Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

Xicheng Zhang

doi:10.1016/j.spa.2010.05.015

Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

Xicheng Zhang^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

51 Citations (Scopus)

Abstract

We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin's transformation to derive a stochastic flow of C¹-diffeomorphisms for non-degenerate SDEs with Hlder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation z̈_t = c₀0ż_t - z_t ³ + Θ(zt)ẇ_t,(z₀,z ₀)=(z,u)∈ ℝ², where c₀ ∈ ℝ,Θ ∈ C∞(ℝ) has a bounded first order derivative, and wt is a one dimensional Brownian white noise.

Original language	English
Pages (from-to)	1929-1949
Number of pages	21
Journal	Stochastic Processes and their Applications
Volume	120
Issue number	10
DOIs	https://doi.org/10.1016/j.spa.2010.05.015
Publication status	Published - Sept 2010
Externally published	Yes

Keywords

Bismut formula, Stochastic Hamiltonian system
Stochastic flow of diffeomorphisms

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10.1016/j.spa.2010.05.015

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abstract = "We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin's transformation to derive a stochastic flow of C1-diffeomorphisms for non-degenerate SDEs with Hlder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation {\"z}t = c00{\.z}t - zt 3 + Θ(zt)ẇt,(z0,z 0)=(z,u)∈ ℝ2, where c0 ∈ ℝ,Θ ∈ C∞(ℝ) has a bounded first order derivative, and wt is a one dimensional Brownian white noise.",

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AU - Zhang, Xicheng

PY - 2010/9

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N2 - We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin's transformation to derive a stochastic flow of C1-diffeomorphisms for non-degenerate SDEs with Hlder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation z̈t = c00żt - zt 3 + Θ(zt)ẇt,(z0,z 0)=(z,u)∈ ℝ2, where c0 ∈ ℝ,Θ ∈ C∞(ℝ) has a bounded first order derivative, and wt is a one dimensional Brownian white noise.

AB - We first consider the stochastic differential equations (SDE) without global Lipschitz conditions, and give sufficient conditions for the SDEs to be strictly conservative. In particular, a criteria for stochastic flows of diffeomorphisms defined by SDEs with non-global Lipschitz coefficients is obtained. We also use Zvonkin's transformation to derive a stochastic flow of C1-diffeomorphisms for non-degenerate SDEs with Hlder continuous drifts. Next, we prove a Bismut type formula for certain degenerate SDEs. Lastly, we apply our results to stochastic Hamiltonian systems, which in particular covers the following stochastic nonlinear oscillator equation z̈t = c00żt - zt 3 + Θ(zt)ẇt,(z0,z 0)=(z,u)∈ ℝ2, where c0 ∈ ℝ,Θ ∈ C∞(ℝ) has a bounded first order derivative, and wt is a one dimensional Brownian white noise.

KW - Bismut formula, Stochastic Hamiltonian system

KW - Stochastic flow of diffeomorphisms

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ER -

Stochastic flows and Bismut formulas for stochastic Hamiltonian systems

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Keywords

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