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An invariance principle for branching diffusions in bounded domains

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Abstract

We study branching diffusions in a bounded domain D of \(\mathbb {R}^d\) in which particles are killed upon hitting the boundary \(\partial D\). It is known that any such process undergoes a phase transition when the branching rate \(\beta \) exceeds a critical value: a multiple of the first eigenvalue of the generator of the diffusion. We investigate the system at criticality and show that the associated genealogical tree, when the process is conditioned to survive for a long time, converges to Aldous’ Continuum Random Tree under appropriate rescaling. The result holds under only a mild assumption on the domain, and is valid for all branching mechanisms with finite variance, and a general class of diffusions.

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Notes

  1. We say that \(D\subset \mathbb {R}^d\) is a {Lipschitz / \(C^k\) / \(C^{k,\alpha }\)} domain (for \(k\in \mathbb {Z}_{\ge 0}\) and \(\alpha \in [0,1]\)) if, at each point \(x_0\in \partial D\), there exists \(r>0\) and a {Lipschitz / \(C^k\) / \(C^{k,\alpha }\)} function \(\gamma :\mathbb {R}^{d-1}\rightarrow \mathbb {R}\) such that relabelling and reorienting axes as necessary, \(D\cap B(x_0,r)=\{x\in B(x_0,r): x^d>\gamma (x^1,\ldots , x^{d-1})\}\).

  2. We define \(H_0^1(D)\) to be the closure of \(C_c^\infty (D)\) (the space of infinitely differentiable functions with compact support strictly inside D) with respect to the norm \(\Vert u||_{H^1(D)}:=\Vert u\Vert _{L^2} +\sum _{i=1}^d \Vert D^{x_i}(u)\Vert _{L^2}\), where \(D^{x_i}u\) is the ith partial derivative of u in the weak sense.

  3. Indeed, since \(\hat{p}(t):=\sup \{p(u,\varepsilon ): u\ge t\}\) converges monotonically to 0 as \(t\rightarrow \infty \), we can choose \(g(t,\varepsilon ) \le \sqrt{t}\) but still converging to \(\infty \), such that \(g(t,\varepsilon )\hat{p}(\sqrt{t})\rightarrow 0\) as \(t\rightarrow \infty \). Then since \(p(t/g(t,\varepsilon ),\varepsilon )\le \hat{p}( t/g(t,\varepsilon ))\le \hat{p}(\sqrt{t})\), the function g will satisfy (5.9).

  4. That is, a point on the tree corresponding to one individual dying, and being replaced with a strictly positive number of offspring.

  5. For example, if w is the oldest younger sibling of \(v_t\) (so \(v_t=uj\) and \(w=u(j+1)\) for some \(j\ge 1\)) then \(\sigma (w)\) would be \((t-s)\) minus the minimum of \((t-s)\) and the total length of the subtree rooted at \(v_s\).

  6. In fact, since \((\bar{S}_t^n)_t\) may have jumps, we also need to verify an extra condition in order to apply the functional central limit theorem. However this condition, see [32, § VIII, Eq.(3.23)], is simply that we have, for every \(t>0\),

    $$\begin{aligned} \frac{\lambda }{m-1}\int _0^{t} \varphi (V_{s-}(w))^2\mathbb {E}\big ((A-1)^2 \mathbb {1}_{\{(A-1)^2 \varphi (V_{s-}(w))^2>n\varepsilon \}}\big ) \, ds \rightarrow 0 \end{aligned}$$

    almost surely as \(n\rightarrow \infty \) (where the expectation \(\mathbb {E}\) is only over A). Since A has finite variance, this does indeed hold.

