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 QN

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

jxjp denotes the p-adic absolute value of x 2 Qp.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

jxjp denotes the p-adic absolute value of x 2 Qp. x = (x1, . . . , xn) 2 QN

p , I set kxkp := maxi jxijp = pord(x), where

• rd(x) := mini ord (xi).
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

jxjp denotes the p-adic absolute value of x 2 Qp. x = (x1, . . . , xn) 2 QN

p , I set kxkp := maxi jxijp = pord(x), where

• rd(x) := mini ord (xi).

BN

γ (a) :=

n x 2 Qn

p; kx akp pγo

.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

jxjp denotes the p-adic absolute value of x 2 Qp. x = (x1, . . . , xn) 2 QN

p , I set kxkp := maxi jxijp = pord(x), where

• rd(x) := mini ord (xi).

BN

γ (a) :=

n x 2 Qn

p; kx akp pγo

. I work with functions from QN

p to C.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

jxjp denotes the p-adic absolute value of x 2 Qp. x = (x1, . . . , xn) 2 QN

p , I set kxkp := maxi jxijp = pord(x), where

• rd(x) := mini ord (xi).

BN

γ (a) :=

n x 2 Qn

p; kx akp pγo

. I work with functions from QN

p to C.

D(QN

p ) denotes the Bruhat-Schwartz space.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

jxjp denotes the p-adic absolute value of x 2 Qp. x = (x1, . . . , xn) 2 QN

p , I set kxkp := maxi jxijp = pord(x), where

• rd(x) := mini ord (xi).

BN

γ (a) :=

n x 2 Qn

p; kx akp pγo

. I work with functions from QN

p to C.

D(QN

p ) denotes the Bruhat-Schwartz space.

D0(QN

p ) denotes the space of distributions.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 4 / 51

Notation

Set χp(y) = exp(2πifygp) for y 2 Qp. The map χp() is an additive character on Qp, i.e. a continuos map from Qp into the unit circle satisfying χp(y0 + y1) = χp(y0)χp(y1), y0, y1 2 Qp.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation

Set χp(y) = exp(2πifygp) for y 2 Qp. The map χp() is an additive character on Qp, i.e. a continuos map from Qp into the unit circle satisfying χp(y0 + y1) = χp(y0)χp(y1), y0, y1 2 Qp. Let B (x, y) be a symmetric non-degenerate Qpbilinear form on QN

p QN p . Thus q(x) := B (x, x), x 2 QN p is a non-degenerate

quadratic form on QN

p . We recall that

B (x, y) = 1 2 fq(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 χp(y) = exp(2πifygp) for y 2 Qp. The map χp() is an additive character on Qp, i.e. a continuos map from Qp into the unit circle satisfying χp(y0 + y1) = χp(y0)χp(y1), y0, y1 2 Qp. Let B (x, y) be a symmetric non-degenerate Qpbilinear form on QN

p QN p . Thus q(x) := B (x, x), x 2 QN p is a non-degenerate

quadratic form on QN

p . We recall that

B (x, y) = 1 2 fq(x + y) q(x) q(y)g . (1) Example: B (x, y) = ∑i xiyi.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation

Set χp(y) = exp(2πifygp) for y 2 Qp. The map χp() is an additive character on Qp, i.e. a continuos map from Qp into the unit circle satisfying χp(y0 + y1) = χp(y0)χp(y1), y0, y1 2 Qp. Let B (x, y) be a symmetric non-degenerate Qpbilinear form on QN

p QN p . Thus q(x) := B (x, x), x 2 QN p is a non-degenerate

quadratic form on QN

p . We recall that

B (x, y) = 1 2 fq(x + y) q(x) q(y)g . (1) Example: B (x, y) = ∑i xiyi. We identify the Qp-vector space QN

p with its algebraic dual

• QN

p

by means of B (, ).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 5 / 51

Notation

We now identify the dual group (i.e. the Pontryagin dual) of

• QN

p , +

• with
• QN

p

by taking x (x) = χp (B (x, x)).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation

We now identify the dual group (i.e. the Pontryagin dual) of

• QN

p , +

• with
• QN

p

by taking x (x) = χp (B (x, x)). The Fourier transform is de…ned by (Fg)(ξ) =

Z

QN

p

g (x) χp (B (x, ξ)) dµ (x) , for g 2 L1, where dµ (x) is a Haar measure on QN

p .

