varepsilon }(u)}{\mathrm {d}}u\right) =\left( {\mathbb {E}}\left[ e^{\xi S(\infty )}\right] t\right) \end{aligned}$$

holds, and

$$\begin{aligned} \sup _{\begin{array}{c} 0{<}\varepsilon {<}1\\ 0{\le }t{\le }T \end{array}} {\mathbb {E}}\left[ e^{\xi S_{\varepsilon }(t)}\right] {<}{+}\infty . \end{aligned}$$

Proof of Proposition 7

Let $T_1$ and $T_2$ be two stopping times bounded by N, $\theta {>}0$, and verifying $0{\le }T_2{-}T_1{\le }\theta$. Using Relation (46) and the strong Markov property of Poisson processes, we have that

$$\begin{aligned}&{\mathbb {E}}\left[ \int _{T_1}^{T_2} e^{\xi S^x_{\varepsilon }(u)}{\mathrm {d}}u\right] = \varepsilon {\mathbb {E}}\left[ \int _{T_1/\varepsilon }^{T_2/\varepsilon } e^{\xi S^x(u)}{\mathrm {d}}u\right] \\&\quad =\varepsilon {\mathbb {E}}\left[ \int _{0}^{(T_2-T_1)/\varepsilon } e^{\xi S^{S^x(T_1/\varepsilon )}(u)}{\mathrm {d}}u\right] \le \varepsilon {\mathbb {E}}\left[ e^{\xi S^x(T_1/\varepsilon )}{\mathbb {E}}\left[ \int _{0}^{\theta /\varepsilon } e^{\xi S(u)}{\mathrm {d}}u\right] \right] \\&\quad \le \theta e^{\xi x}{\mathbb {E}}\left[ e^{\xi S(N/\varepsilon )}\right] {\mathbb {E}}\left[ e^{\xi S(\infty )}\right] \le \theta e^{\xi x}{\mathbb {E}}\left[ e^{\xi S(\infty )}\right] ^2, \end{aligned}$$

holds, by stochastic monotonicity of (S(t)) of Relation (44).

Aldous’ Criterion, see Theorem VI.4.5 of Jacod and Shiryaev [16] gives that the family of processes

$$\begin{aligned} \left( \int _0^te^{\xi S_{\varepsilon }(u)}{\mathrm {d}}u\right) , \end{aligned}$$

is tight when $\varepsilon$ goes to 0. For $p{\ge }1$ and a fixed vector $(t_i){\in }{\mathbb {R}}_+^p$, the ergodic theorem for the Markov process (S(t)) gives the almost-sure convergence of

$$\begin{aligned}&\lim _{\varepsilon \rightarrow 0}\left( \int _0^{t_i}e^{\xi S_{\varepsilon }(u)}{\mathrm {d}}u, i=1,\ldots ,p\right) =\lim _{\varepsilon \rightarrow 0}\left( \varepsilon \int _0^{t_i/\varepsilon }e^{\xi S(u)}{\mathrm {d}}u,i=1,\ldots ,p\right) \\&\quad =\left( {\mathbb {E}}\left[ e^{\xi S(\infty )}\right] t_i,i=1,\ldots ,p\right) . \end{aligned}$$

Hence, due to the tightness property, the convergence also holds in distribution for the processes

$$\begin{aligned} \lim _{\varepsilon \rightarrow 0}\left( \int _0^te^{\xi S_{\varepsilon }(u)}{\mathrm {d}}u\right) =\left( {\mathbb {E}}\left[ e^{\xi S(\infty )}\right] t\right) . \end{aligned}$$

The last part is a direct consequence of the identity $(S^x_{\varepsilon }(t)){{\mathop {=}\limits ^{{\mathrm{dist.}}}}} (S^x(t/\varepsilon ))$, and of Relation (45), which gives

$$\begin{aligned} {\mathbb {E}}\left[ e^{\xi S_{\varepsilon }(t)}\right]= & {} \exp \left( {-}\lambda \int _0^{t/\varepsilon } \left( 1{-}\exp \left( \xi e^{-s}\right) \right) {\mathrm {d}}s\right) \\&\le \exp \left( {-}\lambda \int _0^{+\infty } \left( 1{-}\exp \left( \xi e^{-s}\right) \right) {\mathrm {d}}s\right) . \end{aligned}$$

The proposition is proved. $\square$

1.2 Interacting Shot-Noise Processes

Recall that $(R_{\varepsilon }(t))$ defined by Relation (24) is a shot-noise process with intensity equal to the shot-noise process $(S_{\varepsilon }(t))$.

We start with a simple result on moments of functionals of Poisson processes.

