Abstract
In this work we study the long-time behavior for subcritical measure-valued branching processes with immigration on the space of tempered measures. Under some reasonable assumptions on the spatial motion, the branching and immigration mechanisms, we prove the existence and uniqueness of an invariant probability measure for the corresponding Markov transition semigroup. Moreover, we show that it converges with exponential rate to the unique invariant measure in the Wasserstein distance as well as in a distance defined in terms of Laplace transforms. Finally, we consider an application of our results to super-Lévy processes as well as branching particle systems on the lattice with noncompact spins.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Barczy, M., Li, Z., Pap, G.: Stochastic differential equation with jumps for multi-type continuous state and continuous time branching processes with immigration. ALEA Lat. Am. J. Probab. Math. Stat. 12(1), 129–169 (2015). MR3340375
Bezborodov, V., Kondratiev, Y., Kutoviy, O: Lattice birth-and-death processes. Moscow Mathematical Journal 19(1), 7–36 (2019)
Bramson, M., Cox, J.T., Greven, A.: Ergodicity of critical spatial branching processes in low dimensions. Ann. Probab. 21(4), 1946–1957 (1993). MR1245296
Bramson, M., Cox, J.T., Greven, A.: Invariant measures of critical spatial branching processes in high dimensions, vol. 25. MR1428499 (1997)
Chen, Y.T.: Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration. Ann. Probab. 43(6), 3359–3467 (2015). MR3433584
Dawson, D.A.: The critical measure diffusion process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 40(2), 125–145 (1977). MR0478374
Dawson, D.A., Li, Z.: Stochastic equations, flows and measure-valued processes. Ann. Probab. 40(2), 813–857 (2012). MR2952093
Dynkin, E.B.: An introduction to branching measure-valued processes, CRM Monograph Series, vol. 6, American Mathematical Society, Providence, RI. MR1280712 (1994)
Engel, K., Nagel, R.: One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics, vol. 194. Springer-Verlag, New York (2000). With contributions by Brendle, S., Campiti, M., Hahn, T., Metafune, G., Nickel, G., Pallara, D., Perazzoli, C., Rhandi, A., Romanelli, S., Schnaubelt, R., MR1721989
Alison M.: Etheridge, Asymptotic behaviour of measure-valued critical branching processes. Proc. Amer. Math. Soc. 118(4), 1251–1261 (1993). MR1100650
Alison M.: An introduction to superprocesses, University Lecture Series, vol. 20, American Mathematical Society, Providence, RI. MR1779100 (2000)
Ethier, S.N., Griffiths, R.C.: The transition function of a measure-valued branching diffusion with immigration, Stochastic processes, pp 71–79. Springer, New York (1993). MR1427302
Ethier, Stewart, Kurtz, Thomas G.: Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiles, New York (1986). Characterization and convergence
Feller, W.: Diffusion processes in genetics. In: Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability, 1950, University of California Press, Berkeley and Los Angeles. MR0046022, pp 227–246 (1951)
Friesen, M., Jin, P., Jonas, K., Rüdiger, B.: Exponential ergodicity for stochastic equations of nonnegative processes with jumps. arXiv:1902.02833 (2019)
Friesen, M., Jin, P., Rüdiger, B.: Stochastic equation and exponential ergodicity in Wasserstein distances for affine processes. Ann. Appl. Probab. 30(5), 2165–2195 (2020). MR4149525
Friesen, M., Kondratiev, Y.: Stochastic averaging principle for spatial birth-and-death evolutions in the continuum. J. Stat. Phys. 171(5), 842–877 (2018). MR3800897
Hammer, M., Höpfner, R., Berg, T.: Ergodic branching diffusions with immigration:, properties of invariant occupation measure, identification of particles under high-frequency observation, and estimation of the diffusion coefficient at nonparametric rates, arXiv:1905.02656 (2019)
He, H., Li, Z., Yang, X.: Stochastic equations of super-Lèvy processes with general branching mechanism. Stochastic Process. Appl. 124(4), 1519–1565 (2014). MR3163212
Henry-Labordère, P., Tan, X., Touzi, N.: A numerical algorithm for a class of BSDEs via the branching process. Stochastic Process. Appl. 124(2), 1112–1140 (2014). MR3138609
Höpfner, R., Löcherbach, E.: Remarks on ergodicity and invariant occupation measure in branching diffusions with immigration. Ann. Inst. H. Poincaré Probab. Statist. 41(6), 1025–1047 (2005). MR2172208
