On the spatial asymptotic behavior of stochastic flows in Euclidean space
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 \(\gamma.\)
stochastic partial differential equation
scientific article; zbMATH DE number 1359642
scientific article