Lemma 23

If ${\mathcal {Q}}$ is a Poisson point process on ${\mathbb {R}}_+$ with a positive Radon intensity measure $\mu$ and f is a Borelian function such that

$$\begin{aligned} I_k(f){\mathop {=}\limits ^{{\mathrm{def.}}}}\int f(u)^k\mu ({\mathrm {d}}u) < {+}\infty , 1{\le }k{\le } 4, \end{aligned}$$

then

$$\begin{aligned} {\mathbb {E}}\left[ \left( \int f(u) {\mathcal {Q}}({\mathrm {d}}u)\right) ^4\right] =\left( I_4{+}6I_{1}^2I_{2}{+}4I_{1}I_{3}{+}3I_{2}^2{+}I_{1}^4\right) (f). \end{aligned}$$

Proof

It is enough to prove the inequality for non-negative bounded Borelian functions f with compact support on ${\mathbb {R}}_+$.

The formula for the Laplace transform of Poisson point processes, see Proposition 1.5 of Robert [27], gives for $\xi {\ge }0$,

$$\begin{aligned} {\mathbb {E}}\left[ \exp \left( \xi \int _0^{+\infty } f(u) {\mathcal {Q}}({\mathrm {d}}u)\right) \right] = \exp \left( \int _0^{+\infty } \left( e^{\xi f(u)}{-}1\right) \mu ({\mathrm {d}}u)\right) . \end{aligned}$$

The proof is done in a straightforward way by differentiating the last identity with respect to $\xi$ four times and then set $\xi {=}0$. $\square$

Proposition 24

The inequality

$$\begin{aligned} \sup _{\varepsilon {\in }(0,1), t\ge 0}{\mathbb {E}}\left[ R_\varepsilon (t)^4\right] <{+}\infty \end{aligned}$$

holds.

Proof

Denote, for $t{\ge }0$,

$$\begin{aligned} J_{k,\varepsilon }(t){\mathop {=}\limits ^{{\mathrm{def.}}}} \int _0^t e^{-\gamma k (t-u)/\varepsilon }\frac{S_{\varepsilon }(u)}{\varepsilon }{\mathrm {d}}u, \end{aligned}$$

the identity $(S_{\varepsilon }(t)){{\mathop {=}\limits ^{{\mathrm{dist.}}}}} (S(t/\varepsilon ))$ and Relation (43) coupled with Fubini’s Theorem give the relations

$$\begin{aligned}&J_{k,\varepsilon }(t){=}\int _0^{t/\varepsilon } e^{-\gamma k (t/\varepsilon -u)}S(u){\mathrm {d}}u {=}\frac{1}{\gamma k{-}1}\int _0^{t/\varepsilon } \left( e^{-(t/\varepsilon -v)}{-}e^{-\gamma k (t/\varepsilon -v)}\right) {\mathcal {N}}_\lambda ({\mathrm {d}}v)\\&\quad {\mathop {=}\limits ^{{\mathrm{dist.}}}} \frac{1}{\gamma k{-}1}\int _0^{t/\varepsilon } \left( e^{-v}{-}e^{-\gamma kv}\right) {\mathcal {N}}_\lambda ({\mathrm {d}}v) \le \frac{1}{|\gamma k{-}1|} \overline{J}_{k} \end{aligned}$$

with

$$\begin{aligned} \overline{J}_{k}{\mathop {=}\limits ^{{\mathrm{def.}}}} \int _0^{{+}\infty } \left( e^{-k \gamma v}{+}e^{-v}\right) {\mathcal {N}}_\lambda ({\mathrm {d}}v). \end{aligned}$$

Relation (22) applied to $R_\varepsilon (t)$ gives

$$\begin{aligned} R_\varepsilon (t)=\int _0^t e^{-\gamma (t-u)/\varepsilon } {\mathcal {P}}_2\left( \left( 0,\frac{S_{\varepsilon }(u{-})}{\varepsilon }\right] ,{\mathrm {d}}u\right) . \end{aligned}$$

The quantity $S_{\varepsilon }(u)$ is a functional of the point process ${\mathcal {P}}_1$ and is therefore independent of the Poisson point process ${\mathcal {P}}_2$. Lemma 23 gives therefore that

$$\begin{aligned} {\mathbb {E}}\left[ R_\varepsilon (t)^4\mid {\mathcal {P}}_1\right] = J_{4,\varepsilon }(t){+}6J_{1,\varepsilon }(t)^2J_{2,\varepsilon }(t){+}4J_{1,\varepsilon }(t)J_{3,\varepsilon }(t){+}3J_{2,\varepsilon }(t)^2{+}J_{1,\varepsilon }(t)^4, \end{aligned}$$

hence,

$$\begin{aligned} {\mathbb {E}}\left[ R_\varepsilon (t)^4\right] \le {\mathbb {E}}\left[ \frac{\overline{J}_{4}}{|4\gamma {-}1|}{+}\frac{6\overline{J}_{1}^2\overline{J}_{2}}{|\gamma {-}1|^2|2\gamma {-}1|}{+}\frac{4\overline{J}_{1}\overline{J}_{3}}{|\gamma {-}1||3\gamma {-}1|}{+}\frac{3\overline{J}_{2}^2}{|2\gamma {-}1|^2}{+}\frac{\overline{J}_{1}^4}{|\gamma {-}1|^4}\right] . \end{aligned}$$

