Large Deviations for Stochastic Processes

The notes are devoted to results on large deviations for sequences of Markov processes following closely the book by Feng and Kurtz ([FK06]). We outline how convergence of Fleming's nonlinear semigroups (logarithmically transformed nonlinear semigroups) implies large deviation principles analogous to the use of convergence of linear semigroups in weak convergence. The latter method is based on the range condition for the corresponding generator. Viscosity solution methods however provide applicable conditions for the necessary nonlinear semigroup convergence. Once having established the validity of the large deviation principle one is concerned to obtain more tractable representations for the corresponding rate function. This in turn can be achieved once a variational representation of the limiting generator of Fleming's semigroup can be established. The obtained variational representation of the generator allows for a suitable control representation of the rate function. The notes conclude with a couple of examples to show how the methodology via Fleming's semigroups works. These notes are based on the mini-course 'Large deviations for stochastic processes' the author held during the workshop 'Dynamical Gibs-non-Gibbs transitions' at EURANDOM in Eindhoven, December 2011, and at the Max-Planck institute for mathematics in the sciences in Leipzig, July 2012.

[1]  R. Buck,et al.  Bounded continuous functions on a locally compact space. , 1958 .

[2]  M. Sion.  On general minimax theorems , 1958 .

[3]  H. Trotter.  Approximation of semi-groups of operators , 1958 .

[4]  R. Khas'minskii.  A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides , 1966 .

[5]  Srinivasa R. S. Varadhan,et al.  Asymptotic probabilities and differential equations , 1966 .

[6]  A. Borovkov.  Boundary-Value Problems for Random Walks and Large Deviations in Function Spaces , 1967 .

[7]  Ken-iti Sato.  On the generators of non-negative contraction semi-groups in Banach lattices , 1968 .

[8]  P. Billingsley,et al.  Convergence of Probability Measures , 1970, The Mathematical Gazette.

[9]  Thomas G. Kurtz,et al.  Extensions of Trotter's operator semigroup approximation theorems , 1969 .

[10]  T. Kurtz.  A general theorem on the convergence of operator semigroups , 1970 .

[11]  J. B. Cooper,et al.  The strict topology and spaces with mixed topologies , 1971 .

[12]  Michael G. Crandall,et al.  GENERATION OF SEMI-GROUPS OF NONLINEAR TRANSFORMATIONS ON GENERAL BANACH SPACES, , 1971 .

[13]  R. Giles.  A generalization of the strict topology , 1971 .

[14]  Mark S. C. Reed,et al.  Method of Modern Mathematical Physics , 1972 .

[15]  Thomas G. Kurtz.  Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics , 1973 .

[16]  George Papanicolaou,et al.  A limit theorem with strong mixing in banach space and two applications to stochastic differential equations , 1973 .

[17]  Thomas G. Kurtz,et al.  A limit theorem for perturbed operator semigroups with applications to random evolutions , 1973 .

[18]  S. Varadhan,et al.  Asymptotic evaluation of certain Markov process expectations for large time , 1975 .

[19]  Thomas G. Kurtz,et al.  Semigroups of Conditioned Shifts and Approximation of Markov Processes , 1975 .

[20]  S. Varadhan,et al.  On a variational formula for the principal eigenvalue for operators with maximum principle. , 1975, Proceedings of the National Academy of Sciences of the United States of America.

[21]  Daniel W. Stroock,et al.  Diffusion processes associated with Lévy generators , 1975 .

[22]  A. A. Mogul'skii.  Large Deviations for Trajectories of Multi-Dimensional Random Walks , 1977 .

[23]  K. Musiał.  Radon-Nikodym theorem for Banach space valued measures , 1976 .

[24]  Robert F. Anderson,et al.  Small Random perturbation of dynamical systems with reflecting boundary , 1976, Nagoya Mathematical Journal.

[25]  A. D. Venttsel.  Rough Limit Theorems on Large Deviations for Markov Stochastic Processes. IV , 1977 .

[26]  丸山 徹.  Convex Analysisの二,三の進展について , 1977 .