Iscoe, I.: A weighted occupation time for a class of measure-valued branching processes. Probab. Theory Relat. Fields 71(1), 85–116 (1986). MR814663
Jin, P., Kremer, J., Rüdiger, B.: Existence of limiting distribution for affine processes. J. Math. Anal. Appl. 486(2), 123912 (2020). 31. MR4060087
Jiřina, M.: Stochastic branching processes with continuous state space. Czechoslovak Math. J. 8(83), 292–313 (1958). MR0101554
Kawazu, K., Watanabe, S.: Branching processes with immigration and related limit theorems. Teor. Verojatnost. i Primenen. 16, 34–51 (1971). MR0290475
Keller-Ressel, M., Mijatović, A.: On the limit distributions of continuous-state branching processes with immigration. Stochastic Process. Appl. 122(6), 2329–2345 (2012). MR2922631
Kondratiev, Y., Kutoviy, O., Pirogov, S. : Correlation functions and invariant measures in continuous contact model. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 11(2), 231–258 (2008). MR2426716
Kondratiev, Y., Pirogov, S., Zhizhina, E.: A quasispecies continuous contact model in a critical regime. J. Stat. Phys. 163(2), 357–373 (2016). MR3478314
Konno, N., Shiga, T.: Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79(2), 201–225 (1988). MR958288
Kyprianou, A.E., Palau, S.: Extinction properties of multi-type continuous-state branching processes. Stochastic Process. Appl. 128(10), 3466–3489 (2018). MR3849816
Le Gall, J.F.: Spatial branching processes, random snakes and partial differential equations, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel (1999). MR1714707
Li, Z.: Ergodicities and exponential ergodicities of Dawson-Watanabe type processes. Teor. Veroyatn. Primen. 66(2), 342–368 (2021). MR4252929
Li, Z.: Measure-valued branching Markov processes, Probability and its Applications (New York). Springer, Heidelberg (2011). MR2760602
Li, Z., Ma, C.: Asymptotic properties of estimators in a stable Cox-Ingersoll-Ross model. Stochastic Process. Appl. 125(8), 3196–3233 (2015). MR3343292
Mayerhofer, E., Stelzer, R., Vestweber, J.: Geometric ergodicity of affine processes on cones. Stochastic Process. Appl. 130(7), 4141–4173 (2020). https://doi.org/10.1016/j.spa.2019.11.012
Mytnik, L., Xiong, J.: Well-posedness of the martingale problem for superprocess with interaction. Illinois J. Math. 59(2), 485–497 (2015). MR3499521
Perkins, E., Watanabe, D.: superprocesses and measure-valued diffusions, Lectures on probability theory and statistics (Saint-Flour, 1999), Lecture Notes in Math. 1781, Springer, Berlin, pp. 125–324. MR1915445 (2002)
Reimers, M.: One-dimensional stochastic partial differential equations and the branching measure diffusion. Probab. Theory Related Fields 81(3), 319–340 (1989). MR983088
Shiga, T.: A stochastic equation based on a Poisson system for a class of measure-valued diffusion processes. J. Math. Kyoto Univ. 30(2), 245–279 (1990). MR1068791
Silverstein, M.L.: Continuous state branching semigroups. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 14, 96–112 (1969/1970). MR0266321
Stannat, W.: On transition semigroups of (A,Ψ)-superprocesses with immigration. Ann. Probab. 31(3), 1377–1412 (2003). MR1989437
Stannat, W.: Spectral properties for a class of continuous state branching processes with immigration. J. Funct. Anal. 201(1), 185–227 (2003). MR1986159
Villani, C.: Optimal transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin (2009). Old and new. MR2459454
Watanabe, S.: A limit theorem of branching processes and continuous state branching processes. J. Math. Kyoto Univ. 8, 141–167 (1968). MR0237008
Xiong, J.: Super-Brownian motion as the unique strong solution to an SPDE. Ann. Probab. 41(2), 1030–1054 (2013). MR3077534
Zhang, X., Glynn, P.W. : Affine Jump-Diffusions: Stochastic Stability and Limit Theorems, arXiv:1811.00122 (2018)
Funding
Open Access funding provided by the IReL Consortium.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix : A
Appendix : A
1.1 A.1 Proof of Theorem 2.3
Proof
Assertion (a) is a particular case of [33, Theorem 6.3]. For later use we sketch the most important step in the proof. Define a new transition semigroup \(\widetilde {p}_{t}^{\xi }(x,dy) = e^{-\alpha t} \frac {h(y)}{h(x)}p_{t}^{\xi }(x,dy)\) and branching mechanism \(\widetilde {\phi }(x,f) = h(x)^{-1}\phi (x,hf) - \alpha f(x)\). Then it was shown that there exists a unique locally bounded solution ut(x,f) to
This solution defines a cumulant semigroup Utf(x) = ut(x,f), i.e.