Again with Proposition 7 we obtain that, for $k{\ge }1$, the variable $\overline{J}_{k}$ has finite moments of all orders, therefore by Cauchy-Shwartz’ Inequality, the right-hand side of the last inequality is finite. The proposition is proved. $\square$

Appendix C: Equilibrium of Fast Processes

For $w{\in }K_W$, recall that the Markov process $(X^w(t),Z^w(t))$ of Definition 2 is such that

$$\begin{aligned} {\mathrm {d}}X^w(t)&\displaystyle = {-}X^w(t){\mathrm {d}}t{+}w{\mathcal {N}}_{\lambda }({\mathrm {d}}t){-}g\left( X^w(t{-})\right) {\mathcal {N}}_{\beta ,X^w}\left( {\mathrm {d}}t\right) \end{aligned}$$

(47)

$$\begin{aligned} {\mathrm {d}}Z^w(t)&\displaystyle = ({-}\gamma {\odot }Z^w(t){+}k_0){\mathrm {d}}t{+}k_1(Z^w(t{-})){\mathcal {N}}_{\lambda }({\mathrm {d}}t){+}k_2(Z^w(t{-})){\mathcal {N}}_{\beta ,X^w}({\mathrm {d}}t). \end{aligned}$$

(48)

Proposition 25

Under the conditions of Sections 2.2.2 and 2.2.3 , the Markov process $(X^w(t),Z^w(t))$ solution of the SDEs (47) and (48) has a unique invariant distribution $\Pi _w$, i.e. the unique probability distribution $\mu$ on ${\mathbb {R}}{\times }{\mathbb {R}}_+^\ell$ such that

$$\begin{aligned} \left\langle \mu ,B_w^F(f)\right\rangle =\int _{{\mathbb {R}}{\times }{\mathbb {R}}_+^\ell } B_w^F(f)(x,z)\mu ({\mathrm {d}}x,{\mathrm {d}}z)=0, \end{aligned}$$

(49)

for any $f{\in }{\mathcal {C}}_b^1({\mathbb {R}}{\times }{\mathbb {R}}_+^\ell )$, where $B^F_w$ is the operator defined by Relation (18).

Proof

We denote by $(X^w_n,Z^w_n)$ the embedded Markov chain of the Markov process $(X^w(t),Z^w(t))$, i.e. the sequence of states visited by $(X^w(t),Z^w(t))$ after each jump, of either ${\mathcal {N}}_{\lambda }$ or ${\mathcal {N}}_{\beta ,X}$.

The proof of the proposition is done in three steps. We first show that the return time of $(X^w(t),Z^w(t))$ to a compact set of ${\mathbb {R}}{\times }{\mathbb {R}}_+^\ell$ is integrable. Then we prove that the Markov chain $(X^w_n,Z^w_n)$ is Harris ergodic, and consequently that it has a unique invariant measure. For a general introduction on Harris Markov chains, see Nummelin [23] and Meyn and Tweedie [22]. Finally, the proof of the proposition uses the classical framework of stationary point processes.

1.1 Integrability of Return Times to a Compact Subset

Suppose that $w{\ge }0$. The conditions of Sect. 2.2.2 on the functions $\beta$ and g, and Relation (47) show that $X^w(t){\ge }{-}c_0$, for all $t{\ge }0$, if $X^w(0){\ge }{-}c_0$, with $c_0{=}c_\beta {+}c_g$. The state space of the Markov process $(X^w(t),Z^w(t))$ can be taken as ${\mathcal {S}}{{\mathop {=}\limits ^{{\mathrm{def.}}}}}[{-}c_0,{+}\infty ){\times }{\mathbb {R}}_+^\ell$.

Define, for $(x,z){\in }{\mathcal {S}}$ and $0{<}a{\le }1$,

$$\begin{aligned} H(x,z){\mathop {=}\limits ^{{\mathrm{def.}}}} x{+}a \Vert z\Vert , \text { with } \Vert z\Vert {\mathop {=}\limits ^{{\mathrm{def.}}}} \sum _{i=1}^{\ell } z_i, \end{aligned}$$

we get that

$$\begin{aligned} B_w^F(H)(x,z)= & {} {-}x+\left( {-}a\sum _{i=1}^{\ell } \gamma _{i}z_{i}{+}k_{0,i}\right) +\lambda \left( w{+}a\sum _{i=1}^{\ell } k_{1,i}(z)\right) \\&+\, \beta (x)\left( {-}g(x){+}a\sum _{i=1}^{\ell } k_{2,i}(z)\right) \end{aligned}$$

hence, with the assumptions of Sect. 2.2.3 and 2.2.2 on the function $k_.$ and $\beta$, and $a\le 1$,

