Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes

Abstract

We consider oscillatory systems of interacting Hawkes processes introduced in Ditlevsen and Löcherbach (Stoch Process Appl 2017, http://arxiv.org/abs/1512.00265) to model multi-class systems of interacting neurons together with the diffusion approximations of their intensity processes. This diffusion, which incorporates the memory terms defining the dynamics of the Hawkes process, is hypo-elliptic. It is given by a high-dimensional chain of differential equations driven by 2-dimensional Brownian motion. We study the large population, i.e., small noise limit of its invariant measure for which we establish a large deviation result in the spirit of Freidlin and Wentzell.

Appendix

Proof of Theorem 6

The proof follows the lines of the proof of Theorem 1 in Chapter 6 of Lee and Markus [15]. As there, we write $f( x, u) = b(x) + \sigma ( x) u$. We fix $x_0 \in \Gamma$ and write

$$\begin{aligned} A (t) =\left( \frac{\partial f }{\partial x}\right) _{| x= x_t^{x_0}, u = 0}, \; B (t) = \left( \frac{\partial f }{\partial u}\right) _{| x= x_t^{x_0}, u = 0} = \sigma (x_t^{x_0}). \end{aligned}$$

Let $A := A ( 0)$ and $B = B(0)$. Then, it is easy to see that the columns of $B, AB, A^2 B, \ldots , A^{n - 1 } B$ span ${\mathbb {R}}^n$. We start by considering the equation

$$\begin{aligned} {\dot{Y}} = A Y + Bu, Y(0) = 0. \end{aligned}$$

(6.6)

Denote by $Y^u (t), t \le \delta$, a solution to (6.6) driven by $u(t), t \le \delta$. The above system is controllable, since $B, AB, A^2 B, \ldots , A^{n - 1 } B$ span ${\mathbb {R}}^n$. As a consequence, for every M and for any $\delta < 1$ there exist controls $u_1, u_2, \ldots , u_n$ with $\Vert u_i \Vert _\infty \le M$ such that

$$\begin{aligned} Y^{u_1} (\delta ) = r e_1, \ldots , Y^{u_n } (\delta ) = r e_n, \end{aligned}$$

(6.7)

where $e_1, \ldots , e_n$ are the unit vectors of ${\mathbb {R}}^n$ (Corollary 1 of Chapter 2 of Lee and Markus [15]) and where $r > 0$ is suitably small.

We wish now to replace system (6.6) by the time-dependent system

$$\begin{aligned} \dot{W} = A (t) W + B (t) u, W(0) = 0, t \le \delta . \end{aligned}$$

(6.8)

Write $W_k ( t)$ for the solution of $\dot{W}_k (t) = A ( t) W_k ( t) + B(t) u_k ( t)$, where the $u_k (t)$ are given in (6.7). Then, $W_k (t)$ is explicitly given by

$$\begin{aligned} W_k ( t) = \Phi (t) \int _0^t \Phi ^{-1} (s) B(s) u_k ( s) ds, \end{aligned}$$

with $\Phi (t)$ the matrix solution of $\dot{\Phi } (t) = A(t) \Phi (t)$, $\Phi (0) = Id$. Writing $Y_k (t) = Y^{u_k } (t)$, we obtain similarly

$$\begin{aligned} Y_k ( t) = \overline{\Phi }(t) \int _0^t \overline{\Phi }^{-1} (s) B u_k ( s) ds, \end{aligned}$$

with $\overline{\Phi }(t) = e^{ A t }$ (recall that $A = A(0)$). We wish to show that $\Vert Y_k ( t) - W_k (t) \Vert$ is small for t sufficiently small. For that sake, note that there exists a constant C such that for all $t \le \delta$,

$$\begin{aligned} \Vert \Phi (t) \Vert , \Vert \overline{\Phi }(t) \Vert , \Vert \Phi ^{-1} (t) \Vert , \Vert \overline{\Phi }^{-1} (t) \Vert , \Vert B( t) \Vert , \Vert B\Vert \le C. \end{aligned}$$

Since

$$\begin{aligned} \Phi (t) = Id + \int _0^t A(s) \Phi (s) ds, \; \overline{\Phi }(t) = Id + \int _0^t A \overline{\Phi }(s) ds, \end{aligned}$$

it follows from this that $\Vert \Phi (t) - \overline{\Phi }(t) \Vert \rightarrow 0$ as $t \rightarrow 0$.

