×

Strong solutions of stochastic equations with singular time dependent drift. (English) Zbl 1072.60050

Existence and uniqueness of solutions to stochastic differential equations in domains \(G \subset {\mathbb R}^d\) with unit local diffusion and singular time dependent drift \(b\) up to an explosion time are proved under the assumption of local \(L_{q^-}L_p \equiv L_q({\mathbb R}, L_p)\)-integrability on \(b\) in \({\mathbb R} \times G\) with \(d/p + 2/q < 1\). Strong Feller properties are also proved for this case. In the case when \(b\) is the gradient in \(x\) of a nonnegative function \(\psi\) blowing up as \(G \ni x\to \partial G\), it is proved that the conditions \(2D_t \psi \leq K \psi\), \(2 D_t \psi + \Delta \psi \leq K e^{\varepsilon \psi}\), \(\varepsilon \in [0, 2)\), imply that the explosion time is infinite and the distributions of the solution have sub-Gaussian tails.

MSC:

60H20 Stochastic integral equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
35R60 PDEs with randomness, stochastic partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albeverio, Comm. Math. Phys., 81, 501 (1981) · Zbl 0487.60066
[2] Albeverio, J. Math. Phys., 18, 907 (1977) · Zbl 0368.60091
[3] Albeverio, The Gibbsian case. J. Funct. Anal., 157, 242 (1998) · Zbl 0931.58019
[4] Albeverio, S., Kondratiev, Y.G., Röckner, M.: Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interactions. In: Finite and Infinite Dimensional Analysis in Honor of Leonard Gross, H. H. Kuo et al., (eds.), Contemporary Mathematics, Vol. 317, Amer. Math. Soc., 2003 · Zbl 1029.60063
[5] Albeverio, J. Funct. Anal., 68, 130 (1986) · Zbl 0633.47027
[6] Bliedtner, J., Hansen, W.: Potential Theory. Springer, Berlin, 1986 · Zbl 0706.31001
[7] Cépa, ESAIM Probab. Statist., 5, 203 (2001) · Zbl 1002.60093
[8] Fukushima, Phys. Rep., 77, 255 (1982)
[9] Fukushima, Lect. Notes Math., 923, 146 (1982)
[10] Fukushima, M.: Energy forms and diffusion processes. Mathematics and Physics, Lectures on Recent Results, L. Streit, (ed.), World Scientific Publ., Singapore, 1984
[11] Gyöngy, Probab. Theory Relat. Fields, 105, 143 (1996) · Zbl 0847.60038
[12] Gyöngy, I., Krylov, N.V.: On the rate of convergence of splitting-up approximations for SPDEs. pp. 301-321 in Progress in Probability, Vol. 56, Birkhauser Verlag, Basel, 2003 · Zbl 1038.60057
[13] Ikeda, N., Watanabe, S.: Stochastic differential equations and diffusion processes. North-Holland Publishing Company, Amsterdam Oxford New York, 1981 · Zbl 0495.60005
[14] Khasminskii, Teorija Verojatnosteiıee Primenenija., 4, 309 (3)
[15] Kondratiev, Yu., Kuna, T., Kutovyi, A.: On relations between a priori bounds for measures on configuration spaces. BiBoS Preprint 2002. To appear in IDAQP
[16] Kondratiev, Y.G., Konstantinov, A.Y., Röckner, M.: Uniqueness of diffusion generators for two types of particle systems with singular interactions. BiBoS-Preprint, publication in preparation, 2003, pp. 9 · Zbl 1055.60074
[17] Krylov, Differentsialnye Uravneniya, 3, 315 (2)
[18] Krylov, SIAM J. Math. Anal., 32, 1117 (5) · Zbl 0979.35060
[19] Krylov, J. Funct. Anal., 183, 1 (1) · Zbl 0980.60091
[20] Klein, J. Stat. Phys., 71, 1043 (1993) · Zbl 0935.82519
[21] Ladyzhenskaya, O.A., Solonnikov, V.A., Ural’tceva, N.N.: Linear and quasi-linear parabolic equations. Nauka, Moscow, 1967 in Russian; English translation: American Math. Soc., Providence, 1968 · Zbl 0164.12302
[22] Liptser, R.S., Shiryaev, A.N.: Statistics of random processes. ‘‘Nauka’’, Moscow, 1974 in Russian; English translation: Springer-Verlag, Berlin and New York, 1977
[23] Portenko, N.I.: Generalized Diffusion Processes. Nauka, Moscow, 1982 In Russian; English translation: Amer. Math. Soc., Providence, Rhode Island, 1990 · Zbl 0727.60089
[24] Skorokhod, J. Appl. Math. Stoch. Anal., 9, 427 (1996) · Zbl 0867.60032
[25] Skorokhod, Methods Funct. Anal. Topology, 4, 54 (1999) · Zbl 0955.60062
[26] Skorokhod, J. Appl. Math. Stoch. Anal., 16, 45 (2003) · Zbl 1027.60061
[27] Veretennikov, Math. USSR Sb., 39, 387 (1981) · Zbl 0462.60063
[28] Yamada, J. Math. Kyoto Univ., 11, 155 (1971) · Zbl 0236.60037
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.