[27]  W. Fleming.  Exit probabilities and optimal stochastic control , 1977 .

[28]  M. Freidlin,et al.  THE AVERAGING PRINCIPLE AND THEOREMS ON LARGE DEVIATIONS , 1978 .

[29]  P. Meyer,et al.  Probabilities and potential C , 1978 .

[30]  David Aldous,et al.  Stopping Times and Tightness. II , 1978 .

[31]  A Variational Formula for the Growth Rate of a Positive Operator Semigroup , 1979 .

[32]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[33]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[34]  P. Priouret,et al.  Petites perturbations de systemes dynamiques avec reflexion , 1983 .

[35]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[36]  D. Stroock.  An Introduction to the Theory of Large Deviations , 1984 .

[37]  P. Lions,et al.  Stochastic differential equations with reflecting boundary conditions , 1984 .

[38]  Harold J. Kushner,et al.  Approximation and Weak Convergence Methods for Random Processes , 1984 .

[39]  S. Varadhan.  Large Deviations and Applications , 1984 .

[40]  M. Freidlin,et al.  Random Perturbations of Dynamical Systems , 1984 .

[41]  Hitoshi Ishii,et al.  A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities , 1985 .

[42]  P. Lyons.  Neumann Type Boundary Conditions for Hamilton-Jacobi Equations. , 1985 .

[43]  S. Sheu.  Stochastic Control and Exit Probabilities of Jump Processes , 1985 .

[44]  P. Souganidis,et al.  PDE-viscosity solution approach to some problems of large deviations , 1986 .

[45]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[46]  J. Gärtner,et al.  Large deviations from the mckean-vlasov limit for weakly interacting diffusions , 1987 .

[47]  N. Vakhania,et al.  Probability Distributions on Banach Spaces , 1987 .

[48]  J. Lynch,et al.  Large Deviations for Processes with Independent Increments , 1987 .

[49]  G. Barles,et al.  Exit Time Problems in Optimal Control and Vanishing Viscosity Method , 1988 .

[50]  H. Ishii.  On uniqueness and existence of viscosity solutions of fully nonlinear second‐order elliptic PDE's , 1989 .

[51]  L. Evans.  The perturbed test function method for viscosity solutions of nonlinear PDE , 1989, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[52]  Jürgen Gärtner,et al.  Large Deviations, Free Energy Functional and Quasi-Potential for a Mean Field Model of Interacting Diffusions , 1989 .

[53]  W. Fleming,et al.  Asymptotic expansions for Markov processes with lévy generators , 1989 .

[54]  P. Protter.  Stochastic integration and differential equations : a new approach , 1990 .

[55]  P. Lions,et al.  Viscosity solutions of fully nonlinear second-order elliptic partial differential equations , 1990 .

[56]  A. D. Wentzell,et al.  Limit Theorems on Large Deviations for Markov Stochastic Processes , 1990 .

[57]  George L. O'Brien,et al.  Capacities, Large Deviations and Loglog Laws , 1991 .

[58]  G. Barles,et al.  Convergence of approximation schemes for fully nonlinear second order equations , 1991 .

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

[60]  M. Pinsky,et al.  Lectures on Random Evolution , 1991 .

[61]  A. A. Pukhal'skii,et al.  On functional principle of large deviations , 1991 .

[62]  H. Spohn.  Large Scale Dynamics of Interacting Particles , 1991 .

[63]  L. Evans.  Measure theory and fine properties of functions , 1992 .

[64]  Carl Graham,et al.  McKean-Vlasov Ito-Skorohod equations, and nonlinear diffusions with discrete jump sets , 1992 .

[65]  Daniel Tataru,et al.  Viscosity solutions of Hamilton-Jacobi equations with unbounded nonlinear terms , 1992 .

[66]  ON LARGE DEVIATIONS IN THE AVERAGING PRINCIPLE FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS. II , 1992 .

[67]  Richard B. Sowers,et al.  Large Deviations for a Reaction-Diffusion Equation with Non-Gaussian Perturbations , 1992 .