$$\begin{aligned}&B_w^F(H)(x,z)\le {-}x{-}a{\underline{\gamma }}\Vert z\Vert +\ell C_k +\lambda (w{+}\ell a C_k){+}C_\beta \left( 1 + x\right) \ell a C_k\\&\quad \le (\ell a C_\beta C_k{-}1)x-a{\underline{\gamma }}\Vert z\Vert +\left( \ell C_k {+} \lambda w {+} \lambda \ell C_k {+} \ell C_\beta C_k\right) \\&\quad \le (\ell a C_\beta C_k{-}1)x{-}a{\underline{\gamma }}\Vert z\Vert {+}C, \end{aligned}$$

where ${\underline{\gamma }}{>}0$ is the minimum of the coordinates of $\gamma$ and C is a constant independent of x, z and a. We fix $0{<}a{\le }1$ sufficiently small so that $\ell aC_\beta C_k{<}1$ and $K{>}c_0$ such that

$$\begin{aligned} C{<}{\underline{\gamma }}{K}/{2}{-}1 \text { and } C{<}(1{-}\ell aC_\beta C_k){K}/{2}{-}1. \end{aligned}$$

If $H(x,z){>}K$ then $\max (x,a\Vert z\Vert ){>}K/2$ and therefore $B_w^F(H)(x,z){\le }{-}1$, H is therefore a Lyapounov function for $B^F_w$. One deduces that the same result holds for the return time of Markov chain, $(X^w_n,Z^w_n)$ in the set $I_K=\{(x,z):H(x,z){\le }K\}$.

1.2 Harris Ergodicity of $(X^w_n,Z^w_n)$

Proposition 5.10 of Nummelin [23] is used to show that $I_K$ is a recurrent set. A regeneration property would be sufficient to conclude. In particular, we can prove that $I_K$ is a small set, that is, there exists some positive, non-trivial, Radon measure $\nu$ on ${\mathcal {S}}$ such that,

$$\begin{aligned} {\mathbb {P}}_{(x_0, z_0)}\left( (X_1^w,Z_1^w){\in }S\right) \ge \nu (S), \end{aligned}$$

(50)

for any Borelian subset S of ${\mathcal {S}}$ and all $(x_0, z_0){\in }I_K$.

We denote by $s_1$, resp. $t_1$, the first instant of ${\mathcal {N}}_\lambda$, resp. of ${\mathcal {N}}_{\beta ,X^w}$, then, for $(X_0^w,Z_0^w){=}(x_0, z_0){\in }I_K$, by using the deterministic differential equations between jumps, we get

$$\begin{aligned} {\mathbb {P}}_{(x_0, z_0)}\left( s_1{<} t_1\right) {=} {\mathbb {E}}\left[ \exp \left( {-} \int _0^{s_1} \beta (x_0\exp ({-}s)\right) \,{\mathrm {d}}s\right] \ge {\mathbb {E}}\left[ \exp ({-}c_\beta ^1s_1)\right] {=}p_0{{\mathop {=}\limits ^{{\mathrm{def.}}}}}\frac{\lambda }{\lambda {+}c_\beta ^1}, \end{aligned}$$

since $\beta$ is bounded by some constant $c_\beta ^1$ on the interval $[{-}c_0,K]$.

In the following argument, we restrict X to be non-negative, the extension to $[{-}c_0,+\infty ]$ is straightforward. For ${\mathcal {A}}{=}[0,A]{\in }{\mathcal {B}}({\mathbb {R}}_+)$ and ${\mathcal {B}}{=}[0,B]{\in }{\mathcal {B}}({\mathbb {R}}_+^\ell )$, from Equations (47) and (48), we obtain the relation

$$\begin{aligned}&{\mathbb {P}}_{(x_0, z_0)}\left( (X_1^w,Z_1^w){\in }{\mathcal {A}}{\times }{\mathcal {B}}\right) \ge p_0{\mathbb {P}}\left( (X_1^w,Z_1^w){\in }{\mathcal {A}}{\times }{\mathcal {B}}\mid s_1{<}t_1\right) \\&\quad = p_0{\mathbb {P}}\left( \left. \begin{array}{c} \displaystyle x_0e^{-s_1}{+}w {\in }{\mathcal {A}},\\ \displaystyle (z_0{-}k_0){\odot }e^{-\gamma _{i} s_1}{+}k_0{+}k_1\left( (z_0{-}k_0){\odot }e^{-\gamma _{i} s_1}{+}k_0\right) {\in }{\mathcal {B}} \end{array}\right| s_1{<}t_1\right) \\&\quad \ge {\mathbb {P}}\left( \left. \begin{array}{c} \displaystyle x_0e^{-s_1}{+}w {\in }{\mathcal {A}},\\ \displaystyle H\left( (z_0{-}k_0){\odot }e^{-\gamma _{i} s_1}{+}k_0\right) {\in }{\mathcal {B}} \end{array}\right| s_1{<}t_1\right) \\&\quad =p_0 {\mathbb {P}}\left( \begin{array}{c} \displaystyle x_0e^{-{\bar{s}}_1}{+}w {\in }{\mathcal {A}},\\ \displaystyle H\left( (z_0{-}k_0){\odot }e^{-\gamma _{i} {\bar{s}}_1}{+}k_0\right) {\in }{\mathcal {B}} \end{array}\right) , \end{aligned}$$