  7. Interpreting \(\phi (D_t^H)=\phi (((D_t^H)_{ij})_{1\le i<j\le k})\).

References

  1. Abraham, R., Delmas, J.-F., Hoscheit, P.: A note on the Gromov–Hausdorff–Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Probab. 18(14), 21 (2013)

    MathSciNet  MATH  Google Scholar 

  2. Aldous, D.: The continuum random tree. I. Ann. Probab. 19(1), 1–28 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Aldous, D.: The continuum random tree. III. Ann. Probab. 21(1), 248–289 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  4. Asmussen, S., Hering, H.: Branching Processes, Volume 3 of Progress in Probability and Statistics. Birkhäuser Boston Inc, Boston (1983)

    Book  MATH  Google Scholar 

  5. Bañuelos, R.: Intrinsic ultracontractivity and eigenfunction estimates for Schrödinger operators. J. Funct. Anal. 100(1), 181–206 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berestycki, J., Berestycki, N., Schweinsberg, J.: Survival of near-critical branching Brownian motion. J. Stat. Phys. 143(5), 833–854 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berestycki, J., Berestycki, N., Schweinsberg, J.: The genealogy of branching Brownian motion with absorption. Ann. Probab. 41(2), 527–618 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berestycki, J., Berestycki, N., Schweinsberg, J.: Critical branching Brownian motion with absorption: survival probability. Probab. Theory Relat. Fields 160(3–4), 489–520 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Berestycki, J., Berestycki, N., Schweinsberg, J.: Critical branching Brownian motion with absorption: particle configurations. Ann. Inst. Henri Poincaré Probab. Stat. 51(4), 1215–1250 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Berestycki, N.: Recent Progress in Coalescent Theory, Volume 16 of Ensaios Matemáticos [Mathematical Surveys]. Sociedade Brasileira de Matemática, Rio de Janeiro (2009)

    Google Scholar 

  11. Biggins, J.D.: Martingale convergence in the branching random walk. J. Appl. Probab. 14(1), 25–37 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bolthausen, E., Sznitman, A.-S.: On Ruelle’s probability cascades and an abstract cavity method. Commun. Math. Phys. 197(2), 247–276 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brunet, É., Derrida, B., Mueller, A.H., Munier, S.: Noisy traveling waves: effect of selection on genealogies. Europhys. Lett. 76(1), 1–7 (2006)

    Article  MathSciNet  Google Scholar 

  14. Brunet, É., Derrida, B., Mueller, A.H., Munier, S.: Effect of selection on ancestry: an exactly soluble case and its phenomenological generalization. Phys. Rev. E (3) 76(4), 041104, 20 (2007)

    Article  MathSciNet  Google Scholar 

  15. Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Volume 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence (2001)

    MATH  Google Scholar 

  16. Chauvin, B., Rouault, A.: KPP equation and supercritical branching Brownian motion in the subcritical speed area. Application to spatial trees. Probab. Theory Relat. Fields 80(2), 299–314 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  17. Davies, E.B.: Heat Kernels and Spectral Theory, Volume 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  18. Davies, E.B., Simon, B.: Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians. J. Funct. Anal. 59(2), 335–395 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duquesne, T., Le Gall, J.-F.: Random trees, Lévy processes and spatial branching processes. Astérisque 281, vi+147 (2002)

    MATH  Google Scholar 

  20. Engländer, J., Kyprianou, A.E.: Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32(1A), 78–99 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Evans, L.C.: Partial Differential Equations, Volume 19 of Graduate Studies in Mathematics, 2nd edn. American Mathematical Society, Providence (2010)

    Google Scholar 

  22. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, volume 224 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 2nd edn. Springer, Berlin (1983)

    Book  Google Scholar 

  23. Greven, A., Pfaffelhuber, P., Winter, A.: Convergence in distribution of random metric measure spaces (\(\Lambda \)-coalescent measure trees). Probab. Theory Relat. Fields 145(1–2), 285–322 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hardy, R., Harris, S.C.: A spine approach to branching diffusions with applications to \(L^p\)-convergence of martingales. In: Séminaire de Probabilités XLII, volume 1979 of Lecture Notes in Math., pp 281–330. Springer, Berlin (2009)

  25. Harris, J.W., Harris, S.C.: Survival probabilities for branching Brownian motion with absorption. Electron. Commun. Probab. 12, 81–92 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  26. Harris, S.C., Hesse, M., Kyprianou, A.E.: Branching Brownian motion in a strip: survival near criticality. Ann. Probab. 44(1), 235–275 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  27. Harris, S.C., Roberts, M.I.: A strong law of large numbers for branching processes: almost sure spine events. Electron. Commun. Probab. 19(28), 6 (2014)

