TY - JOUR

T1 - A Feynman–Kac formula approach for computing expectations and threshold crossing probabilities of non-smooth stochastic dynamical systems

AU - Mertz, Laurent

AU - Stadler, Georg

AU - Wylie, Jonathan

N1 - Funding Information:
We thank the editor and the two anonymous reviewers for their helpful comments leading to the improvement of the present manuscript. LM expresses his sincere gratitude to the Courant Institute for being supported as Courant Instructor in 2014 and 2015, when this work was initiated. LM is also supported by a faculty discretionary fund from NYU Shanghai, China and the National Natural Science Foundation of China , Research Fund for International Young Scientists, China under the project #1161101053 entitled “Computational methods for non-smooth dynamical systems excited by random forces” and the Young Scientist Program, China under the project #11601335 entitled “Stochastic Control Method in Probabilistic Engineering Mechanics”. LM also thanks the Department of Mathematics of the City University of Hong Kong for the hospitality. JW acknowledges support from the SAR Hong Kong, China grant [CityU 11306115 ] “Dynamics of Noise-Driven Inelastic Particle Systems”.
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2019/10

Y1 - 2019/10

N2 - We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman–Kac formula, where the underlying stochastic processes satisfy variational inequalities modeling elasto-plastic and obstacle oscillators. We then focus on solving them numerically The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.

AB - We present a computational alternative to probabilistic simulations for non-smooth stochastic dynamical systems that are prevalent in engineering mechanics. As examples, we target (1) stochastic elasto-plastic problems, which involve transitions between elastic and plastic states, and (2) obstacle problems with noise, which involve discrete impulses due to collisions with an obstacle. We formally introduce a class of partial differential equations related to the Feynman–Kac formula, where the underlying stochastic processes satisfy variational inequalities modeling elasto-plastic and obstacle oscillators. We then focus on solving them numerically The main challenge in solving these equations is the non-standard boundary conditions which describe the behavior of the underlying process on the boundary. We illustrate how to use our approach to compute expectations and other statistical quantities, such as the asymptotic growth rate of variance in asymptotic formulae for threshold crossing probabilities.

KW - Engineering mechanics

KW - Feynman–Kac formula

KW - Finite difference scheme

KW - PDEs with non-standard boundary conditions

KW - Stochastic variational inequalities

UR - http://www.scopus.com/inward/record.url?scp=85066397048&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85066397048&partnerID=8YFLogxK

U2 - 10.1016/j.physd.2019.05.003

DO - 10.1016/j.physd.2019.05.003

M3 - Article

AN - SCOPUS:85066397048

VL - 397

SP - 25

EP - 38

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

ER -