A. Arapostathis, A. Biswas, and V. Borkar,
Controlled equilibrium selection in stochastically perturbed
dynamics, Ann. Probab. 46 (2018), 2749-2799.
K. Athreya and C. Hwang, Gibbs measures
asymptotics, Sankhyã A 72 (2018), 191-207.
G. Barrera and M. Jara, Thermalisation for
small random perturbations of dynamical systems, Ann. Appl.
Probab. 30 (2020), 1164-1208.
P. Benner, J. Li, and T. Penzl, Numerical
solution of large-scale Lyapunov equations, Riccati equations, and
linear-quadratic optimal control problems, Numer. Linear
Algebra Appl. 15 (2008), 755-777.
F. Bolley, I. Gentil, and A. Guillin,
Convergence to equilibrium in Wasserstein distance for
Fokker-Planck equations, J. Funct. Anal. 263 (2012),
2430-2457.
A. Biswas and V. Borkar, Small noise
asymptotics for invariant densities for a class of diffusions: a control
theoretic view, J. Math. Anal. Appl. 360 (2009),
476-484.
W. Coffey and Y. Kalmykov, The Langevin
Equation: with Applications in Physics, Chemistry and Electrical
Engineering, 3rd ed., World Sci., 2012.
A. Dalalyan, Theoretical guarantees for
approximate sampling from smooth and log-concave densities,
J. Roy. Statist. Soc. Ser. B. Statist. Methodol. 79 (2017), 651-676.
M. Day, Recent progress on the small parameter
exit problem, Stochastics 20 (1987), 121-150.
M. Day and T. Darden, Some regularity results
on the Ventcel-Freidlin quasi-potential function, Appl.
Math. Optim. 13 (1985), 259-282.
A. Duncan, N. Nüsken, and G. Pavliotis, Using
perturbed underdamped Langevin dynamics to efficiently sample from
probability distributions, J. Statist. Phys. 169 (2017),
1098-1131.
A. Durmus and È Moulines, Quantitative bounds
of convergence for geometrically ergodic Markov chain in the Wasserstein
distance with application to the Metropolis adjusted Langevin
algorithm, Statist. Comput. 25 (2015), 5-19.
A. Eberle, A. Guillin, and R. Zimmer,
Couplings and quantitative contraction rates for Langevin
dynamics, Ann. Probab. 47 (2019), 1982-2010.
A. Eberle, A. Guillin, and R. Zimmer,
Quantitative Harris-type theorems for diffusions and McKean-Vlasov
processes, Trans. Amer. Math. Soc. 371 (2019),
7135-7173.
M. Freidlin and A. Wentzell, Random
Perturbations of Dynamical Systems, 3rd ed., Springer,
2012.
R. Gareth and J. Rosenthal, Hitting time and
convergence rate bounds for symmetric Langevin diffusions,
Methodol. Comput. Appl. Probab. 21 (2019), 921-929.
W. Huang, M. Ji, Z. Liu, and Y. Yi,
Concentration and limit behaviors of stationary
measures, Phys. D 369 (2018), 1-17.
C. Hwang, Laplace's method revisited: weak
convergence of probability measures, Ann. Probab. 8 (1980),
1177-1182.
C. Hwang, S. Hwang-Ma, and S. Sheu,
Accelerating Gaussian diffusions, Ann. Appl.
Probab. 3 (1993), 897-913.
M. Ji, Z. Shen, and Y. Yi, Quantitative
concentration of stationary measures, Phys. D 399 (2019),
73-85.
Y Kabanov, R. Liptser, and A. Shiryaev, On the
variation distance for probability measures defined on a filtered
space, Probab. Theory Related Fields 71 (1986), 19-35.
A. Kulik, Ergodic Behavior of Markov Processes
with Applications to Limit Theorems, De Gruyter Stud. Math.
67, De Gruyter, 2018.
P. Lancaster and M. Tismenetsky, The Theory of
Matrices, 2nd ed., Academic Press, 1985.
T. Lelièvre, F. Nier, and G. Pavliotis,
Optimal non-reversible linear drift for the convergence to
equilibrium of a diffusion, J. Statist. Phys. 152 (2013),
237-274.
N. Madras and D. Sezer, Quantitative bounds
for Markov chain convergence: Wasserstein and total variation
distances, Bernoulli 16 (2010), 882-908.
X. Mao, Stochastic Differential Equations and
Applications, 2nd ed., Horwood, 2008.
T. Mikami, Asymptotic expansions of the
invariant density of a Markov process with a small
parameter, Ann. Inst. H. Poincaré Probab. Statist. 24
(1988), 403-424.
T. Mikami, Asymptotic analysis of invariant
density of randomly perturbed dynamical systems, Ann.
Probab. 18 (1990), 524-536.
V. Panaretos and Y. Zemel, An Invitation to
Statistics in Wasserstein Space, Springer, 2020.
G. Pavliotis, Stochastic Processes and
Applications: Diffusion Processes, the Fokker-Planck and Langevin
Equations, Springer, 2014.
Y. Pomeau and J. Piasecki, The Langevin
equation, C. R. Phys. 18 (2017), 570-582.
A. Scottedward, B. Tenison, and K. Poolla,
Numerical solution of the Lyapunov equation by approximate
power iteration, Linear Algebra Appl. 236 (1996),
205-230.
S. Sheu, Asymptotic behavior of the invariant
density of a diffusion Markov process with small diffusion,
SIAM J. Math. Anal. 17 (1986), 451-460.
C. Villani, Optimal Transport: Old and
New, Grundlehren Math. Wiss. 338, Springer, 2009.
S. Wu, C. Hwang, and M. Chu, Attaining the
optimal Gaussian diffusion acceleration, J. Statist. Phys.
155 (2014), 571-590. |