Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Moderate deviation principles for weakly interacting particle systems

Abstract

Moderate deviation principles for empirical measure processes associated with weakly interacting Markov processes are established. Two families of models are considered: the first corresponds to a system of interacting diffusions whereas the second describes a collection of pure jump Markov processes with a countable state space. For both cases the moderate deviation principle is formulated in terms of a large deviation principle (LDP), with an appropriate speed function, for suitably centered and normalized empirical measure processes. For the first family of models the LDP is established in the path space of an appropriate Schwartz distribution space whereas for the second family the LDP is proved in the space of \(l_2\) (the Hilbert space of square summable sequences)-valued paths. Proofs rely on certain variational representations for exponential functionals of Brownian motions and Poisson random measures.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Baladron, J., Fasoli, D., Faugeras, O., Touboul, J.: Mean-field description and propagation of chaos in networks of Hodgkin-Huxley and Fitzhugh-Nagumo neurons. J. Math. Neurosci. 2(1), 10 (2012)

  2. 2.

    Borkar, V.S., Sundaresan, R.: Asymptotics of the invariant measure in mean field models with jumps. Stoch. Syst. 2(2), 322–380 (2012)

  3. 3.

    Borovkov, A.A., Mogul’skii, A.A.: Probabilities of large deviations in topological spaces. II. Sib. Math. J. 21(5), 653–664 (1980)

  4. 4.

    Boué, M., Dupuis, P.: A variational representation for certain functionals of Brownian motion. Ann. Probab. 26(4), 1641–1659 (1998)

  5. 5.

    Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1/n limit of interacting classical particles. Commun. Math. Phys. 56(2), 101–113 (1977)

  6. 6.

    Budhiraja, A., Dupuis, P.: A variational representation for positive functionals of infinite dimensional Brownian motion. Probab. Math. Stat. 20(1), 39–61 (2000)

  7. 7.

    Budhiraja, A., Dupuis, P., Fischer, M.: Large deviation properties of weakly interacting processes via weak convergence methods. Ann. Probab. 40(1), 74–102 (2012)

  8. 8.

    Budhiraja, A., Dupuis, P., Fischer, M., Ramanan, K.: Limits of relative entropies associated with weakly interacting particle systems. Electron. J. Probab. 20(80), 1–22 (2015)

  9. 9.

    Budhiraja, A., Dupuis, P., Ganguly, A.: Moderate deviation principles for stochastic differential equations with jumps. Ann. Probab. (2015) (to appear)

  10. 10.

    Budhiraja, A., Dupuis, P., Maroulas, V.: Large deviations for infinite dimensional stochastic dynamical systems. Ann. Probab., 1390–1420 (2008)

  11. 11.

    Budhiraja, A., Dupuis, P., Maroulas, V.: Variational representations for continuous time processes. Ann. de l’Institut Henri Poincaré(B), Probab. et Stat. 47(3), 725–747 (2011)

  12. 12.

    Budhiraja, A., Kira, E., Saha, S.: Central limit results for jump-diffusions with mean field interaction and a common factor. Math (2014). arXiv:1405.7682

  13. 13.

    Budhiraja, A., Wu, R.: Some fluctuation results for weakly interacting multi-type particle systems. Stoch. Process. Appl. 126(8), 2253–2296 (2016)

  14. 14.

    Chong, C., Klüppelberg, C.: Partial mean field limits in heterogeneous networks (2015). arXiv:1507.01905 (preprint)

  15. 15.

    Collet, F.: Macroscopic limit of a bipartite Curie-Weiss model: a dynamical approach. J. Stat. Phys. 157(6), 1301–1319 (2014)

  16. 16.

    Dawson, D.A.: Critical dynamics and fluctuations for a mean-field model of cooperative behavior. J. Stat. Phys. 31(1), 29–85 (1983)

  17. 17.

    Dawson, D.A., Gärtner, J.: Large deviations from the McKean-Vlasov limit for weakly interacting diffusions. Stochastics 20(4), 247–308 (1987)

  18. 18.

    Del Moral, P., Hu, S., Wu, L.: Moderate deviations for interacting processes. Stat. Sinica 25(3), 921–951 (2015)

  19. 19.

