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Contents of PMS, Vol. 42, Fasc. 2,
pages 319 - 356
DOI: 10.37190/0208-4147.00069
Published online 6.12.2022
 

Quenched asymptotics for symmetric L\'evy processes interacting with Poissonian fields

Z.-H. Chen
J. Wang

Abstract:

We establish the quenched large time asymptotics for the Feynman-Kac functional \[E_x\left[\exp\left(-\int_0^tV^w(Z_s)\,ds\right)\right]\] associated with a pure-jump symmetric Lévy process \((Z_t)_{t\ge0}\) in general Poissonian random potentials \(V^w\) on \(R^d\), which is closely related to the large time asymptotic behavior of solutions to the nonlocal parabolic Anderson problem with Poissonian interaction. In particular, when the density function with respect to the Lebesgue measure of the associated Lévy measure is given by \[\rho(z)= \frac{1}{|z|^{d+\alpha}}I_{\{|z|\le 1\}}+ e^{-c|z|^\theta}I_{\{|z|> 1\}}\] for some \(\alpha\in (0,2)\), \(\theta\in (0,\infty]\) and \(c>0\), an explicit quenched asymptotics is derived for potentials with the shape function given by \(\varphi(x)=1\wedge |x|^{-d-\beta}\) for \(\beta\in (0,\infty]\) with \(\beta\neq 2\), and it is completely different for \(\beta>2\) and \(\beta<2\). We also discuss the quenched asymptotics in the critical case (e.g.,  \(\beta=2\) in the example above). The work fills the gaps of the related work for pure-jump symmetric Lévy processes in Poissonian potentials, where only the case that the shape function is compactly supported (e.g., \(\beta=\infty\) in the example above) has been handled in the literature.

2010 AMS Mathematics Subject Classification: Primary 60G52; Secondary 60J25, 60J55, 60J35, 60J75.

Keywords and phrases: symmetric Lévy process, Poissonian potential, quenched asymptotic, nonlocal parabolic Anderson problem, spectral theory.

R. A. Carmona and S. A. Molchanov, Stationary parabolic Anderson models and intermittency, Probab. Theory Related Fields 102 (1995), 433-453.

L. Chaumont and G. Uribe Bravo, Markovian bridges: Weak continuity and pathwise constructions, Ann. Probab. 39 (2011), 609-647.

X. Chen, Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related parabolic Anderson models, Ann. Probab. 40 (2012), 1436-1482.

Z.-Q. Chen and T. Kumagai, Heat kernel estimates for stable-like processes on d-sets, Stoch. Process. Appl. 108 (2003), 27-62.

Z.-Q. Chen, P. Kim and T. Kumagai, Global heat kernel estimates for symmetric jump processes, Trans. Amer. Math. Soc. 363 (2011), 5021-5055.

Z.-Q. Chen, P. Kim and R. Song, Heat kernel estimates for Dirichlet fractional Laplacian, J. Eur. Math. Soc. 124 (2010), 1307-1329.

M. D. Donsker and S. R. S. Varadhan, Asymptotics for the Wiener sausage, Comm. Pure Appl. Math. 28 (1975), 525-565.

R. Fukushima, From the Lifshitz tail to the quenched survival asymptotics in the trapping problem, Electron. Comm. Probab. 14 (2009), 435-446.

R. Fukushima, Second order asymptotics for Brownian motion in a heavy tailed Poissonian potential, Markov Process. Related Fields 17 (2011), 447-482.

J. P. Gärtner, W. König and S. A. Molchanov, Almost sure asymptotics for the continuous parabolic Anderson model, Probab. Theory Related Fields 118 (2000), 547-573.

K. Kaleta and K. Pietruska-Pałuba, The quenched asymptotics of nonlocal Schrödinger operators with Poissonian potentials, Potential Anal. 52 (2020), 161-202.

Y. Kasahara, Tauberian theorems of exponential type, J. Math. Kyoto Univ. 18 (1978), 209-219.

K.-Y. Kim and P. Kim, Two-sided estimates for the transition densities of symmetric Markov processes dominated by stable-like processes in C1,η open sets, Stoch. Process. Appl. 124 (2014), 3055-3083.

W. König, The Parabolic Anderson Model: Random Walk in Random Potential, Birkhäuser, Cham, 2016.

V. Knopova and R. L. Schilling, A note on the existence of transition probability densities for Lévy processes, Forum Math. 25 (2013), 125-149.

H. Ôkura, On the spectral distributions of certain integro-differential operators with random potential, Osaka J. Math. 16 (1979), 633-666.

H. Ôkura, Some limit theorems of Donsker–Varadhan type for Markov processes expectations, Z. Wahrsch. Verw. Gebiete 57 (1981), 419-440.

H. Ôkura, An asymptotic property of a certain Brownian motion expectataion for large time, Proc. Japan Acad. Ser. A. Math. Sci. 57 (1981), 155-159.

L. A. Pastur, The behavior of certain Wiener integrals as t → ∞ and the density of states of Schrödinger equations with random potential, Teoret. Mat. Fiz. 32 (1977), 88-95 (in Russian).

A.-S. Sznitman, Brownian asymptotics in a Poisson environment, Probab. Theory Related Fields 95 (1993), 155-174.

A.-S. Sznitman, Brownian Motion, Obstacles and Random Media, Springer, Berlin, 1998.

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