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation

We now identify the dual group (i.e. the Pontryagin dual) of

• QN

p , +

• with
• QN

p

by taking x (x) = χp (B (x, x)). The Fourier transform is de…ned by (Fg)(ξ) =

Z

QN

p

g (x) χp (B (x, ξ)) dµ (x) , for g 2 L1, where dµ (x) is a Haar measure on QN

p .

Let L

• QN

p

• be the space of continuous functions g in L1 whose

Fourier transform Fg is in L1.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 6 / 51

Notation

We now identify the dual group (i.e. the Pontryagin dual) of

• QN

p , +

• with
• QN

p

by taking x (x) = χp (B (x, x)). The Fourier transform is de…ned by (Fg)(ξ) =

Z

QN

p

g (x) χp (B (x, ξ)) dµ (x) , for g 2 L1, where dµ (x) is a Haar measure on QN

p .

Let L

• QN

p

• be the space of continuous functions g in L1 whose

Fourier transform Fg is in L1. The measure dµ (x) can be normalized uniquely in such manner that (F(Fg))(x) = g(x) for every g belonging to L

• QN

p

• . Notice that

dµ (x) = C(q)dNx where C(q) is a positive constant and dNx is the Haar measure on QN

p normalized by the condition vol(BN 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 , covering in particular the case in which G = ∆ + m2α for α 2 (0, 1) and m 0.

• 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 , covering in particular the case in which G = ∆ + m2α for α 2 (0, 1) and m 0. ∆ + m2α X = F for α 2 (0, 1) and m 0.

• 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 , covering in particular the case in which G = ∆ + m2α for α 2 (0, 1) and m 0. ∆ + m2α X = F for α 2 (0, 1) and m 0. For α = 1

2, F the Gaussian white noise, X is the Nelson’s Euclidean

free …eld over Rd.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 7 / 51

Some comments on the Archimedean case

Albeverio, Sergio; Wu, Jiang Lun Euclidean random …elds obtained by convolution from generalized white noise. J. Math. Phys. 36 (1995),

• no. 10, 5217–5245.

Albeverio, Sergio; Gottschalk, Hanno; Wu, Jiang-Lun Convoluted generalized white noise, Schwinger functions and their analytic continuation to Wightman functions. Rev. Math. Phys. 8 (1996), no. 6, 763–817.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 8 / 51

A true non-Archimedean analog of the Schwartz space

The p-adic analog should have the form

• Lα + m2

Φ = z, where Lα is a p-adic Laplacian (an elliptic pseudodi¤erential

• perator), the mass m is a positive real number, and Φ (f , T) is the

Euclidean quantum …eld ( which is a generalized random process parametrized by functions f belonging to a suitable space X and T 2 X

, T plays the role of the ω.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 9 / 51

A true non-Archimedean analog of the Schwartz space

The p-adic analog should have the form

• Lα + m2

Φ = z, where Lα is a p-adic Laplacian (an elliptic pseudodi¤erential

• perator), the mass m is a positive real number, and Φ (f , T) is the

Euclidean quantum …eld ( which is a generalized random process parametrized by functions f belonging to a suitable space X and T 2 X

, T plays the role of the ω.

What is the meaning of

• Lα + m2

Φ (f , T)?

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 9 / 51

A true non-Archimedean analog of the Schwartz space

The p-adic analog should have the form

• Lα + m2

Φ = z, where Lα is a p-adic Laplacian (an elliptic pseudodi¤erential

• perator), the mass m is a positive real number, and Φ (f , T) is the

Euclidean quantum …eld ( which is a generalized random process parametrized by functions f belonging to a suitable space X and T 2 X

, T plays the role of the ω.

What is the meaning of

• Lα + m2

Φ (f , T)?

• Lα + m2

Φ (f , T) := Φ

• Lα + m2

f , T

• , for f 2 X, T 2 X

.

Then

• Lα + m2

X X.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 9 / 51

A true non-Archimedean analog of the Schwartz space

The p-adic analog should have the form

• Lα + m2

Φ = z, where Lα is a p-adic Laplacian (an elliptic pseudodi¤erential

• perator), the mass m is a positive real number, and Φ (f , T) is the

Euclidean quantum …eld ( which is a generalized random process parametrized by functions f belonging to a suitable space X and T 2 X

, T plays the role of the ω.