    Dupuis, P., Ellis, R.S.: A weak convergence approach to the theory of large deviations, vol. 902. Wiley, New York (2011)

  20. 20.

    Dupuis, P., Ramanan, K., Wu, W.: Sample path large deviation principle for mean field weakly interacting jump processes (2012) (preprint)

  21. 21.

    Gel’fand, I.M., Vilenkin, N.Y.: Generalized functions, vol. 4. New York (1964)

  22. 22.

    Graham, C., Méléard, S.: Stochastic particle approximations for generalized Boltzmann models and convergence estimates. Ann. Probab. 25(1), 115–132 (1997)

  23. 23.

    Hitsuda, M., Mitoma, I.: Tightness problem and stochastic evolution equation arising from fluctuation phenomena for interacting diffusions. J. Multivar. Anal. 19(2), 311–328 (1986)

  24. 24.

    Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. Elsevier, London (1981)

  25. 25.

    Kallianpur, G., Xiong, J.: Stochastic differential equations in infinite dimensional spaces. Lect. Notes Monogr. Ser. 26, iii–342 (1995)

  26. 26.

    Kolokoltsov, V.N.: Nonlinear Markov Processes and Kinetic Equations, vol. 182. Cambridge University Press, Cambridge (2010)

  27. 27.

    Kurtz, T.G., Xiong, J.: Particle representations for a class of nonlinear SPDEs. Stoch. Process. Appl. 83(1), 103–126 (1999)

  28. 28.

    Kurtz, T.G., Xiong, J.: A stochastic evolution equation arising from the fluctuations of a class of interacting particle systems. Commun. Math. Sci. 2(3), 325–358 (2004)

  29. 29.

    Leonard, C.: Large deviations for long range interacting particle systems with jumps. Ann. de l’IHP Prob. et Stat. 31, 289–323 (1995)

  30. 30.

    McKean, H.P.: A class of Markov processes associated with nonlinear parabolic equations. Proc. Natl. Acad. Sci. USA 56(6), 1907–1911 (1966)

  31. 31.

    McKean, H.P.: Propagation of chaos for a class of non-linear parabolic equations. In: Stochastic Differential Equations (Lecture Series in Differential Equations, Session 7, Catholic Univ., 1967), pp. 41–57. Air Force Office Sci. Res., Arlington (1967)

  32. 32.

    Méléard, S.: Convergence of the fluctuations for interacting diffusions with jumps associated with Boltzmann equations. Stochastics 63(3–4), 195–225 (1998)

  33. 33.

    Mitoma, I.: Tightness of probabilities on C([0, 1]; \({\cal S}^{\prime }\)) and D([0, 1]; \({\cal S}^{\prime }\)). Ann. Probab. 11(4), 989–999 (1983)

  34. 34.

    Oelschlager, K.: A martingale approach to the law of large numbers for weakly interacting stochastic processes. Ann. Probab. 12(2), 458–479 (1984)

  35. 35.

    Shiga, T., Tanaka, H.: Central limit theorem for a system of Markovian particles with mean field interactions. Probab. Theory Relat. Fields 69(3), 439–459 (1985)

  36. 36.

    Spiliopoulos, K., Sowers, R.B.: Default clustering in large pools: large deviations. SIAM J. Financial Math. 6(1), 86–116 (2015)

  37. 37.

    Sznitman, A.-S.: Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated. J. Funct. Anal. 56(3), 311–336 (1984)

  38. 38.

    Sznitman, A.S.: Topics in propagation of chaos. In: Ecole d’Eté de Probabilités de Saint-Flour XIX—1989, pp. 165–251. Springer, New York (1991)

  39. 39.

    Tanaka, H.: Limit theorems for certain diffusion processes with interaction. North Holl. Math. Libr. 32, 469–488 (1984)

Download references

Author information

Correspondence to Amarjit Budhiraja.

Additional information

The research is supported in part by the National Science Foundation (DMS-1305120), the Army Research Office (W911NF-14-1-0331) and DARPA (W911NF-15-2-0122).

