Non-Archimedean White Noise, Pseudodierential Stochastic Equations, - PowerPoint PPT Presentation

Non-Archimedean White Noise, Pseudodi¤erential Stochastic Equations, and Massive Euclidean Fields. W. A. Zúñiga-Galindo The Center for Research and Advanced Studies of the National Polytechnic Institute, Mexico . p-ADICS.2015, 07-12.09.2015, Belgrade, Serbia W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 1 / 51

Abstract W. A. Zúñiga-Galindo, Non-Archimedean White Noise, Pseudodi¤erential Stochastic Equations of Klein-Gordon Type, and Massive Euclidean Fields, arXiv:1501.00707. We construct p -adic Euclidean random …elds Φ over Q N p , for arbitrary N , these …elds are solutions of p -adic stochastic pseudodi¤erential equations. From a mathematical perspective, the Euclidean …elds are generalized stochastic processes parametrized by functions belonging to a nuclear countably Hilbert space, these spaces are introduced in this article, in addition, the Euclidean …elds are invariant under the action of certain group of transformations. We also study the Schwinger functions of Φ . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 2 / 51

Presentation Plan Notation Some comments on the Archimedean case A new class of non-Archimedean nuclear spaces Non-Archimedean white noise Euclidean random …elds as convoluted generalized white noise Schwinger Functions Final comments W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 3 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . p , I set k x k p : = max i j x i j p = p � ord ( x ) , where x = ( x 1 , . . . , x n ) 2 Q N ord ( x ) : = min i ord ( x i ) . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . p , I set k x k p : = max i j x i j p = p � ord ( x ) , where x = ( x 1 , . . . , x n ) 2 Q N ord ( x ) : = min i ord ( x i ) . n p ; k x � a k p � p γ o B N x 2 Q n γ ( a ) : = . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . p , I set k x k p : = max i j x i j p = p � ord ( x ) , where x = ( x 1 , . . . , x n ) 2 Q N ord ( x ) : = min i ord ( x i ) . n p ; k x � a k p � p γ o B N x 2 Q n γ ( a ) : = . I work with functions from Q N p to C . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . p , I set k x k p : = max i j x i j p = p � ord ( x ) , where x = ( x 1 , . . . , x n ) 2 Q N ord ( x ) : = min i ord ( x i ) . n p ; k x � a k p � p γ o B N x 2 Q n γ ( a ) : = . I work with functions from Q N p to C . D ( Q N p ) denotes the Bruhat-Schwartz space. W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation j x j p denotes the p -adic absolute value of x 2 Q p . p , I set k x k p : = max i j x i j p = p � ord ( x ) , where x = ( x 1 , . . . , x n ) 2 Q N ord ( x ) : = min i ord ( x i ) . n p ; k x � a k p � p γ o B N x 2 Q n γ ( a ) : = . I work with functions from Q N p to C . D ( Q N p ) denotes the Bruhat-Schwartz space. D 0 ( Q N p ) denotes the space of distributions. W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation Set χ p ( y ) = exp ( 2 π i f y g p ) for y 2 Q p . The map χ p ( � ) is an additive character on Q p , i.e. a continuos map from Q p into the unit circle satisfying χ p ( y 0 + y 1 ) = χ p ( y 0 ) χ p ( y 1 ) , y 0 , y 1 2 Q p . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation Set χ p ( y ) = exp ( 2 π i f y g p ) for y 2 Q p . The map χ p ( � ) is an additive character on Q p , i.e. a continuos map from Q p into the unit circle satisfying χ p ( y 0 + y 1 ) = χ p ( y 0 ) χ p ( y 1 ) , y 0 , y 1 2 Q p . Let B ( x , y ) be a symmetric non-degenerate Q p � bilinear form on Q N p � Q N p . Thus q ( x ) : = B ( x , x ) , x 2 Q N p is a non-degenerate quadratic form on Q N p . We recall that B ( x , y ) = 1 2 f q ( x + y ) � q ( x ) � q ( y ) g . (1) W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation Set χ p ( y ) = exp ( 2 π i f y g p ) for y 2 Q p . The map χ p ( � ) is an additive character on Q p , i.e. a continuos map from Q p into the unit circle satisfying χ p ( y 0 + y 1 ) = χ p ( y 0 ) χ p ( y 1 ) , y 0 , y 1 2 Q p . Let B ( x , y ) be a symmetric non-degenerate Q p � bilinear form on Q N p � Q N p . Thus q ( x ) : = B ( x , x ) , x 2 Q N p is a non-degenerate quadratic form on Q N p . We recall that B ( x , y ) = 1 2 f q ( x + y ) � q ( x ) � q ( y ) g . (1) Example: B ( x , y ) = ∑ i x i y i . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation Set χ p ( y ) = exp ( 2 π i f y g p ) for y 2 Q p . The map χ p ( � ) is an additive character on Q p , i.e. a continuos map from Q p into the unit circle satisfying χ p ( y 0 + y 1 ) = χ p ( y 0 ) χ p ( y 1 ) , y 0 , y 1 2 Q p . Let B ( x , y ) be a symmetric non-degenerate Q p � bilinear form on Q N p � Q N p . Thus q ( x ) : = B ( x , x ) , x 2 Q N p is a non-degenerate quadratic form on Q N p . We recall that B ( x , y ) = 1 2 f q ( x + y ) � q ( x ) � q ( y ) g . (1) Example: B ( x , y ) = ∑ i x i y i . � � � by We identify the Q p -vector space Q N Q N p with its algebraic dual p means of B ( � , � ) . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation � � Q N p , + We now identify the dual group (i.e. the Pontryagin dual) of � � � by taking x � ( x ) = χ p ( B ( x , x � )) . Q N with p W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation � � Q N p , + We now identify the dual group (i.e. the Pontryagin dual) of � � � by taking x � ( x ) = χ p ( B ( x , x � )) . Q N with p The Fourier transform is de…ned by Z for g 2 L 1 , ( F g )( ξ ) = g ( x ) χ p ( B ( x , ξ )) d µ ( x ) , Q N p where d µ ( x ) is a Haar measure on Q N p . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation � � Q N p , + We now identify the dual group (i.e. the Pontryagin dual) of � � � by taking x � ( x ) = χ p ( B ( x , x � )) . Q N with p The Fourier transform is de…ned by Z for g 2 L 1 , ( F g )( ξ ) = g ( x ) χ p ( B ( x , ξ )) d µ ( x ) , Q N p where d µ ( x ) is a Haar measure on Q N p . � � be the space of continuous functions g in L 1 whose Q N Let L p Fourier transform F g is in L 1 . W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation � � Q N p , + We now identify the dual group (i.e. the Pontryagin dual) of � � � by taking x � ( x ) = χ p ( B ( x , x � )) . Q N with p The Fourier transform is de…ned by Z for g 2 L 1 , ( F g )( ξ ) = g ( x ) χ p ( B ( x , ξ )) d µ ( x ) , Q N p where d µ ( x ) is a Haar measure on Q N p . � � be the space of continuous functions g in L 1 whose Q N Let L p Fourier transform F g is in L 1 . The measure d µ ( x ) can be normalized uniquely in such manner that � � Q N ( F ( F g ))( x ) = g ( � x ) for every g belonging to L . Notice that p d µ ( x ) = C ( q ) d N x where C ( q ) is a positive constant and d N x is the Haar measure on Q N p normalized by the condition vol ( B N 0 ) = 1. W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Some comments on the Archimedean case A program of constructing Euclidean random …elds of Markovian type by solving pseudo-stochastic partial di¤erential equations of the form LX = F with F a Euclidean noise and L a suitable invariant pseudodi¤erential operator was started in in the 70’s by D. Surgailis and S. Albeverio and R. Høegh-Krohn, among others. W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 7 / 51

Some comments on the Archimedean case A program of constructing Euclidean random …elds of Markovian type by solving pseudo-stochastic partial di¤erential equations of the form LX = F with F a Euclidean noise and L a suitable invariant pseudodi¤erential operator was started in in the 70’s by D. Surgailis and S. Albeverio and R. Høegh-Krohn, among others. Albeverio and Wu studied random …elds of the form X = G � F , � � ∆ + m 2 � � α for covering in particular the case in which G = α 2 ( 0 , 1 ) and m � 0. W. A. Zúñiga-Galindo (CINVESTAV) p-adic Massive Euclidean Fields p-ADICS.2015 7 / 51