What is the meaning of

• Lα + m2

Φ (f , T)?

• Lα + m2

Φ (f , T) := Φ

• Lα + m2

f , T

• , for f 2 X, T 2 X

.

Then

• Lα + m2

X X. Take Lα = Dα be the Vladimirov operator and m > 0, then D (Qp) is not invariant under the action of Dα + m2!

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 9 / 51

A true non-Archimedean analog of the Schwartz space

S(RN) is a nuclear space (we can use the Bochner-Minlos Theorem)

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 10 / 51

A true non-Archimedean analog of the Schwartz space

S(RN) is a nuclear space (we can use the Bochner-Minlos Theorem) S(RN) is invariant under the action of di¤erential operators with polynomial coe¢cients.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 10 / 51

A true non-Archimedean analog of the Schwartz space

S(RN) is a nuclear space (we can use the Bochner-Minlos Theorem) S(RN) is invariant under the action of di¤erential operators with polynomial coe¢cients. D(QN

p ) is a nuclear space, but D(QN p ) is not invariant under the

action of pseudodi¤erential operators with “polynomial symbols”.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 10 / 51

A true non-Archimedean analog of the Schwartz space

S(RN) is a nuclear space (we can use the Bochner-Minlos Theorem) S(RN) is invariant under the action of di¤erential operators with polynomial coe¢cients. D(QN

p ) is a nuclear space, but D(QN p ) is not invariant under the

action of pseudodi¤erential operators with “polynomial symbols”. We construct an space HC (∞), which is nuclear and invariant under the action of a large class of pseudodi¤erential operators. In addition, it contains a dense copy of D(QN

p ).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 10 / 51

Symmetries Archimedean vs. non-Archimedean

The Green function of ∆ + m2α is G(x) = F 1

ξ!x

B @ 1 h jξj2 + m2 iα 1 C A in S0(RN) =

Z

RN

exp (2πiξ x) h jξj2 + m2 iα dNξ (formally), where jξj2 = ∑N

i=1 ξ2 i and ξ x = ∑N i=1 ξixi.

The quadratic form attache to ξ x is exactly jξj2.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 11 / 51

Symmetries Archimedean vs. non-Archimedean

If g 2 GLN is a symmetry of G(x), then gT Eg = E, where E is the identity matrix. Then the group of symmetries of G(x) is O(N) =

• g 2 GLN; gT Eg = E
• (because if g preserves jξj2 then g

preserves ξ x.)

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 12 / 51

Symmetries Archimedean vs. non-Archimedean

If g 2 GLN is a symmetry of G(x), then gT Eg = E, where E is the identity matrix. Then the group of symmetries of G(x) is O(N) =

• g 2 GLN; gT Eg = E
• (because if g preserves jξj2 then g

preserves ξ x.) Notice that any symmetry of G(x y) has the form x0 + g, with x0 2 RN and g 2 O(N).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 12 / 51

Symmetries Archimedean vs. non-Archimedean

If g 2 GLN is a symmetry of G(x), then gT Eg = E, where E is the identity matrix. Then the group of symmetries of G(x) is O(N) =

• g 2 GLN; gT Eg = E
• (because if g preserves jξj2 then g

preserves ξ x.) Notice that any symmetry of G(x y) has the form x0 + g, with x0 2 RN and g 2 O(N). Which is the p-adic counterpart of ∆?

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 12 / 51

Symmetries Archimedean vs. non-Archimedean

If g 2 GLN is a symmetry of G(x), then gT Eg = E, where E is the identity matrix. Then the group of symmetries of G(x) is O(N) =

• g 2 GLN; gT Eg = E
• (because if g preserves jξj2 then g

preserves ξ x.) Notice that any symmetry of G(x y) has the form x0 + g, with x0 2 RN and g 2 O(N). Which is the p-adic counterpart of ∆? (Lγϕ) (x) = F 1

ξ!x(jl(ξ)jγ p Fx!ξ ϕ) where γ > 0 and

l(ξ) 2 Qp [ξ1, , ξN] satis…es l(ξ) = 0 , ξ = 0.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 12 / 51