    MathSciNet  MATH  Google Scholar 

  28. Harris, S.C., Roberts, M.I.: The many-to-few lemma and multiple spines. Ann. Inst. Henri Poincaré Probab. Stat. 53(1), 226–242 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hering, H.: Limit theorem for critical branching diffusion processes with absorbing barriers. Math. Biosci. 19, 355–370 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  30. Hering, H.: Multigroup branching diffusions. In: Branching processes Conf., Saint Hippolyte, Que., 1976), volume 5 of Advanced Probab. Related Topics, pp. 177–217. Dekker, New York (1978)

  31. Hering, H.: Uniform primitivity of semigroups generated by perturbed elliptic differential operators. Math. Proc. Camb. Philos. Soc. 83(2), 261–268 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  32. Jacod, J., Shiryaev, A.N.: Limit Theorems for stochastic processes. volume 288 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1987)

    Google Scholar 

  33. Kesten, H.: Branching Brownian motion with absorption. Stoch. Processes Appl. 7(1), 9–47 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  34. Kolmogorov, A .N.: Zur Lösung einer biologischen Aufgabe. Izv. Nauč. Issled. Inst. Mat. i Meh. Tomsk. Gosudarstv. Univ. 2, 1–6 (1938)

    MATH  Google Scholar 

  35. Kyprianou, A.E.: Travelling wave solutions to the K-P-P equation: alternatives to Simon Harris’ probabilistic analysis. Ann. Inst. Henri Poincaré Probab. Stat. 40(1), 53–72 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  36. Le Gall, J.-F.: Random trees and applications. Probab. Surv. 2, 245–311 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lyons, R.: A simple path to Biggins’ martingale convergence for branching random walk. In: Classical and modern branching processes (Minneapolis, MN, 1994), volume 84 of IMA Vol. Math. Appl., pp. 217–221. Springer, New York (1997)

  38. McKean, H.P.: Application of Brownian motion to the equation of Kolmogorov–Petrovskii–Piskunov. Comm. Pure Appl. Math. 28(3), 323–331 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  39. Miermont, G.: Invariance principles for spatial multitype Galton–Watson trees. Ann. Inst. Henri Poincaré Probab. Stat. 44(6), 1128–1161 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  40. Miermont, G.: Tessellations of random maps of arbitrary genus. Ann. Sci. Éc. Norm. Supér. (4) 42(5), 725–781 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  41. Pinsky, R.G.: On the convergence of diffusion processes conditioned to remain in a bounded region for large time to limiting positive recurrent diffusion processes. Ann. Probab. 13(2), 363–378 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  42. Revuz, D., Yor, M.: Continuous Martingales and Brownian motion, volume 293 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 3rd edn. Springer, Berlin (1999)

    Google Scholar 

  43. Roberts, M. I.: Spine changes of measure and branching diffusions. PhD thesis, University of Bath. http://people.bath.ac.uk/mir20/thesis.pdf (2010)

  44. Rudin, W.: Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics, 3rd edn. McGraw-Hill Book Co., New York-Auckland-Düsseldorf (1976)

    MATH  Google Scholar 

  45. Sevast’yanov, B.A.: Branching stochastic processes for particles diffusing in a bounded domain with absorbing boundaries. Teor. Veroyatnost. i Primenen. 3, 121–136 (1958)

    MathSciNet  MATH  Google Scholar 

  46. Watanabe, S.: On the branching process for Brownian particles with an absorbing boundary. J. Math. Kyoto Univ. 4, 385–398 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

I would particularly like to thank Nathanaël Berestycki, for suggesting this problem and for many helpful discussions. I am also grateful to the anonymous referee for numerous useful comments and suggestions.

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  1. Eidgenossische Technische Hochschule Zurich, Zurich, Switzerland

    Ellen Powell

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Powell, E. An invariance principle for branching diffusions in bounded domains. Probab. Theory Relat. Fields 173, 999–1062 (2019). https://doi.org/10.1007/s00440-018-0847-8

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