where $H(z){=} z{+}k_1(z)$, ${\bar{s}}_1{{\mathop {=}\limits ^{{\mathrm{dist.}}}}}(s_1{\mid }s_1{\le }t_1)$. By using the fact that $k_1$ is in ${\mathcal {C}}_b^1({\mathbb {R}}_+^\ell ,{\mathbb {R}}_+^\ell )$ by the conditions of Sect. 2.2.3 and in the same way as Example of Sect. 4.3.3 page 98 of Meyn and Tweedie [22], we can prove that the random variable

$$\begin{aligned} \left( x_0e^{-{\bar{s}}_1}{+}w,H\left( (z_0{-}k_0){\odot }e^{-\gamma _{i} {\bar{s}}_1}{+}k_0\right) \right) \end{aligned}$$

has a density, uniformly bounded below by a positive function h on ${\mathbb {R}}_+{\times }{\mathbb {R}}_+^{\ell }$, so that

$$\begin{aligned} {\mathbb {P}}_{(x_0, z_0)}\left( (X_1^w,Z_1^w){\in }A{\times }B\right) \ge \int _{A{\times }B} h(x,z)\,{\mathrm {d}}x{\mathrm {d}}z, \forall A{\in }{\mathcal {B}}({\mathbb {R}}_+), B{\in }{\mathcal {B}}({\mathbb {R}}_+^\ell ), \end{aligned}$$

for all $(x_0,z_0){\in }I_K)$. This relation is then extended to all Borelian subsets S of ${\mathcal {S}}$, so that Relation (50) holds. Proposition 5.10 of Nummelin [23] gives therefore that $(X^w_n,Z^w_n)$ is Harris ergodic.

If $w{<}0$, the last two steps can be done in a similar way. In this case, the process $(-X^w(t))$ satisfies an analogous equation with the difference that the process ${\mathcal {N}}_{\beta ,X^w}$ does not jump when ${-}X^w(t){>}c_\beta$ since $\beta (x){=}0$ for $x{\le }{-}c_\beta$.

1.3 Characterization of $\Pi _w$

Let ${\widehat{\Pi }}_w$ be the invariant probability distribution of $(X^w_n,Z^w_n)$. With the above notations,

$$\begin{aligned} {\mathbb {E}}_{{\widehat{\Pi }}_w}\left[ \min (s_1,t_1)\right] \le {\mathbb {E}}_{{\widehat{\Pi }}_w}\left[ s_1\right] {=}\frac{1}{\lambda }<{+}\infty , \end{aligned}$$

the probability defined by the classical cycle formula,

$$\begin{aligned} \frac{1}{{\mathbb {E}}_{{\widehat{\Pi }}_w}\left[ \min (s_1,t_1)\right] }{\mathbb {E}}_{{\widehat{\Pi }}_w}\left[ \int _0^{\min (s_1,t_1)}f(X^w(u),Z^w(u))\,{\mathrm {d}}u\right] , \end{aligned}$$

for any bounded Borelian function on ${\mathbb {R}}{\times }{\mathbb {R}}_+^\ell$ is an invariant distribution for the process $(X^w(t),Z^w(t))$.

Proposition 9.2 of Ethier and Kurtz [7] shows that any distribution is invariant for $(X^w(t),Z^w(t))$ if and only if it satisfies Relation (49). It remains to prove the uniqueness of the invariant distribution, using the fact that the embedded Markov chain has a unique invariant distribution.

Although this is a natural result, we have not been able to find a reference in the literature. Most results are stated for discrete time, the continuous time is usually treated by looking at the process on a “discrete skeleton”, i.e. at instants multiple of some positive constant. See Proposition 3.8 of Asmussen [1] for example. As this technique is not adapted to our system, we derive a different proof using the Palm measure of the associated stationary point process.

If $\mu$ is some invariant distribution of the Markov process $(X^w(t),Z^w(t))$, we build a stationary version $((X^w(t),Z^w(t)), t{\in }{\mathbb {R}})$ of it on the whole real line. In particular, we have that $(X^w(t),Z^w(t)){{\mathop {=}\limits ^{{\mathrm{dist.}}}}}\mu$, for all $t{\in }{\mathbb {R}}$.

