Celebratio Mathematica

L. H. Jensen and S. R. S. Varadhan: Large deviations of the asymmetric exclusion process in one dimension. Preprint, 2000.

S. R. S. Varadhan: “Large deviations for random walks in a random environment,” Comm. Pure Appl. Math. 56 : 8 (August 2003), pp. 1222–1245. Dedicated to the memory of Jürgen K. Moser. MR 1989232 Zbl 1042.60071 article

S. R. S. Varadhan: “Large deviations for the asymmetric simple exclusion process,” pp. 1–27 in Stochastic analysis on large scale interacting systems. Edited by T. Funaki and H. Osada. Advanced Studies in Pure Mathematics 39. Math. Soc. Japan (Tokyo), 2004. MR 2073328 Zbl 1114.60026 incollection

Hirofumi Osada	Related
Tadahisa Funaki	Related

S. R. S. Varadhan: “Random walks in a random environment,” Proc. Indian Acad. Sci. Math. Sci. 114 : 4 (2004), pp. 309–318. MR 2067696 Zbl 1077.60078 ArXiv math/0503089 article

E. Kosygina, F. Rezakhanlou, and S. R. S. Varadhan: “Stochastic homogenization of Hamilton–Jacobi–Bellman equations,” Comm. Pure Appl. Math. 59 : 10 (2006), pp. 1489–1521. MR 2248897 Zbl 1111.60055 article

We study the homogenization of some Hamilton–Jacobi–Bellman equations with a vanishing second-order term in a stationary ergodic random medium under the hyperbolic scaling of time and space. Imposing certain convexity, growth, and regularity assumptions on the Hamiltonian, we show the locally uniform convergence of solutions of such equations to the solution of a deterministic “effective” first-order Hamilton–Jacobi equation. The effective Hamiltonian is obtained from the original stochastic Hamiltonian by a minimax formula. Our homogenization results have a large-deviations interpretation for a diffusion in a random environment.

Fraydoun Rezakhanlou	Related
Elena Kosygina	Related

S. R. S. Varadhan: “Homogenization,” Math. Student 76 : 1–4 (2007), pp. 129–136. MR 2522935 Zbl 1182.35023 article

S. R. S. Varadhan: “Homogenization of random Hamilton–Jacobi–Bellman equations,” pp. 397–403 in Probability, geometry and integrable systems. Edited by M. Pinsky and B. Birnir. Mathematical Sciencies Research Institute Publications 55. Cambridge University Press, 2008. MR 2407606 Zbl 1160.35334 incollection

We consider nonlinear parabolic equations of Hamilton–Jacobi–Bellman type. The Lagrangian is assumed to be convex, but with a spatial dependence which is stationary and random. Rescaling in space and time produces a similar equation with a rapidly varying spatial dependence and a small viscosity term. Motivated by corresponding results for the linear elliptic equation with small viscosity, we seek to find the limiting behaviour of the solution of the Cauchy (final value) problem in terms of a homogenized problem, described by a convex function of the gradient of the solution. The main idea is to use the principle of dynamic programming to write a variational formula for the solution in terms of solutions of linear problems. We then show that asymptotically it is enough to restrict the optimization to a subclass, one for which the asymptotic behavior can be fully analyzed. The paper outlines these steps and refers to the recently published work of Kosygina, Rezakhanlou and the author for full details.

Björn Birnir	Related
Mark A. Pinsky	Related

E. Kosygina and S. R. S. Varadhan: “Homogenization of Hamilton–Jacobi–Bellman equations with respect to time-space shifts in a stationary ergodic medium,” Comm. Pure Appl. Math. 61 : 6 (2008), pp. 816–847. MR 2400607 Zbl 1144.35008 article

We consider a family \( \{u_{\varepsilon}(t,x,\omega)\} \), \( \varepsilon < 0 \), of solutions to the equation \[ \frac{\partial u_{\varepsilon}}{\partial t} + \frac{\varepsilon}{2}\,\Delta u_{\varepsilon} + H\Bigl(\frac{t}{\varepsilon}, \frac{x}{\varepsilon}, \nabla u_{\varepsilon},\,\omega\Bigr) = 0 \] with the terminal data \( u_{\varepsilon}(T,x,\omega) = U(x) \). Assuming that the dependence of the Hamiltonian \( H(t,x,p,\omega) \) on time and space is realized through shifts in a stationary ergodic random medium, and that \( H \) is convex in \( p \) and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of \( u_{\varepsilon}(t,x,\omega) \) as \( \varepsilon\to 0 \) to the solution \( u(t,x) \) of a deterministic averaged equation \[ \frac{\partial u}{\partial t} + \overline{H}(\nabla u) = 0, \qquad u(T,x) = U(x) .\] The “effective” Hamiltonian \( \overline{H} \) is given by a variational formula.

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S. R. Srinivasa Varadhan

Large deviation and homogenization

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