W eierstraß-Institut für Angewandte Analysis und Stochastik Colloquium Equations Diff´ erentielles Stochastiques Toulon, October 20, 2003 Barbara Gentz Large deviations and Wentzell–Freidlin theory Mohrenstr. 39 – 10117 Berlin – Germany gentz@wias-berlin.de www.wias-berlin.de/people/gentz
O u t li n e Large deviations – Introduction – Sample-path large deviations for Brownian motion – Sample-path large deviations for stochastic differential equations Diffusion exit from a domain – Introduction – Relation to PDEs (reminder) – The concept of a quasipotential – Asymptotic behaviour of fi rst-exit times and locations References Slides available at http://www.wias-berlin.de/people/gentz/misc.html Colloq. Equations Diff. Stoch. October 20, 2003 1 (24)
I n t r odu c t i on : Sm a ll r a ndom p e r t u r b a t i on s Consider small random perturbation of ODE (with same initial cond.) We expect for small . Depends on deterministic dynamics noise intensity time scale Colloq. Equations Diff. Stoch. October 20, 2003 2 (24)
I n t r odu c t i on : Sm a ll r a ndom p e r t u r b a t i on s Indeed, for Lipschitz continuous and Gronwall’s lemma shows Remains to estimate : Use re fl ection principle : Reduce to using independence Colloq. Equations Diff. Stoch. October 20, 2003 3 (24)
I n t r odu c t i on : Sm a ll r a ndom p e r t u r b a t i on s For with ( equipped with sup norm ) and as Event is atypical: Occurrence a large deviation Question: Rate of convergence as a function of ? Generally not possible, but exponential rate can be found A i m : Find functional s.t. for Provides local description Colloq. Equations Diff. Stoch. October 20, 2003 4 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : Th e e ndpo i n t Sp ec i a l ca s e : Scaled Brownian motion, Consider endpoint instead of whole path Use Laplace method to evaluate integral as C a u t i on : Limit does not necessarily exist R e m e d y : Use interior and closure Large deviation principle Colloq. Equations Diff. Stoch. October 20, 2003 5 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : S c h il d e r ’ s t h e o r e m S c h il d e r ’ s Th e o r e m (1966) Scaled BM satis fi es a (full) large deviation principle with good rate function if with otherwise That is Rate function: is lower semi-continuous Good rate function: has compact level sets Upper and lower large-deviation bound: for all R e m a r k s In fi nite-dimensional version of Laplace method (almost surely) re fl ects ( ) Colloq. Equations Diff. Stoch. October 20, 2003 6 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : E x a mp l e s E x a mp l e I : Endpoint again . . . ( ) cost to force BM to be in at time Note: Typical spreading of is E x a mp l e II : BM leaving a small ball cost to force BM to leave before E x a mp l e III : BM staying in a cone (similarly . . . ) Colloq. Equations Diff. Stoch. October 20, 2003 7 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : Lo w e r bound To s ho w : Lower bound for open sets for all open L e mm a (local variant of lower bound) for all with , all Lemma lower bound Standard proof of Lemma: uses Cameron–Martin–Girsanov formula C a m e r on – M a r t i n – G i r s a no v f o r mu l a (special case, ) –BM –BM where Colloq. Equations Diff. Stoch. October 20, 2003 8 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : P r oo f o f C a m e r on – M a r t i n – G i r s a no v f o r mu l a F i r s t s t e p are exponential martingales wrt. S ec ond s t e p is –independent of increments are independent Increments are Gaussian is BM with respect to Colloq. Equations Diff. Stoch. October 20, 2003 9 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : P r oo f o f t h e l o w e r bound , , with , Estimate integral by Jensen’s inequality Finally note Colloq. Equations Diff. Stoch. October 20, 2003 10 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : A pp r o x i m a t i on b y po l y gon s ( upp e r bound ) Approximate by the random polygon joining To s ho w : is a good approximation to (standard estimate) Difference is negligible: for all Colloq. Equations Diff. Stoch. October 20, 2003 11 (24)
L a r g e d e v i a t i on s f o r B r o w n i a n mo t i on : P r oo f o f t h e upp e r bound closed, , , negligible term being a polygon yields ( i.i.d.) By Chebychev’s inequality, for being arbitrary and the lower semi-continuity of show Colloq. Equations Diff. Stoch. October 20, 2003 12 (24)