We denote by $(S_n,n{\in }{\mathbb {Z}})$ the non-decreasing sequence of the jumps (due to ${\mathcal {N}}_\lambda$ and ${\mathcal {N}}_{\beta ,X^w}$), with the convention $S_0{\le }0{<}S_1$ The sequence $((X^w(S_n),Z^w(S_n)), n{\ge }0)$ has the same distribution as the process $((X^w_n,Z^w_n), n{\ge }0)$, the Markov chain with initial state $(X^w(S_0),Z^w(S_0))$. Since, for any $t{\in }{\mathbb {R}}$,

$$\begin{aligned} \left( (X^w(s{+}t),Z^w(s{+}t)), s{\in }{\mathbb {R}}\right) {\mathop {=}\limits ^{{\mathrm{dist.}}}}\left( (X^w(s),Z^w(s)), s{\in }{\mathbb {R}}\right) , \end{aligned}$$

the marked point process ${\mathcal {T}}{{\mathop {=}\limits ^{{\mathrm{def.}}}}}\left( S_n,(X^w(S_n),Z^w(S_n)\right) , n{\in }{\mathbb {Z}})$ is a stationary point process, i.e.

$$\begin{aligned} ((S_n,X^w(S_n),Z^w(S_n), n{\in }{\mathbb {Z}}) {\mathop {=}\limits ^{{\mathrm{dist.}}}} ((S_n{-}t,X^w(S_n),Z^w(S_n), n{\in }{\mathbb {Z}}), \quad \forall t{\in }{\mathbb {R}}. \end{aligned}$$

The Palm measure of ${\mathcal {T}}$ is a probability distribution ${\widehat{Q}}$ such that the sequence $((S_n{-}S_{n-1},X^w(S_n),Z^w(S_n), n{\in }{\mathbb {Z}})$ is stationary. See Chapter 11 of Robert [27] for a quick presentation of stationary point processes and Palm measures.

Under ${\widehat{Q}}$, the Markov chain $((X^w(S_n),Z^w(S_n)), n{\ge }0)$ is at equilibrium. Using Harris ergodicity, we have proved in the previous section that the Markov chain $((X^w_n,Z^w_n), n{\ge }0)$ has a unique invariant measure. Considering that both sequences $((X^w_n,Z^w_n), n{\ge }0)$ and $((X^w(S_n),Z^w(S_n)), n{\ge }0)$ have the same distribution, we have that ${\widehat{Q}}\left( {\mathbb {R}}_+^{{\mathbb {Z}}}, \cdot \right)$ is uniquely determined.

Moreover, remembering that,

$$\begin{aligned} {\widehat{Q}}\left( S_n{-}S_{n-1}{>}t\right) = {\mathbb {E}}_{{\widehat{Q}}}\left[ \exp \left( {-}\int _{0}^t \beta \left( X^w(S_{n-1})e^{-s}\right) {\mathrm {d}}s \right) \right] \end{aligned}$$

We have that ${\widehat{Q}}$ is entirely determined by the ergodic distribution of the embedded Markov chain and consequently that the Palm measure ${\widehat{Q}}$ is unique. By Proposition 11.5 of Robert [27], the distribution of ${\mathcal {T}}$ is expressed with ${\widehat{Q}}$.

We have, for every bounded function f,

$$\begin{aligned} {\mathbb {E}}_{\mu }\left[ f(X^w(0),Z^w(0)) \right] = {\mathbb {E}}_{\mathcal {T}}\left[ f(X^w(S_0)e^{S_0},Z^w(S_0){\odot }e^{\gamma S_0})\right] , \end{aligned}$$

which uniquely determines the invariant distribution $\mu$.

The proposition is proved. $\square$

Appendix D: Averaging Principles for Discrete Models of Plasticity

In this section, we present a general discrete model of plasticity, state the associated averaging principle theorem and give a sketch of its proof. We will only point out the differences with the proof of the main result of this paper, Theorem 4.

For this model of plasticity, the membrane potential X, the plasticity processes Z and the synaptic weight W are integer-valued variables. This system is illustrated in Sect. 7 of Robert and Vignoud [28] for calcium-based models. It amounts to represent these three quantities X, Z and W as multiple of a “quantum” instead of a continuous variable. The leaking mechanism in particular, the term corresponding to ${-}\gamma Y(t){\mathrm {d}}t$ in the continuous model, $Y{\in }\{X,Z,W\}$ and $\gamma {>}0$, in the SDEs, is represented by the fact that each quantum leaves the system at a fixed rate $\gamma$.

The main advantage of this model is that simple analytical expressions of the invariant distribution are available.

