## Large deviation of diffusion processes with discontinuous drift and their occupation times.(English)Zbl 1044.60065

Summary: For the system of $$d$$-dim stochastic differential equations, $dX^\varepsilon(t) =b\bigl(X^\varepsilon(t)\bigr)d t+\varepsilon dW(t),\;t\in[0,1],\quad X^\varepsilon(0) =x^0\in \mathbb R^d,$ where $$b$$ is smooth except possibly along the hyperplane $$x_1=0$$, we shall consider the large deviation principle for the law of the solution diffusion process and its occupation time as $$\varepsilon\to 0$$. In other words, we consider $$P(\| X^\varepsilon- \varphi\|<\delta$$, $$\| u^\varepsilon- \psi\|<\delta)$$ where $$u^\varepsilon (t)$$ and $$\psi(t)$$ are the occupation times of $$X^\varepsilon$$ and $$\varphi$$ in the positive half space $$\{x\in \mathbb R^d:x_1>0\}$$, respectively. As a consequence, a unified approach of the lower level large deviation principle for the law of $$X^\varepsilon(\cdot)$$ can be obtained.

### MSC:

 60J60 Diffusion processes 60F10 Large deviations
Full Text:

### References:

 [1] Alanyali, M. and Hajek, B. (1998). On large deviations of Markov processes with discontinuous statistics. Ann. Appl. Probab. 8 45-66. · Zbl 0936.60021 [2] Alanyali, M. and Hajek, B. (1998). On large deviations in load sharing network. Ann. Appl. Probab. 8 67-97. · Zbl 0938.60097 [3] Azencott, R. (1980). Grandes Deviations et Applications. Probabilités de Saint-Flour VIII. Lecture Notes in Math. 774 1-176. Springer, Berlin. · Zbl 0435.60028 [4] Azencott, R. and Ruget, G. (1977). Melanges d’ équations differentialles et grands écarts a la loi des grand nombres.Wahrsch. Verw. Gebiete 38 1-54. · Zbl 0372.60082 [5] Blinovskii, V. M. and Dobrushin, R. L. (1994). Process level large deviations for a class of piecewise homogeneous random walks. In The Dynkin Festschrift: Markov Processes and Their Applications (M. I. Friedlin, ed.) 1-59. Birkhäuser, Boston. · Zbl 0819.60029 [6] Boué, M., Dupuis, P. and Ellis, R. (1997). Large deviations for diffusions with discontinuous statistics. · Zbl 0949.60046 [7] Chiang, T. S. and Sheu, S. J. (1997). Large deviations of small perturbation of some unstable systems. Stochastic Anal. Appl. 15 31-50. · Zbl 0873.60013 [8] Dembo, A. and Zeitouni, O. (1992). Large Deviations Techniques and Applications. Jones and Bartlett, Boston. · Zbl 0793.60030 [9] Dupuis, P. and Ellis, R. (1992). Large deviations for Markov process with discontinuous statistics II: Random walks. Probab. Theory Related. Fields 91 153-194. · Zbl 0746.60025 [10] Dupuis, P. and Ellis, R. (1995). The large deviation principle for a general class of queueing systems I. Trans. Amer. Math. Soc. 347 2689-2751. JSTOR: · Zbl 0869.60022 [11] Dupuis, P. and Ellis, R. (1997). A Weak Convergence Approach to the Theory of Large Deviations. Wiley, NewYork. · Zbl 0904.60001 [12] Freidlin, M. and Sheu, S. J. (2000). Diffusion processes on graphs: stochastic differential equations, large deviation principle. Probab. Theory Related Fields 116. · Zbl 0957.60088 [13] Freidlin, M. I. and Wentzell A. D. (1984). Random Perturbations of Dynamical Systems. Springer, NewYork. · Zbl 0522.60055 [14] Ikeda, N. and Watanabe, S. (1981). Stochastic Differential Equations and Diffusion Processes. North-Holland. · Zbl 0495.60005 [15] Korostelev, A. P. and Leonov, S. L. (1992). Actional functional for diffusion process with discontinuous drift. Theory Probab. Appl. 37 543-550. · Zbl 0788.60094 [16] Korostelev, A. P. and Leonov, S. L. (1992). Actional functional for diffusions in discontinuous media. Probab. Theory Related Fields 94 317-333. · Zbl 0767.60023 [17] Nagot, I. (1995). Grandes d éviations pour les processus d’apprentissages lent a statistiques discontinues sur une surface. Th ese de Docteur en Sciences, Univ. Paris XI Orsay, U. F. R. Scientifique d’Orsay. [18] Pinsky, M. and Ross G. (1995). Positive Harmonic Functions and Diffusion. Cambridge Stud. Adv. Math. 45. · Zbl 0858.31001 [19] Schilder, M. (1966). Some asymptotic formulae for Wiener integrals. Trans. Amer. Math. Soc. 125 63-85. JSTOR: · Zbl 0156.37602 [20] Varadhan, S. (1980). Lectures on Diffusion Problems and Partial Differential Equations. TATA Institute of Fundamental Research, Bombay. · Zbl 0489.35002 [21] Varadhan, S. (1984). Large Deviations and Applications. SIAM, Philadelphia. · Zbl 0549.60023 [22] Yor, M. (1978). Sur la continuité des temps locaux associés a certaines semi-martingales. Astérisque 52-53 23-35.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.