Fix $\varepsilon > 0$ such that $\tilde{e}_1, \ldots , \tilde{e}_n$ still span ${\mathbb {R}}^n$ for all $\tilde{e}_k \in B_\varepsilon ( r e_k ), 1 \le k \le n$. Then, there exists $\delta ^*$ such that for all $\delta \le \delta ^*$, $W_k ( \delta ) \in B_\varepsilon ( Y_k ( \delta ))$, for all $1 \le k \le n,$ and therefore, the following holds.

$$\begin{aligned}&\text{ The } \text{ solutions } \text{ of } \dot{W}_k (t) = A ( t) W_k ( t) + B(t) u_k ( t),\; W_k ( 0 ) = 0, \; 1 \le k \le n, \nonumber \\&\quad \text{ are } \text{ such } \text{ that } W_1 (\delta ), \ldots , W_n ( \delta ) \text{ span } {\mathbb {R}}^n. \end{aligned}$$

(6.9)

We are now able to conclude the proof, following the lines of Lee and Markus [15]. Consider $x ( t, \xi )$ which is the solution of

$$\begin{aligned} d x( t, \xi ) = b ( x( t, \xi ) ) dt + \sigma ( x( t , \xi ) ) \dot{h} (t, \xi ) dt, \; x(0, \xi ) = x_0, \end{aligned}$$

following the control $\dot{h} ( t, \xi ) = \xi _1 u_1 (t) + \cdots + \xi _n u_n ( t)$, for $| \xi _i | \le 1, 1 \le i \le n$. It is clear that $x ( t, 0 ) = x^{x_0}(t)$. Hence, if we can prove that $Z (t) = \left( \frac{\partial x (t, \xi ) }{\partial \xi }\right) _{ | \xi = 0 }$ is non-degenerate at $t = \delta ,$ we are done, using the inverse function theorem. But

$$\begin{aligned} \frac{ \partial x (t, x) }{\partial t} = f( x (t, \xi ), \dot{h} (t, \xi ) ) \end{aligned}$$

and thus

$$\begin{aligned} \frac{\partial }{\partial t} \frac{\partial x (t, \xi ) }{\partial \xi } = f_x ( x(t, \xi ), \dot{h} (t, \xi ) ) \frac{\partial x}{\partial \xi } + f_u ( x (t, \xi ), \dot{h} (t, \xi ) ) \frac{\partial \dot{h} }{\partial \xi }. \end{aligned}$$

Notice that $x (t, 0) = x^{x_0 }_{t }$ and $\dot{h} (t, 0) = 0$. Thus, we obtain

$$\begin{aligned} \dot{Z} (t) = A ( t) Z (t) + B(t) U (t), \end{aligned}$$

where $U (t) = (u_1 (t), \ldots , u_n (t))$. Writing $z_1, \ldots , z_n$ for the columns of Z(t),  this gives

$$\begin{aligned} \dot{z}_k ( t) = A(t) z_k ( t) + B(t) u_k (t), \; z_k ( 0) = 0. \end{aligned}$$

The solutions of this system are given by (6.9), and they are such that $z_k ( \delta ), 1 \le k \le n$, span ${\mathbb {R}}^n$. Therefore, $Z ( \delta )$ is non-degenerate, and this concludes the proof. $\square$

Proof of Proposition 5

The proof follows closely the ideas of Chapter 5.7 of [8].

1) For all $x \in \partial B_\varepsilon (K)$, by small-time local controllability, there exists a smooth path $\psi ^x$ of length $t^x$ such that $\psi ^x (t^x) \in K= \{x^* \} \cup \bigcup _{l=2}^L K_l$ and such that $\psi ^x (t)$ does not leave $B_{2 \bar{\varepsilon } /3 }( K)$ for all $t \le t^x$. Moreover, this path can be chosen such that $I_{x, t^x} ( \psi ^x ) \le h/2$.

2) For all $x_0 \in K$, there exists $z \in \partial B_\varepsilon (K)$ and a path $\psi ^{x_0}$ of length $t^{x_0}$ steering $x_0$ to z,  during $[0, t^{x_0} ]$, without leaving $B_{2 \bar{\varepsilon } /3}(K)$, at a cost $I_{x_0, t^{x_0}} ( \psi ^{x_0 } ) \le h/2$.

3) We concatenate the two paths $\psi ^x$ and then $\psi ^{x_0}$ to obtain a new trajectory $\Psi ^x$ of length $T^x = t^x + t^{x_0}$ steering x to $z \in \partial B_\varepsilon (K)$. Let then

$$\begin{aligned} T_0 := \inf _{ x \in \partial B_{\varepsilon }(K ) } T^x > 0 \end{aligned}$$

and put

$$\begin{aligned} {{\mathcal {O}}} := \bigcup _{ x \in \partial B_\varepsilon (K) } \{ \varphi \in C ( [ 0, T_0], {\mathbb {R}}^n ) : \Vert \varphi - \Psi ^x \Vert _\infty < \varepsilon / 2 \}, \end{aligned}$$

which is an open set. Then

$$\begin{aligned} \liminf _{N \rightarrow \infty } \frac{1}{N} \log \inf _{x \in \partial B_\varepsilon (K) } Q^N_x ( {{\mathcal {O}}} ) \ge - h, \end{aligned}$$

which implies the assertion since $Q^N_x ( Y^N \in {{\mathcal {O}}} ) \le Q^N_x ( \sigma _0 \ge T_0 ) \le \frac{E^N_x \sigma _0}{T_0}$. $\square$