Definition 26

The SDEs for the discrete model are

$$\begin{aligned} {\left\{ \begin{array}{ll} \quad {\mathrm {d}}X(t) &{}= \displaystyle {-}{\mathcal {N}}_{I,X}({\mathrm {d}}t) +W(t{-}){\mathcal {N}}_{\lambda }({\mathrm {d}}t){-}{\mathcal {N}}_{I,\beta X}({\mathrm {d}}t),\\ \quad {\mathrm {d}}Z(t) &{}= \displaystyle {-}{\mathcal {N}}_{I,\gamma Z}({\mathrm {d}}t)+ B_1{\mathcal {N}}_{\lambda }({\mathrm {d}}t)+B_2{\mathcal {N}}_{I,\beta X}({\mathrm {d}}t),\\ \quad {\mathrm {d}}\Omega _a(t)&{}=\displaystyle {-}\alpha \Omega _a(t){\mathrm {d}}t{+}n_{a,0}(Z(t)){\mathrm {d}}t\\ &{} {+}\displaystyle n_{a,1}(Z(t{-})){\mathcal {N}}_{\lambda }({\mathrm {d}}t){+}n_{a,2}(Z(t{-})){\mathcal {N}}_{I,\beta X}({\mathrm {d}}t),\quad a{\in }\{p,d\},\\ \quad {\mathrm {d}}W(t) &{}=\displaystyle -{\mathcal {N}}_{I,\mu W}({\mathrm {d}}t){+}A_p{\mathcal {N}}_{I,\Omega _{p}}({\mathrm {d}}t)-A_d{\mathbb {1}}_{\left\{ W(t-){\ge }A_d\right\} }{\mathcal {N}}_{I,\Omega _{d}}({\mathrm {d}}t), \end{array}\right. } \end{aligned}$$

(51)

where $\beta ,\gamma ,\mu$ are non-negative real numbers, $B_1$, $B_2{\in }{\mathbb {N}}^{\ell }$ and, for $a{\in }\{p,d\}$, $A_a{\in }{\mathbb {N}}$. The functions $n_{a,i}$ are assumed to be bounded by $C_n$.

For $a{\in }\{p,d\}$, the function I of ${\mathcal {N}}_{I,G}$ for $G{\in }\{X,\beta X, \gamma Z, \mu W, \Omega _{p}, \Omega _{d}\}$, defined by relation (6), is the identity function $I(x){=}x$, $x{\in }{\mathbb {R}}$ and ${\mathcal {N}}_\lambda$ is a Poisson process on ${\mathbb {R}}_+$ with rate $\lambda$. All associated Poisson processes are assumed to be independent.

Definition 27

For a fixed w, the process of the fast variables $(X^w(t),Z^w(t))$ on ${\mathbb {N}}{\times }{\mathbb {N}}^{\ell }$ of the SDEs is the Markov process whose transition rates are given by, for $(x,z){\in }{\mathbb {N}}{\times }{\mathbb {N}}^{\ell }$,

$$\begin{aligned} (x,z)\longrightarrow {\left\{ \begin{array}{ll} (x{+}w,z{+}B_1) &{} \lambda , \\ (x{-}1,z) &{} x, \end{array}\right. } \,\, \longrightarrow {\left\{ \begin{array}{ll} (x,z{-}1) &{} \gamma z, \\ (x{-}1,z{+}B_2) &{} \beta x. \end{array}\right. } \end{aligned}$$

Theorem 28

(Averaging Principle for a Discrete Model) If the assumptions of Definition 26 are verified, the family of scaled processes $(W_{\varepsilon }(t))$ associated to Relations (51) is converging in distribution, as $\varepsilon$ goes to 0, to the càdlàginteger-valued process (w(t)) satisfying the ODE

$$\begin{aligned} \quad {\mathrm {d}}w(t) = -{\mathcal {N}}_{I,\gamma w}({\mathrm {d}}t) {+}A_p{\mathcal {N}}_{I,\omega _{p}}({\mathrm {d}}t){-}A_d{\mathbb {1}}_{\left\{ w(t{-}){\ge }A_d\right\} }{\mathcal {N}}_{I,\omega _{d}}({\mathrm {d}}t), \end{aligned}$$

(52)

and, for $a{\in }\{p,d\}$,

$$\begin{aligned}&\frac{{\mathrm {d}}\omega _a}{{\mathrm {d}}t}(t)= {-}\alpha \omega _a(t){+}\\&\quad \int _{{\mathbb {N}}{\times }{\mathbb {N}}^\ell } \left( n_{a,0}(z){+}\lambda n_{a,1}(z){+}\beta (x) n_{a,2}(z)\right) \Pi _{w(t)}({\mathrm {d}}x,{\mathrm {d}}z), \end{aligned}$$

where $\Pi _{w}$ is the invariant distribution of the Markov process of Definition 27.

Proof

Again, we have to show that, on a fixed finite interval, the process (W(t)) is bounded with high probability. A coupled process that stochastically bounds from above the discrete process is also defined.

