Compactness and large deviations Chiranjib Mukherjee TU Munich - PowerPoint PPT Presentation

Flashback on Donsker-Varadhan theory from the seventies Our theory Compactness and large deviations Chiranjib Mukherjee TU Munich Joint work with S.R. S. Varadhan (New York) Berlin-Padova workshop, October, 2014

Flashback on Donsker-Varadhan theory from the seventies Our theory A weak LDP for occupation measures Rate function is Legendre dual of principle eigenvalue We have a d -dimensional Brownian motion ( β t ) t , d ≥ 2.

Flashback on Donsker-Varadhan theory from the seventies Our theory A weak LDP for occupation measures Rate function is Legendre dual of principle eigenvalue We have a d -dimensional Brownian motion ( β t ) t , d ≥ 2. For a continuous function V in a bounded domain B , expect integrals to grow exponentially: � �� � � t E exp V ( β s ) d s ∼ exp { t λ ( V ) } � �� � 0 � �� � > 0 = t � V ( x ) L t ( d x )

Flashback on Donsker-Varadhan theory from the seventies Our theory A weak LDP for occupation measures Rate function is Legendre dual of principle eigenvalue We have a d -dimensional Brownian motion ( β t ) t , d ≥ 2. For a continuous function V in a bounded domain B , expect integrals to grow exponentially: � �� � � t E exp V ( β s ) d s ∼ exp { t λ ( V ) } � �� � 0 � �� � > 0 = t � V ( x ) L t ( d x ) � t Important object: L t = 1 0 δ β s d s (time spent on Borel sets). t

Flashback on Donsker-Varadhan theory from the seventies Our theory A weak LDP for occupation measures Rate function is Legendre dual of principle eigenvalue We have a d -dimensional Brownian motion ( β t ) t , d ≥ 2. For a continuous function V in a bounded domain B , expect integrals to grow exponentially: � �� � � t E exp V ( β s ) d s ∼ exp { t λ ( V ) } � �� � 0 � �� � > 0 = t � V ( x ) L t ( d x ) � t Important object: L t = 1 0 δ β s d s (time spent on Borel sets). t Equivalently: then, exponential decay of probabilities: � � � � L t ≃ f 2 d x on B − tI ( f 2 ) � f � 2 = 1 , f ∈ H 1 ∼ exp 0 ( B ) P I ( f 2 ) = 1 2 ||∇ f || 2 2 Donsker-varadhan rate function.

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces.

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus).

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope.

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work:

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: Statistical mechanics:

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ),

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ), still exponential growth?? � �� � t � t � 1 exp d r d sV ( β s − β r ) E t 0 0 � �� � = t � � R d V ( x − y ) L t ( d x ) L t ( d y ) R d

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ), still exponential growth?? � �� � t � t � 1 exp d r d sV ( β s − β r ) ∼ exp { t λ } ?? λ > 0 . E t 0 0 � �� � = t � � R d V ( x − y ) L t ( d x ) L t ( d y ) R d

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ), still exponential growth?? � �� � t � t � 1 exp d r d sV ( β s − β r ) ∼ exp { t λ } ?? λ > 0 . E t 0 0 � �� � = t � � R d V ( x − y ) L t ( d x ) L t ( d y ) R d Need: � L t ∼ f 2 d x on R d � � � − tI ( f 2 ) � f � 2 = 1 , f ∈ H 1 ( R d ). P ∼ exp

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ), still exponential growth?? � �� � t � t � 1 exp d r d sV ( β s − β r ) ∼ exp { t λ } ?? λ > 0 . E t 0 0 � �� � = t � � R d V ( x − y ) L t ( d x ) L t ( d y ) R d Need: � L t ∼ f 2 d x on R d � � � − tI ( f 2 ) � f � 2 = 1 , f ∈ H 1 ( R d ). P ∼ exp No full LDP exists, and projection on torus does not save us.

Flashback on Donsker-Varadhan theory from the seventies Our theory Theory suffers from the lack of full LDP Physical problems often need statements on the whole space Crucial: theory needs compact state spaces. Workaround: For non-compact spaces, sometimes, one point compactification of R d is enough (wrap a BM around a torus). Success depends on problem and limited scope. Here is a problem where it does not work: 1 Statistical mechanics: For V ∈ C 0 ( R d ) (think of V ( x ) = | x | ), still exponential growth?? � �� � t � t � 1 exp d r d sV ( β s − β r ) ∼ exp { t λ } ?? λ > 0 . E t 0 0 � �� � = t � � R d V ( x − y ) L t ( d x ) L t ( d y ) R d Need: � L t ∼ f 2 d x on R d � � � − tI ( f 2 ) � f � 2 = 1 , f ∈ H 1 ( R d ). P ∼ exp No full LDP exists, and projection on torus does not save us. Need a robust theory via general compactification.

Flashback on Donsker-Varadhan theory from the seventies Our theory Probability measures are not compact Need to identify regions where mass is accumulated What do we want to compactify?

Flashback on Donsker-Varadhan theory from the seventies Our theory Probability measures are not compact Need to identify regions where mass is accumulated What do we want to compactify? First start with M 1 ( R d ). Not compact under the weak topology.

Flashback on Donsker-Varadhan theory from the seventies Our theory Probability measures are not compact Need to identify regions where mass is accumulated What do we want to compactify? First start with M 1 ( R d ). Not compact under the weak topology. Why?

Flashback on Donsker-Varadhan theory from the seventies Our theory Probability measures are not compact Need to identify regions where mass is accumulated What do we want to compactify? First start with M 1 ( R d ). Not compact under the weak topology. Why? Mass may escape and leak out or spread too flat.