Large deviations for multiscale diffusion via weak convergence methods

We study the large deviations principle for locally periodic SDEs with small noise and fast oscillating coefficients. There are three regimes depending on how fast the intensity of the noise goes to zero relative to homogenization parameter. We use weak convergence methods which provide convenient representations for the action functional for all regimes. Along the way, we study weak limits of controlled SDEs with fast oscillating coefficients. We derive, in some cases, a control that nearly achieves the large deviations lower bound at prelimit level. This control is useful for designing efficient importance sampling schemes for multiscale small noise diffusion.

[1]  Kai Liu.  Stochastic Stability of Differential Equations in Abstract Spaces , 2019 .

[2]  Z. Artstein,et al.  Tracking Fast Trajectories Along a Slow Dynamics: A Singular Perturbations Approach , 1997 .

[3]  Pulak Kumar Ghosh,et al.  Noise-induced transport in a rough ratchet potential. , 2009, The Journal of chemical physics.

[4]  H. Kushner.  Numerical Methods for Stochastic Control Problems in Continuous Time , 2000 .

[5]  Changbong Hyeon,et al.  Can energy landscape roughness of proteins and RNA be measured by using mechanical unfolding experiments? , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[6]  M. Robin.  Long-term average cost control problems for continuous time Markov processes: A survey , 1983 .

[7]  A. Veretennikov,et al.  On Large Deviations in the Averaging Principle for SDEs with a “Full Dependence” , 1999 .

[8]  S. Sheu,et al.  ON THE STRUCTURE OF SOLUTIONS OF ERGODIC TYPE BELLMAN EQUATION RELATED TO RISK-SENSITIVE CONTROL , 2006, math/0602625.

[9]  Paolo Baldi,et al.  Large deviations for diffusion processes with homogenization and applications , 1991 .

[10]  Ioannis Karatzas,et al.  Brownian Motion and Stochastic Calculus , 1987 .

[11]  Konstantinos Spiliopoulos,et al.  Importance Sampling for Multiscale Diffusions , 2011, Multiscale Model. Simul..

[12]  P. Lions,et al.  ON ERGODIC STOCHASTIC CONTROL , 1998 .

[13]  R. Liptser.  Large deviations for two scaled diffusions , 1996 .

[14]  Alexander Yu. Veretennikov.  On large deviations in the averaging principle for SDE's with a "full dependence,'' revisited , 2012 .

[15]  A. Bensoussan,et al.  On Bellman equations of ergodic control in Rn , 1992 .

[16]  J. Lynch,et al.  A weak convergence approach to the theory of large deviations , 1997 .

[17]  R. Zwanzig,et al.  Diffusion in a rough potential. , 1988, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Vivek S. Borkar,et al.  On Existence of Limit Occupational Measures Set of a Controlled Stochastic Differential Equation , 2005, SIAM J. Control. Optim..

[19]  Z. Artstein,et al.  The Value Function of Singularly Perturbed Control Systems , 2000 .

[20]  Vladimir Gaitsgory,et al.  On a Representation of the Limit Occupational Measures Set of a Control System with Applications to Singularly Perturbed Control Systems , 2004, SIAM J. Control. Optim..

[21]  A. Veretennikov.  On large deviations for SDEs with small diffusion and averaging , 2000 .

[22]  Vivek S. Borkar,et al.  Averaging of Singularly Perturbed Controlled Stochastic Differential Equations , 2007 .

[23]  A. Ansari,et al.  Mean first passage time solution of the Smoluchowski equation: Application to relaxation dynamics in myoglobin , 2000 .

[24]  H. Kushner.  Large Deviations for Two-Time-Scale Diffusions, with Delays , 2010 .

[25]  J. Norris.  Long-Time Behaviour of Heat Flow: Global Estimates and Exact Asymptotics , 1997 .

[26]  Simultaneous Effects of Homogenization and Vanishing Viscosity in Fully Nonlinear Elliptic Equations , 2003 .

[27]  Vladimir Gaitsgory,et al.  Multiscale Singularly Perturbed Control Systems: Limit Occupational Measures Sets and Averaging , 2002, SIAM J. Control. Optim..

[28]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[29]  R. Khas'minskii.  Ergodic Properties of Recurrent Diffusion Processes and Stabilization of the Solution to the Cauchy Problem for Parabolic Equations , 1960 .

[30]  P. Dupuis,et al.  A variational representation for certain functionals of Brownian motion , 1998 .

[31]  V. Borkar.  Ergodic Control of Diffusion Processes , 2012 .

[32]  Large deviations for small noise diffusions with discontinuous statistics , 2000 .

[33]  A. Veretennikov,et al.  On the poisson equation and diffusion approximation 3 , 2001, math/0506596.

[34]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[35]  J. L. Jackson,et al.  On the Self‐Diffusion of Ions in a Polyelectrolyte Solution , 1962 .

[36]  S. Ethier,et al.  Markov Processes: Characterization and Convergence , 2005 .

[37]  Martino Bardi,et al.  Viscosity Solutions Methods for Singular Perturbations in Deterministic and Stochastic Control , 2001, SIAM J. Control. Optim..

[38]  J. Onuchic,et al.  Funnels, pathways, and the energy landscape of protein folding: A synthesis , 1994, Proteins.

[39]  H. Kushner.  Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems , 1990 .

[40]  Ying Hu,et al.  Probabilistic approach to homogenization of viscosity solutions of parabolic PDEs , 1999 .

[41]  Richard B. Sowers,et al.  A comparison of homogenization and large deviations, with applications to wavefront propagation , 1999 .

[42]  T. Kurtz,et al.  Existence of Markov Controls and Characterization of Optimal Markov Controls , 1998 .

[43]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[44]  Vivek S. Borkar,et al.  Optimal Control of Diffusion Processes , 1989 .

[45]  Jeffery G. Saven,et al.  Kinetics of protein folding: The dynamics of globally connected rough energy landscapes with biases , 1994 .

[46]  Zvi Artstein,et al.  Singularly perturbed ordinary differential equations with dynamic limits , 1996, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[47]  A. Bensoussan,et al.  Asymptotic analysis for periodic structures , 1979 .