Global flows for stochastic differential equations without global Lipschitz conditions

Fang, Shizan; Imkeller, Peter; Zhang, Tusheng

doi:10.1214/009117906000000412

Cited by 41 publications

(42 citation statements)

References 9 publications

(12 reference statements)

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“…In the general case, using to the boundedness of b and the maximum principle, we get easily a-priori estimates for equation (9) (assuming that there exists a bounded solution u). Then a continuity method (see [20,Lemma 4.3]) allows to get the existence of the solution which verifies equation (9), along with the estimate (11).…”

Section: Theorem 2 Let Us Consider Equationmentioning

confidence: 99%

Well-posedness of the transport equation by stochastic perturbation

2009

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We consider the linear transport equation with a globally Hölder continuous and bounded vector field, with an integrability condition on the divergence. While uniqueness may fail for the deterministic PDE, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be the first explicit example of partial differential equation that become well-posed under the influence of noise. The key tool is a differentiable stochastic flow constructed and analyzed by means of a special transformation of the drift of Itô-Tanaka type.

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Section: Theorem 2 Let Us Consider Equationmentioning

confidence: 99%

Well-posedness of the transport equation by stochastic perturbation

2009

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“…This result was recently generalized in [27] to the case where the drift A 0 is allowed to be only log-Lipschitz continuous. Studies on SDE beyond the Lipschitz setting attracted great interest during the last years, see for instance [10,13,12,19,20,23,24,29,34,35].…”

Section: Introductionmentioning

confidence: 99%

“…The SDE (1.1) is said to be complete if for each x ∈ R d , P(τ x = +∞) = 1; it is said to be strongly complete if P(τ x = +∞, x ∈ R d ) = 1. The goal in [26] is to construct examples for which the coefficients are smooth, but such that the SDE (1.1) is not strongly complete (see [13,25] for positive examples). Now consider…”

Section: Introductionmentioning

confidence: 99%

Stochastic differential equations with coefficients in Sobolev spaces

Fang

Luo

Thalmaier

2010

Journal of Functional Analysis

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We consider the Itô stochastic differential equation dX t = m j =1 A j (X t ) dw j t + A 0 (X t ) dt on R d . The diffusion coefficients A 1 , . . . , A m are supposed to be in the Sobolev space W 1,p loc (R d ) with p > d, and to have linear growth. For the drift coefficient A 0 , we distinguish two cases: (i) A 0 is a continuous vector field whose distributional divergence δ(A 0 ) with respect to the Gaussian measure γ d exists, (ii) A 0 has Sobolev regularity W 1,p loc for some p > 1. Assume R d exp[λ 0 (|δ(A 0 )| + m j =1 (|δ(A j )| 2 + |∇A j | 2 ))] dγ d < +∞ for some λ 0 > 0. In case (i), if the pathwise uniqueness of solutions holds, then the push-forward (X t ) # γ d admits a density with respect to γ d . In particular, if the coefficients are bounded Lipschitz continuous, then X t leaves the Lebesgue measure Leb d quasi-invariant. In case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish existence and uniqueness of stochastic flow of maps.

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“…Motivated by Malliavin [13], the study of SDE (1) with log-Lipschitz coefficients attracted extensive attentions during the last decade (see [2,6,7,12]), and in some special cases, the solution is still a flow of homeomorphisms (cf. [3,4,16]). Notice that the logLipschitz condition is stronger than Hölder continuity.…”

Section: Introductionmentioning

confidence: 99%