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Hairer, Martin; Stuart, Andrew M.; Voss, Jochen

Sampling conditioned hypoelliptic diffusions. (English) Zbl 1219.60062

Ann. Appl. Probab. 21, No. 2, 669-698 (2011).
The authors consider the problem of sampling conditioned paths of the second-order SDE of the form \(m\ddot{x}(t)=f(x(t))-\dot{x}(t)+\dot{w}(t)\) conditioned on \(x(0)=x_-\) and \(x(T)=x_+\). This equation describes, for example, the time evolution of a noisy mechanical system with inertia and friction. Rewriting this equation as a system of first-order SDEs for \(x\) and \(\dot{x}\) leads to a drift which is in general not a gradient and, since the noise only acts on \(\dot{x}\), one obtain a singular diffusion matrix. To solve the problem, a fourth-order parabolic type SPDE with nonlinear drift is derived so that its boundary conditions and the drift term supply that the conditioned distribution of the initial equation is stationary for this SPDE. The nonlinear drift has very weak dissipativity and regularity properties that creates additional technical difficulties.
Reviewer: Yuliya S. Mishura (Kyïv)

Cited in 5 Documents

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J60 Diffusion processes
37H10 Generation, random and stochastic difference and differential equations

Keywords:

stochastic partial differential equations; fourth-order SPDEs; hypoelliptic diffusions; conditioned SPDEs; Sobolev spaces; space-time white noise
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\textit{M. Hairer} et al., Ann. Appl. Probab. 21, No. 2, 669--698 (2011; Zbl 1219.60062)
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