Bridging the Gap between Center and Tail for Multiscale Processes Matthew R. Morse Department of Mathematics and Statistics Boston University BU-Keio 2016, August 16 Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 1 / 19
Introduction A multiscale process is characterized by multiple coupled processes which evolve at separated time scales. Event probabilities on the order of greater than one in one thousand are estimated by the Central Limit Theorem. Event probabilities on the order of less than 10 9 are estimated by the Large Deviations Principle. We focus on the Moderate Deviations Principle, which provides estimates for probabilities between these two extremes. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 2 / 19
Outline 1 Examples of Multiscale Processes Molecular Physics 2 Review of Related Work Multiscale Processes Large Deviations Principle 3 Moderate Deviations Principle 4 Future Work Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 3 / 19
Examples of Multiscale Processes Molecular Physics A Model of Protein Folding [Dupuis, et al., 2011] Define X ε (t) The configuration of a protein at time t V ε (x, x/δ) The potential surface of a protein X ε (t) satisfies the Langevin equation ( dx ε (t) = V ε X ε (t), X ε ) (t) dt + ε 2D dw (t), X ε (0) = x 0. δ where 2D is a diffusion constant, W (t) is standard Brownian motion, and δ and ε are small positive parameters. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 4 / 19
Examples of Multiscale Processes Molecular Physics Graph of a Potential Function V(x) -0.5 0.5 1.0 1.5 2.0 2.5 3.0-2 -1 0 1 2 x V ε (x, x δ ) = ε ( cos( x δ ) + sin( x δ )) + 3 2 (x 2 1) 2, ε = 0.1, δ = 0.01. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 5 / 19
Review of Related Work Multiscale Processes Definition of Multiscale Processes Let Xt ε be a n-dimensional process and Yt ε be a k-dimensional process for 0 t 1 which solve the system dx ε t = c(x ε t, Y ε t ) dt + εσ(x ε t, Y ε t ) dw t, X ε 0 = x 0 dy ε t = 1 ε g(x ε t, Y ε t ) dt + 1 ε τ(x ε t, Y ε t ) db t, Y ε 0 = y 0 where the coefficient functions c(x, y), σ(x, y), g(x, y), and τ(x, y) are twice differentiable, and all functions, derivatives, and second derivatives are bounded, W t and B t are independent m-dimensional Brownian motions, and ε is a small positive parameter. Furthermore, assume that lim y g(x, y) y = for all x and that ττ T is uniformly positive definite. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 6 / 19
Review of Related Work Multiscale Processes Law of Large Numbers [Spiliopoulos, 2014] Consider the Y process with the x variable frozen, that is, dỹt = g(x, Ỹt) dt + τ(x, Ỹt) db t, Ỹ 0 = y 0, where x R n is a fixed parameter. This is associated with the operator L x defined by L x F (y) = F (y)g(x, y) + 1 2 ττt (x, y) : F (y) and the invariant measure µ x (dy). Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 7 / 19
Review of Related Work Multiscale Processes Law of Large Numbers (cont.) Define the averaged drift c(x) = Y c(x, y)µ x (dy) and the averaged process X as the solution to d X t = c( X t ) dt, X0 = x 0. Then as ε 0, X ε t X t in probability. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 8 / 19
Review of Related Work Multiscale Processes Central Limit Theorem Central Limit Theorem As ε 0, 1 ε (X ε t X t ) η t in distribution, where η t is a single scale stochastic process. This is used to estimate probabilities for X ε near the center of the distribution. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 9 / 19
Review of Related Work Large Deviations Principle Definition of Large Deviations Let X ε, ε > 0, be a family of random variables parameterized by ε. X ε satisfies a large deviations principle with rate function I if for every closed set F, lim sup ε log P{X ε F } inf I (x) ε 0 x F and for every open set G, lim inf ε 0 ε log P{X ε G} inf x G I (x). If a family of random variables satisfies a large deviations principle with some rate function, that rate function is unique. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 10 / 19
