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Đối xứng của quá trình Brownian với tiềm năng đã chuẩn hóa trong môi trường ngẫu nhiên di động
Tóm tắt
Được thúc đẩy bởi việc nghiên cứu mô hình polymer có hướng với các bẫy hoặc chất xúc tác Poisson di động và mô hình Anderson parabol ngẫu nhiên với tiềm năng phụ thuộc theo thời gian, chúng tôi nghiên cứu hành vi tiệm cận của \( \mathbb {E}\otimes \mathbb {E}_0 \exp \left\{ \pm \ \theta \int \limits ^t_0 \bar{V}(s,B_s)\hbox {d}s\right\} \quad (t\rightarrow \infty ) \), trong đó \( \theta >0 \) là một hằng số, \( \overline{V} \) là tiềm năng Poisson đã chuẩn hóa với dạng \( \overline{V}(s,x)=\int \limits _{\mathbb {R}^d}\frac{1}{|y-x|^p}\left( \omega _s(\hbox {d}y)-\hbox {d}y\right) , \) và \( \omega _s \) là quá trình giá trị đo lường bao gồm các hạt Brownian độc lập có vị trí ban đầu tạo thành một phép đo ngẫu nhiên Poisson trên \( \mathbb {R}^d \) với độ đo Lebesgue là cường độ của nó. Các giới hạn tỉ lệ khác nhau được thu được theo tham số \( p \) và chiều \( d \). Đối với logarith của mô men mũ âm, khoảng \( \frac{d}{2}2 \)), các mô men mũ trở thành vô hạn cho mọi \( t>0 \).
Từ khóa
Tài liệu tham khảo
Bass, R., Chen, X., Rosen, J.: Large deviations for Riesz potentials of additive processes. Annales de l’Institut Henri Poincare 45, 626–666 (2009)
van den Berg, M., Bolthausen, E., den Hollander, F.: Brownian survival among Poissonian traps with random shapes at critical intensity. Probab. Theory Relat. Fields 132, 163–202 (2005)
Bezerra, S., Tindel, S., Viens, F.: Superdiffusivity for a Brownian polymer in a continuous Gaussian environment. Ann. Probab. 36, 1642–1675 (2008)
Borkar, V.S.: Probability Theory: An Advanced Course. Springer, New York (1995)
Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities in dimfusion-dominated annihilation reactions. Phys. Rev. Lett. 61, 2397 (1988)
Bramson, M., Lebowitz, J.L.: Asymptotic behavior of densities for two-particle annihilating random walks. J. Stat. Phys. 62, 297–372 (1991)
Carmona, R., Molchanov, S.: Parabolic Anderson Problem and Intermittency. American Mathematical Society, Providence (1994)
Carmona, R., Viens, F.G.: Almost-sure exponential behavior of a stochastic Anderson model with continuous space parameter. Stoch. Rep. 62, 251–273 (1998)
Chen, X.: Random Walk Intersections: Large Deviations and Related Topics. Mathematical Surveys and Monographs, 157. American Mathematical Society, Providence (2009)
Chen, X.: Quenched asymptotics for Brownian motion of renormalized Poisson potential and for the related Anderson models. Ann. Probab. 40(4), 1436–1482 (2012)
Chen, X., Kulik, A.M.: Brownian motion and parabolic Anderson model in a renormalized Poisson potential. Ann. Inst. Henri Poincaré Probab. Stat. 48(3), 631–660 (2012)
Chen, X., Kulik, A.M.: Asymptotics of negative exponential moments for annealed Brownian motion in a renormalized Poisson potential. Int. J. Stoch. Anal., Art. ID 803683 (2011)
Chen, X., Rosinski, J.: Spatial Brownian motion in renormalized Poisson potential: a critical case. (preprint) (2013)
Dalang, R.C., Mueller, C.: Intermittency properties in a hyperbolic Anderson problem. Annales de l’Institut Henri Poincare 45, 1150–1164 (2009)
Donoghue, W.: Distributions and Fourier Transforms. Academic Press, New York (1969)
Donsker, M.D., Varadhan, S.R.S.: Asymptotics for the Wiener sausage. Commun. Pure Appl. Math. 28(4), 525–565 (1975)