where \(\widetilde {\lambda }_{t}\) and \(\widetilde {L}_{t}\) are bounded kernels for each t ≥ 0. Finally it was shown that
is the desired solution to Eq. 2.6, i.e. assertion (a) is proved. By uniqueness we find that (Vt)t≥ 0 is a nonlinear semigroup. Invoking (A.7) and Eq. A.8 we find that
where . This proves assertion (b). Assertion (c) can be obtained as follows. Define and let ψn be given by Eq. 2.4 with H2 replaced by \(H_{2}^{(n)}\). By [33, Proposition 9.17] there exists a unique Markov kernel \(P^{(n)}_{t}(\mu ,d\nu )\) satisfying (2.8) with ψ replaced by ψn. Taking the limit \(n \to \infty \) and using Proposition 2.1 proves the existence of Pt(μ,dν) satisfying (2.8). Since Vs+t = VsVt, it is not difficult to see that Pt(μ,dν) is a Markov kernel. □
1.2 A.2 Proof of Lemma 3.1
Proof
Let (Ut)t≥ 0 be the cumulant semigroup given by Eq. A.7 and let \((\widetilde {R}_{t})_{t \geq 0}\) be the semigroup on obtained from
where \(\widetilde {p}_{t}^{\xi }(x,dy) = e^{-\alpha t} \frac {h(y)}{h(x)}p_{t}^{\xi }(x,dy)\), \(\widetilde {b}(x) = b(x) - \alpha \) and
It was shown in [33, Proposition 2.18 and 2.24] that \(U_{t}f \leq \widetilde {R}_{t}f\) and \(\frac {1}{\varepsilon }U_{t}(\varepsilon f)(x) \nearrow \widetilde {R}_{t}f(x)\) as ε ↘ 0 for each .
Observe that \(h(x) \widetilde {R}_{t}(h^{-1}f)(x)\) defines a semigroup of bounded linear operators on Bh(E)+ which satisfies (3.1). Since the operator Bf(x) = Γf(x) − b(x)f(x) is a bounded linear operator on Bh(E), it follows that Eq. 3.1 has a unique solution, i.e. we get \(R_{t}f(x) = h(x)\widetilde {R}_{t}(h^{-1}f)(x)\). Using representation (2.7) we easily find that
Hence using Eq. A.8 and this limit one finds Eq. 3.3. □
1.3 A.3 Proof of Proposition 3.2
Fix μ ∈ Mh(E) and let f ∈ Bh(E)+. By linearity and definition of the adjoint semigroup \((R_{t}^{*})_{t \geq 0}\) it suffices to prove for each t ≥ 0
We start with an auxiliary result.
Lemma A.1
For each f ∈ Bh(E)+ and t ≥ 0 one has
Proof
Write
Next using \(\frac {V_{s}(\varepsilon f)}{\varepsilon } \nearrow R_{s}f\) pointwise and
with
the assertion follows by dominated convergence. □
Based on this observation we complete the proof of Proposition 3.2.
Proof Proof of Proposition 3.2
Fix f ∈ Bh(E)+ and take ε > 0. Applying first monotone convergence and then Eq. 2.8 gives
This proves the assertion. □
1.4 A.4 Some Results on Distances on \(\mathcal {P}(M_{h}(E))\)
Lemma A.2
Let \(\rho , \widetilde {\rho } \in \mathcal {P}(M_{h}(E))\), \(H \in {\mathscr{H}}(\rho , \widetilde {\rho })\), and recall that Pt(μ,dν) denotes the transition kernel given by Theorem 2.3 while \(P_{t}^{*}\) was defined in Eq. 5.1. Then
Proof
Take f ∈ Bh(E)+. Using the definition of \(P_{t}^{*}\) and the fact that H is a coupling of \((\rho , \widetilde {\rho })\) gives
Since f was arbitrary, the assertion is proved. □
Lemma A.3
Let \(\rho , \widetilde {\rho }, \thinspace g \in \mathcal {P}(M_{h}(E))\). Then \(W_{1}(\rho \ast g,\widetilde {\rho }\ast g)\leq W_{1}(\rho ,\widetilde {\rho })\).
Proof
For F : Mh(E)→ℝ define \(\Vert F \Vert _{\text {Lip}} = \sup _{\mu \neq \widetilde {\mu }} \frac {|F(\mu ) - F(\widetilde {\mu })|}{\| \mu - \widetilde {\mu }\|_{TV,h}}\). Using the Kantorovich-Duality we obtain
where we used that \(F_{g}(\mu )={\int \limits }_{M_{h}(E)} F(\mu + \widetilde {\mu })g(d\widetilde {\mu })\) satisfies ∥Fg∥Lip ≤ 1. □
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Friesen, M. Long-Time Behavior for Subcritical Measure-Valued Branching Processes with Immigration. Potential Anal 59, 705–730 (2023). https://doi.org/10.1007/s11118-021-09983-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11118-021-09983-4
Keywords
- Dawson-Watanabe superprocess
- Measure-valued Markov process
- Branching
- Ergodicity
- Invariant measure
- Immigration