Review of Related Work Large Deviations Principle The Laplace Principle [Dupuis and Ellis, 1997] The family X ε, ε > 0, satisfies a Laplace principle with rate function I if for every bounded continuous function a, lim ε log E{exp[ 1 ε 0 ε a(x ε )]} = inf{i (x) + a(x)}. x The Laplace principle and the large deviations principle are equivalent. The large deviations principle (or Laplace principle) is used to estimate probabilities in the tail of the distribution. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 11 / 19
Moderate Deviations Principle Definition of Moderate Deviations Let h(ε) be a function such that as ε 0, h(ε) and εh(ε) 0. Consider ηt ε 1 = (X ε εh(ε) t X t ). A family of stochastic processes X ε satisfies a moderate deviations principle with rate function I if for every bounded continuous function a, 1 lim ε 0 h 2 (ε) log E{exp[ h2 (ε)a(η ε )]} = inf{i (x) + a(x)}. x Previous results by [Guillin, 2003] and others. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 12 / 19
Moderate Deviations Principle Poisson Equation Define Φ(x, y) as the unique solution to L x Φ(x, y) = (c(x, y) c(x)), Y Φ(x, y)µ x (dy) = 0. (Existence and uniqueness due to [Pardoux and Veretennikov, 2003].) Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 13 / 19
Moderate Deviations Principle Statement of Theorem Theorem Let X ε and Y ε as previously defined be a multiscale process. Then X ε satisfies a moderate deviations principle with rate function I (φ) = 1 2 1 0 ( φ s r( X s, φ s )) T q 1 ( X s )( φ s r( X s, φ s ))ds if φ C([0, 1]; R n ) is absolutely continuous, and I (φ) = otherwise, where r(x, η) = c(x)η and q(x) = Y [ ] σσ T (x, y) + [ y Φτ] [ y Φτ] T (x, y) µ x (dy). Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 14 / 19
Moderate Deviations Principle Sketch of Proof The starting point for the proof is a stochastic control representation which follows from [Boué and Dupuis, 1998], which yields 1 h 2 (ε) log E [ exp{ h 2 (ε)a(η ε )} ] [ 1 1 ] = inf E u ε (s) 2 ds + a(η ε,uε ) u ε 2 where u ε is a stochastic control and η ε,uε 0 is a controlled process. Then, 1 Prove that (u ε, η ε,uε ) converges in distribution as ε 0. 2 Define the rate function I in terms of the limit of the control u ε, and prove the Laplace principle lower bound, lim inf ε 0 1 h 2 (ε) log E [ exp { h 2 (ε)a(η ε ) }] inf [I (φ) + a(φ)]. φ 3 Rewrite the rate function in the form given in the theorem statement, and use this to prove the Laplace principle upper bound, lim sup 1 ε 0 h 2 (ε) log E [ exp { h 2 (ε)a(η ε ) }] inf [I (φ) + a(φ)]. φ Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 15 / 19
Future Work Future Work 1 Importance sampling schemes for accelerated Monte Carlo simulation based on the stochastic control representation. 2 Statistical inference for multiscale processes. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 16 / 19
The End Thank You Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 17 / 19
Bibliography Bibliography I M. Boué and P. Dupuis, A Variational Representation for Certain Functionals of Brownian Motion, The Annals of Probability, Vol. 26, No. 4 (1998), pp. 1641 1659. P. Dupuis and R.S. Ellis, A Weak Convergence Approach to the Theory of Large Deviations, John Wiley & Sons, New York, 1997. P. Dupuis, K. Spiliopoulos, and H. Wang, Rare Event Simulation for Rough Energy Landscapes, Proceedings of the 2011 Winter Simulation Conference. A. Guillin, Averaging Principle of SDE with Small Diffusion: Moderate Deviations, The Annals of Probability, Vol. 31, No. 1 (2003), pp. 413 443. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 18 / 19
Bibliography Bibliography II E. Pardoux and A.Yu. Veretennikov, On Poisson Equation and Diffusion Approximation 2, The Annals of Probability, Vol. 31, No. 3 (2003), pp. 1166 1192. K. Spiliopoulos, Fluctuation Analysis and Short Time Asymptotics for Multiple Scales Diffusion Processes, Stochastics and Dynamics, Vol. 14, No. 3, (2014), pp. 1350026. Matthew R. Morse (BU) Moderate Deviations for Multiple Scales BU-Keio 2016, August 16 19 / 19