Drewitz, A., Sousi, P., Sun, R.: Symmetric rearrangements around infinity with applications to Lévy processes. Probab. Theory Relat. Fields 158(3–4), 637–664 (2014)
Drewitz, A., Gärtner, J., Ramírez, A., Sun, R.: Survival probability of a random walk among a Poisson system of moving traps. In: Probability in Complex Physical Systems-In Honour of Erwin Bolthausen and Jürgen Gärtner, pp. 119–158. Springer Proceedings in Mathematics 11 (2012)
Durrett, R.: Stochastic Calculus: A Pratical Introduction. CRC Press, Boca Raton (1996)
Florescu, I., Viens, F.: Sharp estimation of the almost-sure Lyapunov exponent for the Anderson model in continuous space. Probab. Theory Relat. Fields 135, 603–644 (2006)
Fukushima, R.: Second order asymptotics for Brownian motion among a heavy tailed Poissonian potential. Markov Processes and Related Fields (to appear)
Gärtner, J., den Hollander, F.: Intermittency in a catalytic random medium. Ann. Probab. 34, 2219–2287 (2006)
Gärtner, J., den Hollander, F., Mailland, G.: Intermittency on catalysts. Trends in Stochastic analysis, London Math. Soc. Lecture Note Ser., vol. 353, pp. 235–248. Cambridge University Press, Cambridge (2009)
Gärtner, J., den Hollander, F., Mailland, G.: Intermittency of catalysts: voter model. Ann. Probab. 38, 2066–2102 (2010)
Gärtner, J., den Hollander, F., Molchanov, S.A.: Diffusion in an annihilating environment. Nonlinear Anal. 7, 25–64 (2006)
Gärtner, J., König, W.: Moment asymptotics for the continuous parabolic Anderson model. Ann. Appl. Probab. 10, 192–217 (2000)
Gärtner, J., König, W., Molchanov, S.A.: Almost sure asymptotics for the continuous parabolic Anderson model. Probab. Theory Relat. Fields 118, 547–573 (2000)
Germinet, F., Hislop, P., Klein, A.: Localization for Schrödinger operators with Poisson random potential. J. Eur. Math. Soc. 9, 577–607 (2007)
Hardy, G., Pólya, G., Littlewood, J.E.: Inequalities, 2nd edn. Cambridge University Press, Cambridge (1952)
Komorowski, T.: Brownian motion in a Poisson obstacle field. Séminaire Bourbaki 1998/99, 91–111 (2000)
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Pascal principle for diffusion-controlled trapping reactions. Phys. Rev. E 67, 045104(R) (2003)
Moreau, M., Oshanin, G., Bénichou, O., Coppey, M.: Lattice theory of trapping reactions with mobile species. Phys. Rev. E 69, 046101 (2004)
Opic, B., Kufner, A.: Hardy-Type Inequalities. Pitman Research Notes in Math. 219, Longman (1990)
Peres, Y., Sinclair, A., Sousi, P., Stauffer, A.: Mobile geometric graphs: detection, coverage and percolation. In: Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 412–428. SIAM, Philadelphia, PA (2011)
Peres, Y., Sousi, P.: An isoperimetric inequality for the Wiener sausage. Geom. Funct. Anal. 22(4), 1000–1014 (2012)
Povel, T.: Confinement of Brownian motion among Poissonian obstacles in \(\mathbb{R}^d, d\ge 3\). Probab. Theory Relat. Fields 114, 177–205 (1999)
Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Stolz, G.: Non-monotonic random Schrödinger operators: the Anderson model. J. Math. Anal. Appl. 248(1), 173–183 (2000)
Sznitman, A.-L.: Brownian Motion, Obstacles and Random Media. Springer, Berlin (1998)
van den Berg, J., Meester, R., White, D.G.: Dynamic Boolean models. Stoch. Process. Appl. 69, 247–257 (1997)
Yakimiv, A.L.: Probabilistic Applications of Tauberian Theorems. Translated from the Russian original by Andrei V. Kolchin. Modern Probability and Statistics. VSP, Leiden (2005)