Definition 29

The process $(\overline{X}(t), \overline{Z}(t), \overline{\Omega }(t), \overline{W}(t) )$ satisfies the following SDEs

$$\begin{aligned} {\left\{ \begin{array}{ll} \quad {\mathrm {d}}\overline{X}(t) &{}= \displaystyle {-}{\mathcal {N}}_{I,\overline{X}}({\mathrm {d}}t) +\overline{W}(t{-}){\mathcal {N}}_{\lambda }({\mathrm {d}}t){-}{\mathcal {N}}_{I,\beta \overline{X}}({\mathrm {d}}t),\\ \quad {\mathrm {d}}\overline{Z}(t) &{}= \displaystyle {-}{\mathcal {N}}_{I,\gamma \overline{Z}}({\mathrm {d}}t)+ B_1{\mathcal {N}}_{\lambda }({\mathrm {d}}t)+B_2{\mathcal {N}}_{I,\beta \overline{X}}({\mathrm {d}}t),\\ \quad {\mathrm {d}}\overline{\Omega }(t)&{}=\displaystyle {-}\alpha \overline{\Omega }(t){\mathrm {d}}t{+}C_n{\mathrm {d}}t{+}\displaystyle C_n{\mathcal {N}}_{\lambda }({\mathrm {d}}t){+}C_n{\mathcal {N}}_{I,\beta \overline{X}}({\mathrm {d}}t),\\ \quad {\mathrm {d}}\overline{W}(t) &{}=\displaystyle A_p{\mathcal {N}}_{I,\overline{\Omega }}({\mathrm {d}}t), \end{array}\right. } \end{aligned}$$

(53)

where $B_1$, $B_2{\in }{\mathbb {N}}^{\ell }$ and, for $a{\in }\{p,d\}$, $A_p{\in }{\mathbb {N}}$.

It is not difficult to prove that this process is indeed a coupling that verifies the relation $W(t){\le }\overline{W}(t)$, for all $t{\ge }0$ and that the process $(\overline{W}(t))$ is non-decreasing.

From the SDEs governing the scaled version of the coupled system, we obtain

$$\begin{aligned}&{\mathbb {E}}\left[ \overline{W}_{\varepsilon }(t){-}w_0\right] \le A_p{\mathbb {E}}\left[ \int _0^t{\mathcal {N}}_{I,\overline{\Omega }_{\varepsilon ,p}}{\mathrm {d}}s\right] {\le }A_p t {\mathbb {E}}\left[ \sup _{s\le t}\overline{\Omega }_{\varepsilon ,p}(s)\right] \\&\quad \le A_p t {\mathbb {E}}\left[ \omega _{0}{+}C_n\sup _{s\le t}\int _0^se^{-\alpha (s{-}u)}\left( {\mathrm {d}}u + \varepsilon {\mathcal {N}}_{\lambda /\varepsilon }({\mathrm {d}}u){+}\varepsilon {\mathcal {N}}_{I,\beta \overline{X}/\varepsilon }({\mathrm {d}}u)\right) \right] \\&\quad \le A_p t \left( \omega _{0}{+}\frac{C_n}{\alpha }(1{+}\lambda ){+}{\mathbb {E}}\left[ \int _0^t\beta \overline{X}_{\varepsilon }(u)\right] {\mathrm {d}}u\right) \\&\quad \le A_p t \left( \omega _{0}{+}\frac{C_n}{\alpha }(1{+}\lambda ){+}\lambda \beta {\mathbb {E}}\left[ \int _0^t\overline{W}_{\varepsilon }(u){\mathrm {d}}u\right] \right) \le D {+}D\int _0^t{\mathbb {E}}\left[ \overline{W}_{\varepsilon }(u)\right] {\mathrm {d}}u, \end{aligned}$$

for all $t{\le }T$, for some constant $D{\ge }0$. Gronwall’s Lemma gives a uniform bound, with respect to $\varepsilon$,

$$\begin{aligned} {\mathbb {E}}\left[ \sup _{t{\le }T}\overline{W}_{\varepsilon }(t)\right] ={\mathbb {E}}\left[ \overline{W}_{\varepsilon }(T)\right] )\le (D{+}w_0)e^{D T}. \end{aligned}$$

Using Markov inequality, we have then that, for any $\eta {>}0$, the existence of $K_0$ and $n_0$ such that $n{\ge }n_0$, the inequality

$$\begin{aligned} {\mathbb {P}}\left( \sup _{t{\le }T} \overline{W}_{\varepsilon _n}(t){\le } K_0\right) \ge 1{-}\eta \end{aligned}$$

holds. We can then finish the proof in the same way as in Sect. 7.2. The tightness property of the family of càdlàgprocesses $(\overline{W}_{\varepsilon }(t))$, $\varepsilon {\in }(0,1)$ are proved with Aldous’ criterion, see Theorem VI.4.5 of Jacod and Shiryaev [16].

We have to prove the uniqueness of the solution of Relation (52) and the convergence in distribution of the scaled process to the process w(t). For this, we need to have some Lipschitz property on the limiting system, and finite first moments for the invariant distribution of $(X^w(t),Z^w(t))$. This is proved in Sect. 7 of Robert and Vignoud [28] for the case where Z is a one-dimensional process, the extension to multi-dimensional Z is straightforward. $\square$