On the spatial asymptotic behavior of stochastic flows in Euclidean space. (English) Zbl 0942.60053

Imkeller, Peter; Scheutzow, Michael

Ann. Probab. 27, No. 1, 109-129 (1999).

The authors study asymptotic growth rates of stochastic flows on \(\mathbb R^d\) and their derivatives with respect to the spatial parameter under Lipschitz conditions on the local characteristics of the generating semimartingales. It is asked: Can we obtain exact rates of growth by applying the Garsia-Rodemich-Rumsey lemma (GRR lemma) in its majorizing measure version to moment inequalities for the flow, while carefully keeping track of the quality of the constants appearing in the main martingale inequalities of Doob and Burkholder, Davis and Gundy? The main task is to show that the answer to this question is “yes”. The majorizing measure form of the GRR lemma is used to derive moduli of continuity for random fields on \(\mathbb R^d\) with values in some metric space \((M,\rho)\) which satisfy moment conditions of the form \[ E[\rho(\varphi(x), \varphi(y))^p]\leq d(x, y)^p \exp(cp^2), \] either for some fixed \(p\) and the metric \(d(x, y) = |x - y|\) or \(\forall p\geq 1\) and the metric \(d(x, y) = |x - y|\) or \(\forall p\geq 1\) and the metric \(d(x, y) = |x - y|\wedge 1\), \(x, y \in M.\) The moment conditions in the second case are verified for the stochastic flow \(\varphi\) generated by a stochastic differential equation driven by a semimartingale \(F\) whose local characteristics satisfy suitable Lipschitz conditions. Together with a similar estimate on the inverse of the modulus of the flow it is obtained a theorem, which shows that the supremum of \(|\varphi_{st}(x)|\) over \(0\leq s, t\leq T\) is bounded by a random variable \(Y\) multiplied by \(|x|\exp(\gamma(\ln\ln|x|)^{1/2})\) as \(|x|\to\infty\) for some constant \(\gamma > 0,\) and moreover that \(Y\) is integrable with respect to a certain Young function growing faster than any polynomial. The corresponding questions on the growth rate of higher derivatives of the flow \(\varphi\) are considered too. Under differentiability conditions on the local characteristics of \(F,\) the growth rate of any partial derivative of order at least 1 is at most \(Z\exp(\gamma(\ln|x|)^{1/2})\) for some constant \(\gamma > 0,\) where \(Z\) has a similar integrability property as \(Y\) above. Examples show that the rate is optimal, possibly up to the value of the constant \(\gamma.\)

Reviewer: Alexandr Vasil’ev (Odessa)

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

stochastic partial differential equation; stochastic flow; spatial asymptotic growth rate; modulus of continuity; Garsia-Rodemich-Rumsey lemma; random fields; semimartingale

References:

[1]	Arnold, L. and Imkeller, P. (1996). Stratonovich calculus with spatial parameters and anticipative problems in multiplicative ergodic theory. Stochastic Process Appl. 62 19-54. · Zbl 0847.60037 · doi:10.1016/0304-4149(95)00081-X
[2]	Arnold, L. and Imkeller, P. (1998). On the integrability condition in the multiplicative ergodic theorem. Stochastics Stochastics Rep. 64 195-210. · Zbl 1043.60512
[3]	Barlow, M. and Yor, M. (1982). Semi-martingale inequalities via the Garsia-Rodemich- Rumsey lemma, and applications to local times. J. Funct. Anal. 49 198-229. · Zbl 0505.60054 · doi:10.1016/0022-1236(82)90080-5
[4]	Imkeller, P. (1993). Occupation densities of Stratonovich stochastic differential equations with boundary conditions. Stochastics Stochastics Rep. 45 249-264. · Zbl 0796.60060 · doi:10.1080/17442509308833864
[5]	Kunita, H. (1990). Stochastic Flows and Stochastic Differential Equations. Cambridge Univ. Press. · Zbl 0743.60052
[6]	Ledoux, M. and Talagrand, M. (1991). Probability in Banach Spaces. Springer, Berlin. · Zbl 0748.60004
[7]	Mohammed, S. E. A. (1990). The Lyapunov spectrum and stable manifolds for stochastic linear delay equations. Stochastics Stochastics Rep. 29 89-131. · Zbl 0699.60045 · doi:10.1080/17442509008833609
[8]	Mohammed, S. E. A. (1992). Lyapunov exponents and stochastic flows of linear and affine hereditary systems. In Diffusion Processes and Related Problems in Analysis 2 (M. Pinsky and V. Wihstutz, eds.) 141-169. Birkhäuser, Boston. · Zbl 0763.60030
[9]	Mohammed, S. E. A. and Scheutzow, M. (1996). Lyapunov exponents of linear stochastic functional differential equations driven by semimartingales I: the multiplicative ergodic theory. Ann. Inst. H. Poincaré 32 69-105. · Zbl 0862.60044
[10]	Mohammed, S. E. A. and Scheutzow, M. (1998). Spatial estimates for stochastic flows in Euclidean space. Ann. Probab. 26 30-55. · Zbl 0937.60056 · doi:10.1214/aop/1022855411
[11]	Ocone, D. and Pardoux, E. (1989). A generalized It o-Ventzell formula. Application to a class of anticipating stochastic differential equations. Ann. Inst. H. Poincaré 25 39-71. · Zbl 0674.60057