## 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.

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## Notes

Indeed, for any

*j*, \(V (K_j, z ) < \infty \). Suppose that the trajectory achieving the minimal cost to go from \(K_j \) to*z*visits the sets \(K_j, \) followed by \( K_{n_1}, \ldots , K_{n_l}\), before leaving the last of them, \(K_{n_l}\), and reaching the target*z*. It is then sufficient to choose*i*to be equal to the index of the last visited set, that is, \( i := n_l\).

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## Acknowledgements

I would like to thank an anonymous reviewer for his valuable comments and suggestions which helped me to improve the paper. This research has been conducted as part of the project Labex MME-DII (ANR11-LBX-0023-01) and as part of the activities of FAPESP Research, Dissemination and Innovation Center for Neuromathematics (Grant 2013/07699-0, S. Paulo Research Foundation).

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## Appendix

### 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

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

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

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

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

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

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 \( t \le \delta \),

Since

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.

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

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

and thus

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

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

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

and put

which is an open set. Then

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

2) We shall now show that there exists \( T_2\) such that

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

and

Then

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.

By iteration, we obtain

But

for *N* sufficiently large. This implies the desired assertion. \(\square \)

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Löcherbach, E. Large Deviations for Cascades of Diffusions Arising in Oscillating Systems of Interacting Hawkes Processes.
*J Theor Probab* **32**, 131–162 (2019). https://doi.org/10.1007/s10959-017-0789-6

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DOI: https://doi.org/10.1007/s10959-017-0789-6

### Keywords

- Hawkes processes
- Piecewise deterministic Markov processes
- Diffusion approximation
- Sample path large deviations for degenerate diffusions
- Control theory for degenerate diffusions