Appendix: Proof of Remark 3.2(iii)

Appendix: Proof of Remark 3.2(iii)

Suppose that for some \(n \in {\mathbb {N}}\), \(\sum _{m=1}^\infty [a(m)]^{2n} < \infty \). We need to show that \(a(m) \sqrt{m} \Vert \mu ^m(0)-p(0) \Vert \rightarrow 0\) almost surely. To simplify the notation, we will abbreviate \(\mu ^m(0), p(0), \mu ^m_i(0), p_i(0)\) as \(\mu ^m, p, \mu ^m_i, p_i\). It follows from Markov’s inequality that for \({\varepsilon }> 0\),

$$\begin{aligned} {\mathbb {P}}(a(m) \sqrt{m} \Vert \mu ^m-p \Vert > {\varepsilon })\le & {} \left( \frac{a(m) \sqrt{m}}{{\varepsilon }}\right) ^{2n} {\mathbb {E}}\Vert \mu ^m-p \Vert ^{2n} \\= & {} \frac{[a(m)]^{2n}}{{\varepsilon }^{2n}} m^n {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^n. \end{aligned}$$

Since \(\sum _{m=1}^\infty [a(m)]^{2n} < \infty \), by Borel-Cantelli lemma, it suffices to show that for every \(n \in {\mathbb {N}}\) there exists some \(\hat{\gamma }_n \in (0,\infty )\) such that

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^n \le \frac{\hat{\gamma }_n}{m^n}. \end{aligned}$$
(6.1)

We will prove (6.1) when \(n=2\) in detail and then sketch the argument for \(n > 2\). First write

$$\begin{aligned} \mu _i^m-p_i = \frac{1}{m} \sum _{j=1}^m {\varvec{1}}_{\{\xi _j=i\}} - p_i \equiv \frac{1}{m} \sum _{j=1}^m Y_{ij}, \end{aligned}$$

where \(Y_{ij} \doteq {\varvec{1}}_{\{\xi _j=i\}} - p_i\). Note that for all \(\alpha , \beta \in {\mathbb {N}}\),

$$\begin{aligned} |Y_{ij}|\le & {} 1, \quad {\mathbb {E}}Y_{ij} = 0, \quad {\mathbb {E}}Y_{ij}^{2\alpha } \le {\mathbb {E}}Y_{ij}^2 \le p_i, \\ {\mathbb {E}}Y_{ij}^{2\alpha } Y_{kl}^{2\beta }\le & {} {\mathbb {E}}Y_{ij}^2 Y_{kl}^2 \le p_i p_k \quad \text { for } i \ne k. \end{aligned}$$

So we have

$$\begin{aligned} {\mathbb {E}}\left[ \sum _{i=1}^\infty (\mu _i^m-p_i)^2\right] ^2= & {} \frac{1}{m^4} {\mathbb {E}}\left[ \sum _{i=1}^\infty \sum _{j,j'=1}^m Y_{ij} Y_{ij'} \right] ^2\nonumber \\= & {} \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{j,j',l,l'=1}^m Y_{ij} Y_{ij'} Y_{kl} Y_{kl'}. \end{aligned}$$
(6.2)

From independence of \(\{\xi _j\}\) it follows that \(Y_{ij}\) and \(Y_{kl}\) are independent for \(j \ne l\). Hence

$$\begin{aligned} {\mathbb {E}}Y_{ij} Y_{ij'} Y_{kl} Y_{kl'} \ne 0 \text{ only } \text{ if } j,j',l,l' \text{ are } \text{ matched } \text{ in } \text{ pairs } \text{(e.g. } j=j' \text{ and } l=l'). \end{aligned}$$
(6.3)