Proof of Proposition 6

1) Let $S = \inf \{ t \ge 0 : Y^N \in B_{\varepsilon } ( K) \cup D \}$, where $D = (B_{\bar{\varepsilon } }(K) )^c$. We know by Lemma 2 that there exists $T_1 > 0$ such that

$$\begin{aligned} \limsup _{N \rightarrow \infty } \sup _{x \in \overline{B_{\bar{\varepsilon } }(K )}} Q_x^N ( S > T_1 ) < 1. \end{aligned}$$

(6.10)

2) We shall now show that there exists $T_2$ such that

$$\begin{aligned} \liminf _{N \rightarrow \infty } \frac{1}{N} \log \inf _{x \in \overline{B_\varepsilon (K)}} Q_x^N ( \sigma _0 \le T_2 ) \ge - h. \end{aligned}$$

(6.11)

Indeed, like in [8], page 231, we first construct, for all $x \in \overline{B_\varepsilon (K)}$ a smooth path $\psi ^x$ of length $t^x$ such that $\psi ^x (t^x) \in K$ and such that $\psi ^x (t)$ does not leave $B_{2 \bar{\varepsilon } /3}(K )$ for all $t \le t^x$. Moreover, this path can be chosen such that $I_{x, t^x} ( \psi ^x ) \le h/2$.

We then fix $\varepsilon ' > \bar{\varepsilon }$ such that $6 \bar{\varepsilon } < \varepsilon '$ and apply Proposition 4 to $2 \bar{\varepsilon }$ and $\varepsilon '$. This is possible if $\bar{\varepsilon }$ is sufficiently small. Then for any $x_0 \in K$, there exists $z \in \partial B_{2 \bar{\varepsilon } }(K)$ and a path $\psi ^{x_0}$ of length $t^{x_0}$ steering $x_0$ to z, during $[0, t^{x_0} ]$, such that $I_{x_0, t^{x_0 } } ( \psi ^{x_0 } ) \le h/2$. We then concatenate the two paths and obtain a new path $\Psi ^x$ of length $T^x = t^x + t^{x_0}$, steering x to z, at cost $\le h$. Let

$$\begin{aligned} T_2 = \sup _{ x \in \overline{B_\varepsilon (K)} } T^x < \infty \end{aligned}$$

and

$$\begin{aligned} {{\mathcal {O}}} =\bigcup _{ x \in \overline{B_\varepsilon (K)} } \{ \varphi \in C ( [ 0, T_2], {\mathbb {R}}^n ) : \Vert \varphi - \Psi ^x \Vert _\infty < \bar{\varepsilon } / 2 \}. \end{aligned}$$

Then

$$\begin{aligned} \liminf _{N \rightarrow \infty } \frac{1}{N} \log \inf _{x \in \overline{B_\varepsilon (K)}} Q_x^N ( Y^N \in {{\mathcal {O}}} ) \ge - h, \end{aligned}$$

which implies (6.11), since $\varphi \in {{\mathcal {O}}}$ implies that $\sigma _0 ( \varphi ) \le T_2$.

3) We deduce from the above discussion the following.

$$\begin{aligned} \inf _{ x \in \overline{B_{\bar{\varepsilon }}(K ) }} Q^N_x (\sigma _0 \le T := T_1 + T_2 ) \ge \inf _{ x \in \overline{B_{\bar{\varepsilon } }(K )} } Q^N_x ( S \le T_1 ) \cdot \inf _{ x \in \overline{ B_\varepsilon (K )} } Q^N_x ( \sigma _0 \le T_2 ) =: q . \end{aligned}$$

By iteration, we obtain

$$\begin{aligned} \sup _{ x \in \overline{B_{\bar{\varepsilon }} (K ) }} Q^N_x ( \sigma _0 > k T ) \le (1- q )^k, \text{ whence } \sup _{ x \in \partial {B_{ \varepsilon }(K ) }} E^N_x \sigma _0 \le \sup _{ x \in \overline{B_{\bar{\varepsilon } }(K ) }} E^N_x \sigma _0 \le \frac{T}{q}. \end{aligned}$$

But

$$\begin{aligned} q \ge e^{ - Nh } \inf _{ x \in \overline{B_{\bar{\varepsilon } }(K )} } Q^N_x ( S \le T_1 ) \ge c e^{- N h }, \end{aligned}$$

for N sufficiently large. This implies the desired assertion. $\square$