L a r g e d e v i a t i on s f o r s o l u t i on s o f SD E s : Sp ec i a l ca s e ( Lipschitz, bounded growth, identity matrix ) De fi ne by , being the unique solution in to is continuous (use Gronwall’s lemma) De fi ne by C on t r ac t i on p r i n c i p l e (trivial version) good rate fct, governing LDP for good rate fct, governing LDP for if with Identify : otherwise Colloq. Equations Diff. Stoch. October 20, 2003 13 (24)
L a r g e d e v i a t i on s f o r s o l u t i on s o f SD E s : G e n e r a l ca s e A ss ump t i on s , Lipschitz continuous bounded growth: , ellipticity: Th e o r e m (Wentzell–Freidlin) satis fi es a LDP with good rate function if with otherwise R e m a r k If is only positive semi-de fi nite: LDP remains valid with good rate function but identi fi cation of may fail; Colloq. Equations Diff. Stoch. October 20, 2003 14 (24)
L a r g e d e v i a t i on s f o r s o l u t i on s o f SD E s : S ke t c h o f t h e p r oo f f o r t h e g e n e r a l ca s e Dif fi culty: Cannot apply contraction principle directly Introduce Euler approximations Schilder’s Theorem and contraction principle imply LDP for with good rate function if with otherwise To show: (1) is a good approximation to (not dif fi cult but tedious, uses It ˆ o’s formula) (2) approximates : for all Colloq. Equations Diff. Stoch. October 20, 2003 15 (24)
L a r g e d e v i a t i on s f o r s o l u t i on s o f SD E s : V a r a dh a n ’ s L e mm a A ss ump t i on s continuous Tail condition Th e o r e m (Varadhan’s Lemma) R e m a r k s Moment condition for some implies tail condition. In fi nite-dimensional analogue of Laplace method Holds in great generality — as long as satis fi es a LDP with a good rate function Colloq. Equations Diff. Stoch. October 20, 2003 16 (24)
D i ff u s i on e x i t f r om a dom a i n : I n t r odu c t i on No i s e - i ndu ce d e x i t from a domain (bounded, open, smooth boundary) Consider small random perturbation of ODE (with same initial cond.) First-exit time Q u e s t i on s Does leave ? If so: When and where? Expected time of fi rst exit? Concentration of fi rst-exit time and location? To w a r d s a n s w e r s If leaves , so will . Use LDP to estimate deviation . Later on: Assume does not leave . Study noise-induced exit. Colloq. Equations Diff. Stoch. October 20, 2003 17 (24)
D i ff u s i on e x i t f r om a dom a i n : R e l a t i on t o P D E s A ss ump t i on s (from now on) , Lipschitz cont., bounded growth (uniform ellipticity) bounded domain, smooth boundary In fi nitesimal generator of diffusion Th e o r e m For continuous in is the unique solution of the PDE on in is the unique solution of the PDE on R e m a r k s Information on fi rst-exit times and exit locations can be obtained exactly from PDEs In principle . . . Smoothness assumption for can be relaxed to “exterior-ball condition” Colloq. Equations Diff. Stoch. October 20, 2003 18 (24)
D i ff u s i on e x i t f r om a dom a i n : A n e x a mp l e O v e r d a mp e d mo t i on o f a B r o w n i a n p a r t i c l e i n a s i ng l e - w e ll po t e n t i a l , potential deriving from , , for , Drift pushes particle towards bottom Probability of leaving ? Solve the (one-dimensional) Dirichlet problem in for with on for if if if Colloq. Equations Diff. Stoch. October 20, 2003 19 (24)
D i ff u s i on e x i t f r om a dom a i n : A fi r s t r e s u l t C o r o ll a r y (to LDP for ) where such that and cost of forcing a path to connect and in time R e m a r k s Upper and lower LDP bounds coincide limit exists Calculation of asymptotical behaviour reduces to variational problem is solution to a Hamilton–Jacobi equation; extremals solution to an Euler equation Colloq. Equations Diff. Stoch. October 20, 2003 20 (24)
D i ff u s i on e x i t f r om a dom a i n : A ss ump t i on s a nd t h e c on ce p t o f qu a s i po t e n t i a l s A ss ump t i on s has a unique stable equilibrium point in , is asymptotically stable is contained in the basin of attraction of (for the deterministic dynamics) with quasipotential cost of forcing a path starting in to reach eventually R e m a r k s Similar if contains for instance a stable periodic orbit Conditions exclude characteristic boundary Uniform-ellipticity condition can be relaxed; requires additional controllability condition Were , all possible exit points would be equally unlikely If derives from a potential , : Quasipotential satis fi es for all such that A rr h e n i u s l a w : For deriving from a potential, The average time to leave potential well is twice the barrier height noise intensity Colloq. Equations Diff. Stoch. October 20, 2003 21 (24)
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