Abstract

In this paper we study the ergodic problem for viscous Hamilton--Jacobi equations with superlinear Hamiltonian and inward drift. We investigate (i) existence and uniqueness of eigenfunctions associated with the generalized principal eigenvalue of the ergodic problem, (ii) relationships with the corresponding stochastic control problem of both finite and infinite time horizon, and (iii) the precise growth exponent of the generalized principal eigenvalue with respect to a perturbation of the potential function.

Keywords

  1. ergodic problem
  2. viscous Hamilton--Jacobi equation
  3. stochastic ergodic control
  4. generalized principal eigenvalue

MSC codes

  1. 35Q93
  2. 60J60
  3. 93E20

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
A. Arapostathis, V. S. Borkar, and K. S. Kumar, Convergence of the relative value iteration for the ergodic control problem of nondegenerate diffusions under near-monotone costs, SIAM J. Control Optim., 52 (2014), pp. 1--31.
2.
A. Arapostathis, A. Biswas, and S. Saha, Strict monotonicity of principal eigenvalues of elliptic operators in ${\mathbb R}^d$ and risk-sensitive control, J. Math. Pures Appl. (9), to appear.
3.
G. Barles and J. Meireles, On unbounded solutions of ergodic problems in ${\mathbb R}^m$ for viscous Hamilton--Jacobi equations, Comm. Partial Differential Equations, 41 (2016), pp. 1985--2003.
4.
G. Barles, A. Porretta, and T. Tchamba, On the large time behavior of solutions of the Dirichlet problem for subquadratic viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 94 (2010), pp. 497--519.
5.
G. Barles and P. E. Souganidis, Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations, SIAM J. Math. Anal., 32 (2001), pp. 1311--1323.
6.
F. Cagnetti, D. Gomes, H. Mitake, and H. V. Tran, A new method for large time behavior of degenerate viscous Hamilton-Jacobi equations with convex Hamiltonians, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), pp. 183--200.
7.
D. Castorina, A. Cesaroni, and L. Rossi, Large time behavior of solutions to a degenerate parabolic Hamilton-Jacobi-Bellman equation, Commun. Pure Appl. Anal., 15 (2016), pp. 1251--1263.
8.
E. Chasseigne and N. Ichihara, Qualitative properties of generalized principal eigenvalues for superquadratic viscous Hamilton-Jacobi equations, NoDEA Nonlinear Differential Equations Appl., 23 (2016), pp. 1--17.
9.
M. Cirant, On the solvability of some ergodic control problems in $\R^d$, SIAM J. Control Optim., 52 (2014), pp. 4001--4026.
10.
A. Cosso, M. Fuhrman, and H. Pham, Long time asymptotics for fully nonlinear Bellman equations: A backward SDE approach, Stochastic Process. Appl., 126 (2016), pp. 1932--1973.
11.
W. H. Fleming and W. M. McEneaney, Risk-sensitive control on an infinite time horizon, SIAM J. Control Optim., 33 (1995), pp. 1881--1915.
12.
W. H. Fleming and M. Soner, Controlled Markov Processes and Viscosity Solutions, Applications of Mathematics 25, Springer-Verlag, New York, 1993.
13.
Y. Fujita, H. Ishii, and P. Loreti, Asymptotic solutions of viscous Hamilton--Jacobi equations with Ornstein-Uhlenbeck operator, Comm. Partial Differential Equations, 31 (2006), pp. 827--848.
14.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, 2nd ed., Grundlehren Math. Wiss. 224, Springer-Verlag, Berlin, 1988.
15.
Y. Hu and P.-Y. Madec, A probabilistic approach to large time behaviour of viscosity solutions of parabolic equations with Neumann boundary conditions, Appl. Math. Optim., 74 (2016), pp. 345--374.
16.
N. Ichihara, Recurrence and transience of optimal feedback processes associated with Bellman equations of ergodic type, SIAM J. Control Optim., 49 (2011), pp. 1938--1960.
17.
N. Ichihara, Large time asymptotic problems for optimal stochastic control with superlinear cost, Stochastic Process. Appl., 122 (2012), pp. 1248--1275.
18.
N. Ichihara, Criticality of viscous Hamilton-Jacobi equations and stochastic ergodic control, J. Math. Pures Appl., 100 (2013), pp. 368--390.
19.
N. Ichihara, The generalized principal eigenvalue for Hamilton-Jacobi-Bellman equations of ergodic type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 32 (2015), pp. 623--650.
20.
N. Ichihara and S.-J. Sheu, Large time behavior of solutions of Hamilton-Jacobi-Bellman equations with quadratic nonlinearity in gradients, SIAM J. Math. Anal., 45 (2013), pp. 279--306.
21.
H. Kaise and S. J. Sheu, On the structure of solutions of ergodic type Bellman equation related to risk-sensitive control, Ann. Probab., 34 (2005), pp. 284--320.
22.
H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Stud. Adv. Math. 24, 1990.
23.
O. Ley and V. D. Nguyen, Large time behavior for some nonlinear degenerate parabolic equations, J. Math. Pures Appl., 102 (2014), pp. 293--314.
24.
P.-L. Lions, Résolution de problèmes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., 74 (1980), pp. 335--353.
25.
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968.
26.
H. Nagai, Down side risk minimization via a large deviations approach, Ann. Appl. Probab., 22 (2012), pp. 608--669.
27.
R. G. Pinsky, Positive Harmonic Functions and Diffusion, Cambridge Stud. Adv. Math. 45, 1995.
28.
S. Robertson and H. Xing, Large time behavior of solutions to semilinear equations with quadratic growth in the gradient, SIAM J. Control Optim., 53 (2015), pp. 185--212.
29.
P. Souplet and Q. Zhang, Global solutions of inhomogeneous Hamilton-Jacobi equations, J. Anal. Math., 99 (2006), pp. 355--396.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 23 - 52
ISSN (online): 1095-7138

History

Submitted: 9 April 2018
Accepted: 29 October 2018
Published online: 2 January 2019

Keywords

  1. ergodic problem
  2. viscous Hamilton--Jacobi equation
  3. stochastic ergodic control
  4. generalized principal eigenvalue

MSC codes

  1. 35Q93
  2. 60J60
  3. 93E20

Authors

Affiliations

Funding Information

Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-16-CE40-0015-01
Japan Society for the Promotion of Science https://doi.org/10.13039/501100001691 : 15K04935, 18K03343

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.