Symmetries Archimedean vs. non-Archimedean

If g 2 GLN is a symmetry of G(x), then gT Eg = E, where E is the identity matrix. Then the group of symmetries of G(x) is O(N) =

• g 2 GLN; gT Eg = E
• (because if g preserves jξj2 then g

preserves ξ x.) Notice that any symmetry of G(x y) has the form x0 + g, with x0 2 RN and g 2 O(N). Which is the p-adic counterpart of ∆? (Lγϕ) (x) = F 1

ξ!x(jl(ξ)jγ p Fx!ξ ϕ) where γ > 0 and

l(ξ) 2 Qp [ξ1, , ξN] satis…es l(ξ) = 0 , ξ = 0. Lγ is an elliptic operator.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 12 / 51

Symmetries Archimedean vs. non-Archimedean

In the case N = 4 there is a unique elliptic quadratic form, up to linear equivalence, which is l4 (ξ) = ξ2

1 sξ2 2 pξ2 3 + sξ2 4, where

s 2 Zr f0g is a quadratic non-residue, i.e.

• s

p

• = 1.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 13 / 51

Symmetries Archimedean vs. non-Archimedean

In the case N = 4 there is a unique elliptic quadratic form, up to linear equivalence, which is l4 (ξ) = ξ2

1 sξ2 2 pξ2 3 + sξ2 4, where

s 2 Zr f0g is a quadratic non-residue, i.e.

• s

p

• = 1.

The quadratic form q(ξ) = ξ2

1 + ξ2 2 + ξ2 3 + ξ2 4 is NOT an elliptic

quadratic form

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 13 / 51

Symmetries Archimedean vs. non-Archimedean

In the case N = 4 there is a unique elliptic quadratic form, up to linear equivalence, which is l4 (ξ) = ξ2

1 sξ2 2 pξ2 3 + sξ2 4, where

s 2 Zr f0g is a quadratic non-residue, i.e.

• s

p

• = 1.

The quadratic form q(ξ) = ξ2

1 + ξ2 2 + ξ2 3 + ξ2 4 is NOT an elliptic

quadratic form The matrix of q(ξ) is E and the matrix of l4 (ξ) is diag [1, s, p, s]. If q(ξ) is elliptic there is a non-singular matrix g such that g 1diag [1, s, p, s] g = E, then taking determinants s2p = 1 in Qp!.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 13 / 51

Symmetries Archimedean vs. non-Archimedean

In the case N = 4 there is a unique elliptic quadratic form, up to linear equivalence, which is l4 (ξ) = ξ2

1 sξ2 2 pξ2 3 + sξ2 4, where

s 2 Zr f0g is a quadratic non-residue, i.e.

• s

p

• = 1.

The quadratic form q(ξ) = ξ2

1 + ξ2 2 + ξ2 3 + ξ2 4 is NOT an elliptic

quadratic form The matrix of q(ξ) is E and the matrix of l4 (ξ) is diag [1, s, p, s]. If q(ξ) is elliptic there is a non-singular matrix g such that g 1diag [1, s, p, s] g = E, then taking determinants s2p = 1 in Qp!. Which are the p-adic analogs of ∆ + m2α?

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 13 / 51

Symmetries Archimedean vs. non-Archimedean

There are two choices: F 1

ξ!x(

h jl(ξ)jα

p + m2i

Fx!ξ ϕ) (A) F 1

ξ!x(

h jl(ξ)jp + m2iα Fx!ξ ϕ) (B). I did not …nd any substantial di¤erence between them, I chosen the

• ption (A).
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 14 / 51

Symmetries Archimedean vs. non-Archimedean

There are two choices: F 1

ξ!x(

h jl(ξ)jα

p + m2i

Fx!ξ ϕ) (A) F 1

ξ!x(

h jl(ξ)jp + m2iα Fx!ξ ϕ) (B). I did not …nd any substantial di¤erence between them, I chosen the

• ption (A).
• Lα + m2

ϕ = F 1

ξ!x(

h jl(ξ)jα

p + m2i

Fx!ξ ϕ).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 14 / 51

Symmetries Archimedean vs. non-Archimedean

The Green function of Lα + m2, using the classical de…nition for Fourier transform, G(x) = F 1

ξ!x

1 jl(ξ)jα

p + m2

! in D0(QN

p )

=

Z

QN

p

χp (ξ x) jl(ξ)jα

p + m2 dNξ

(formally), where ξ x = ∑N

i=1 ξixi.