[68]  W. Fleming,et al.  Controlled Markov processes and viscosity solutions , 1992 .

[69]  P. Lions,et al.  User’s guide to viscosity solutions of second order partial differential equations , 1992, math/9207212.

[70]  A. A. Mogulskii.  Large Deviations for Processes with Independent Increments , 1993 .

[71]  A. Veretennikov,et al.  Large Deviations for Discrete-Time Processes with Averaging , 1993 .

[72]  A. Puhalskii.  The method of stochastic exponentials for large deviations , 1994 .

[73]  Harald Cram'er,et al.  Sur un nouveau théorème-limite de la théorie des probabilités , 2018 .

[74]  A. Acosta,et al.  Large deviations for vector-valued Lévy processes , 1994 .

[75]  G. O'Brien,et al.  Compactness in the theory of large deviations , 1995 .

[76]  R. McCann.  Existence and uniqueness of monotone measure-preserving maps , 1995 .

[77]  D. Kinderlehrer,et al.  THE VARIATIONAL FORMULATION OF THE FOKKER-PLANCK EQUATION , 1996 .

[78]  G. O'Brien.  Sequences of capacities, with connections to large-deviation theory , 1996 .

[79]  S. Méléard.  Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models , 1996 .

[80]  R. McCann.  A Convexity Principle for Interacting Gases , 1997 .

[81]  O. Kallenberg.  Foundations of Modern Probability , 2021, Probability Theory and Stochastic Modelling.

[82]  Exponential Tightness and Projective Systems in Large Deviation Theory , 1997 .

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

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

[85]  Anatolii A. Puhalskii,et al.  Large deviations of semimartingales: A maxingale problem approach i. limits as solutions to a maxingale problem , 1997 .

[86]  R. Russell.  The Large Deviations of random time-changes , 1997 .

[87]  C. Landim,et al.  Derivation of Cahn-Hilliard equations from Ginzburg-Landau models , 1997 .

[88]  Amir Dembo,et al.  Large Deviations Techniques and Applications , 1998 .

[89]  S. Rachev,et al.  Mass transportation problems , 1998 .

[90]  The Averaging Principle. Fluctuations in Dynamical Systems with Averaging , 1998 .

[91]  Jin Feng.  Martingale problems for large deviations of Markov processes , 1999 .

[92]  On Large Deviations for Stochastic DifferentialEquations with a Small Diffusion and Averaging , 1999 .

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

[94]  T. Kurtz,et al.  Particle representations for a class of nonlinear SPDEs , 1999 .

[95]  A. Acosta.  A general non-convex large deviation result with applications to stochastic equations , 2000 .

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

[97]  T. Kurtz,et al.  Stationary Solutions and Forward Equations for Controlled and Singular Martingale Problems , 2001 .

[98]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[99]  Anatolii A. Puhalskii,et al.  Large Deviations and Idempotent Probability , 2001 .

[100]  Sanjay K. Bose,et al.  Fundamentals of Queueing Networks , 2002 .

[101]  S. Meyn,et al.  Spectral theory and limit theorems for geometrically ergodic Markov processes , 2002, math/0209200.

[102]  A. Guillin.  AVERAGING PRINCIPLE OF SDE WITH SMALL DIFFUSION: MODERATE DEVIATIONS , 2003 .

[103]  C. Villani.  Topics in Optimal Transportation , 2003 .

[104]  M. Crandall,et al.  Hamilton-Jacobi Equations in Infinite Dimensions , 2003 .

[105]  J. García.  An Extension of the Contraction Principle , 2004 .

[106]  L. Ambrosio,et al.  Gradient Flows: In Metric Spaces and in the Space of Probability Measures , 2005 .

[107]  S. Meyn,et al.  Large Deviations Asymptotics and the Spectral Theory of Multiplicatively Regular Markov Processes , 2005, math/0509310.

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

[109]  Charalambos D. Aliprantis,et al.  Positive Operators , 2006 .

[110]  Adam Shwartz,et al.  Large Deviations For Performance Analysis , 2019 .