Using this observation, (6.2) can be written as

$$\begin{aligned} \frac{1}{m^4} {\mathbb {E}}\sum _{r=1}^5 \sum _{(i,k,j,j',l,l') \in {\mathcal {H}}_r} Y_{ij} Y_{ij'} Y_{kl} Y_{kl'} \equiv \sum _{r=1}^5 {\mathcal {T}}_r^m, \end{aligned}$$

where \({\mathcal {H}}_1\), \({\mathcal {H}}_2\), \({\mathcal {H}}_3\), \({\mathcal {H}}_4\) and \({\mathcal {H}}_5\) are collections of \((i,k) \in {\mathbb {N}}^2\) and \((j,j',l,l') \in \{1,\cdots ,m\}^4\) such that \(\{ j = j' \ne l = l' \}\), \(\{ j = l \ne j' = l' \}\), \(\{ j = l' \ne j' = l \}\), \(\{ j = j' = l = l', i = k \}\) and \(\{ j = j' = l = l', i \ne k \}\), respectively. For \({\mathcal {T}}_1^m\), it follows from independence of \(\{\xi _j\}\) that

$$\begin{aligned} {\mathcal {T}}_1^m = \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l=1 \\ j \ne l \end{array}}^m Y_{ij}^2 Y_{kl}^2 = \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l=1 \\ j \ne l \end{array}}^m {\mathbb {E}}Y_{ij}^2 {\mathbb {E}}Y_{kl}^2 \le \frac{1}{m^2} \sum _{i,k=1}^\infty p_i p_k = \frac{1}{m^2}. \end{aligned}$$

For \({\mathcal {T}}_2^m\), using independence of \(\{\xi _j\}\) and Cauchy-Schwarz inequality we have

$$\begin{aligned} {\mathcal {T}}_2^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m Y_{ij} Y_{il'} Y_{kj} Y_{kl'} = \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m {\mathbb {E}}\left( Y_{ij} Y_{kj} \right) {\mathbb {E}}\left( Y_{il'} Y_{kl'} \right) \\&\le \frac{1}{m^4} \sum _{i,k=1}^\infty \sum _{\begin{array}{c} j,l'=1 \\ j \ne l' \end{array}}^m \sqrt{{\mathbb {E}}Y_{ij}^2 {\mathbb {E}}Y_{kj}^2 {\mathbb {E}}Y_{il'}^2 {\mathbb {E}}Y_{kl'}^2} \le \frac{1}{m^2} \sum _{i,k=1}^\infty p_i p_k = \frac{1}{m^2}. \end{aligned}$$

Similarly, \({\mathcal {T}}_3^m \le \frac{1}{m^2}\). For \({\mathcal {T}}_4^m\) and \({\mathcal {T}}_5^m\), we have

$$\begin{aligned} {\mathcal {T}}_4^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{i=1}^\infty \sum _{j=1}^m Y_{ij}^4 \le \frac{1}{m^3} \sum _{i=1}^\infty p_i = \frac{1}{m^3}, \\ {\mathcal {T}}_5^m&= \frac{1}{m^4} {\mathbb {E}}\sum _{\begin{array}{c} i,k=1 \\ i \ne k \end{array}}^\infty \sum _{j=1}^m Y_{ij}^2 Y_{kj}^2 \le \frac{1}{m^3} \sum _{\begin{array}{c} i,k=1 \\ i \ne k \end{array}}^\infty p_i p_k \le \frac{1}{m^3}. \end{aligned}$$

Combining above estimates we can bound (6.2) by \(\frac{3}{m^2}+\frac{2}{m^3}\). This proves (6.1) when \(n=2\).

Note that the above argument relies crucially on (6.3), which helps reduce the 4-fold summation over \(j,j',l,l'\) in (6.2) to one that is at most 2-fold so that we have (6.1) when \(n=2\). When \(n > 2\), the right hand side of (6.2) will consist of a 2n-fold summation over the second subscript of Y’s. It is easy to check that an observation analogous to (6.3) holds and reduces the 2n-fold summation to one that is at most n-fold. From this we can deduce the desired bound (6.1) and complete the proof.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Budhiraja, A., Wu, R. Moderate deviation principles for weakly interacting particle systems. Probab. Theory Relat. Fields 168, 721–771 (2017). https://doi.org/10.1007/s00440-016-0723-3

Download citation

Keywords

  • Moderate deviations
  • Large deviations
  • Laplace principle
  • Variational representations
  • Weakly interacting jump-diffusions
  • Nonlinear Markov processes
  • Mean field asymptotics
  • Schwartz distributions
  • Poisson random measures

Mathematics Subject Classification

  • 60F10
  • 60K35
  • 60J75
  • 60J60