Now there are two di¤erent forms: ∑N

i=1 ξ2 i and l(ξ). The group of

symmetries preserving simultaneously both forms can be trivial or very small.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 15 / 51

Symmetries Archimedean vs. non-Archimedean

Consider the case N = 2, q(ξ) = ξ2

1 + ξ2 2, the quadratic form

attached to ξ x = ∑2

i=1 ξixi, and l(ξ) = ξ2 1 τξ2 2 with

τ 2 Qp r f0g, τ 6= 1, τ is not a square in Qp r f0g.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 16 / 51

Symmetries Archimedean vs. non-Archimedean

Consider the case N = 2, q(ξ) = ξ2

1 + ξ2 2, the quadratic form

attached to ξ x = ∑2

i=1 ξixi, and l(ξ) = ξ2 1 τξ2 2 with

τ 2 Qp r f0g, τ 6= 1, τ is not a square in Qp r f0g. We determine the group of transformations preserving simultaneously q(ξ) and l(ξ).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 16 / 51

Symmetries Archimedean vs. non-Archimedean

Consider the case N = 2, q(ξ) = ξ2

1 + ξ2 2, the quadratic form

attached to ξ x = ∑2

i=1 ξixi, and l(ξ) = ξ2 1 τξ2 2 with

τ 2 Qp r f0g, τ 6= 1, τ is not a square in Qp r f0g. We determine the group of transformations preserving simultaneously q(ξ) and l(ξ). q(ξ) = ξT Eξ where we are identifying ξ = ξ1 ξ2

• and E is the

identity matrix 2 2.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 16 / 51

Symmetries Archimedean vs. non-Archimedean

Consider the case N = 2, q(ξ) = ξ2

1 + ξ2 2, the quadratic form

attached to ξ x = ∑2

i=1 ξixi, and l(ξ) = ξ2 1 τξ2 2 with

τ 2 Qp r f0g, τ 6= 1, τ is not a square in Qp r f0g. We determine the group of transformations preserving simultaneously q(ξ) and l(ξ). q(ξ) = ξT Eξ where we are identifying ξ = ξ1 ξ2

• and E is the

identity matrix 2 2. We are looking for transformations g = g11 g12 g21 g22

• 2 GL2 (Qp)

satisfying:

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 16 / 51

Symmetries Archimedean vs. non-Archimedean

gT Eg = E , gT g = E , gT = g 1 gT 1 τ

• g =

1 τ

• ,

1 τ

• g = g

1 τ

• g11

g12 τg21 τg22

• =

g11 τg12 g21 τg22

• g12 = τg12 , (1 + τ) g12 = 0 , g12 = 0 because τ 6= 1. Similarly

g21 = g22 = 0, thus g = g11

• which is a singular matrix. Hence in

this case the group of symmetries is trivial.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 17 / 51

Symmetries Archimedean vs. non-Archimedean

For this reason we have to work with more general Fourier transforms, so that G(x) = F 1

ξ!x

1 jl(ξ)jα

p + m2

! in D0(QN

p )

=

Z

QN

p

χp (B (x, ξ)) dµ (x) jl(ξ)jα

p + m2

dNξ (formally), may have a chance of having a non-trivial group of symmetries. Of course, it is necessary that the quadratic form B (ξ, ξ) and the form l(ξ) are related nicely in order to have a non-trivial group of symmetries.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 18 / 51

A true non-Archimedean analog of the Schwartz space

We set R+ := fx 2 R : x 0g. We denote by N the set of non-negative integers.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 19 / 51

A true non-Archimedean analog of the Schwartz space

We set R+ := fx 2 R : x 0g. We denote by N the set of non-negative integers. We de…ne for f , g in DR(QN

p ) (or in DC(QN p )) the following scalar

product: hf , gil,α := hf , gil = R

QN

p

h max

• 1, kξkp

i2αl b f (ξ) b g (ξ) dNξ, (2) for a …xed α 2 R+r f0g and l 2 Z.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 19 / 51

A true non-Archimedean analog of the Schwartz space

We set R+ := fx 2 R : x 0g. We denote by N the set of non-negative integers. We de…ne for f , g in DR(QN

p ) (or in DC(QN p )) the following scalar

product: hf , gil,α := hf , gil = R

QN

p

h max

• 1, kξkp

i2αl b f (ξ) b g (ξ) dNξ, (2) for a …xed α 2 R+r f0g and l 2 Z. We also set kf k2

l,α =: kf k2 l = hf , f il. Notice that kkm kkn for

m n. Let denote by HR

• QN

p ; l, α

=: HR (l) the completion of DR(QN

p ) with respect to h, il.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 19 / 51

A true non-Archimedean analog of the Schwartz space

Then HR (n) HR (m) for m n.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 20 / 51

A true non-Archimedean analog of the Schwartz space

Then HR (n) HR (m) for m n. We set HR

• QN

p ; α, ∞

• := HR
• QN

p ; ∞

• := HR (∞) = T

l2N

HR (l) .

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 20 / 51

A true non-Archimedean analog of the Schwartz space

Then HR (n) HR (m) for m n. We set HR

• QN

p ; α, ∞

• := HR
• QN

p ; ∞

• := HR (∞) = T

l2N

HR (l) . Notice that HR (0) = L2

R and that HR (∞) L2

• R. With the topology

induced by the family of seminorms kkl2N, HR (∞) becomes a locally convex space, which is metrizable. Indeed, d (f , g) := max

l2N

• 2l

kf gkl 1 + kf gkl

• is a metric for the topology of HR (∞) considered as a convex

topological space.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 20 / 51

A true non-Archimedean analog of the Schwartz space

Remark

We denote by HC (l), HC (∞) the C-vector spaces constructed from DC(QN

p ). All the above results are valid for these spaces. We shall use d

to denote the metric of HC (∞).

Lemma

HR (∞) endowed with the topology τP is a countably Hilbert space in the sense of Gel’fand and Vilenkin. Furthermore (HR (∞) , τP) is metrizable and complete and hence a Fréchet space.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 21 / 51

A true non-Archimedean analog of the Schwartz space

Lemma

(i) Set DR(QN

p ), d

• for the completion of the metric space

DR(QN

p ), d

• . Then

DR(QN

p ), d

= (HR (∞) , d). (ii) (HR (∞) , d) is a nuclear space.

Proof.

(i) it is established by an argument based on sequences. (ii) We recall that DC(QN

p ) is a nuclear space, and thus DR(QN p ) is a

nuclear space, since any subspace of a nuclear space is also nuclear. Now, since the completion of a nuclear space is also nuclear, by (i), HR (∞) is a nuclear space.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 22 / 51

A true non-Archimedean analog of the Schwartz space

Theorem

HR (∞) is nuclear countably Hilbert space.

Remark

(i) As a nuclear Fréchet space HR (∞) admits a sequence of de…ning Hilbertian norms jjm2N such that (1) jgjm Cm jgjm+1, g 2 HR (∞), with some Cm > 0; (2) the canonical map in,n+1 : HR (n + 1) ! HR (n) is

• f Hilbert-Schmidt type, where HR (n) is the Hilbert space associated with

jjn. (ii) Let H

• R (l) be the dual space of HR (l). By identifying H
• R (l) with

HR (l) and denoting the dual pairing between H

• R (∞) and HR (∞) by

h, i, we have from the results of Gel’fand and Vilenkin that H

• R (∞) = [l2NH
• R (l). We shall consider H
• R (∞) as equipped with the

weak topology.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 23 / 51

A true non-Archimedean analog of the Schwartz space

Lemma

For any l 2 N, we set dνl,N := h max

• 1, kξkp

i2αl dNξ, and L2

l,N :=

• f : QN

p

! C : R

QN

p

• b

f

• 2

dνl,N < ∞

• .

Notice that L2

l,N L2. Then HC (l) = L2 l,N for any l 2 N. A similar result

is valid for HR (l).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 24 / 51

Pseudodi¤erential operators

De…nition

We say that a function a : QN

p ! R+ is a smooth symbol, if it satis…es

the following properties: (i) a is a continuous function; (ii) there exists a positive constant C = C(a) such that a (ξ) C for any ξ 2 QN

p ;

(iii) there exist positive constants C0, C1, α, m0, with m0 2 N, such that C0 kξkα

p a (ξ) C1 kξkα p

for kξkp pm0.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 25 / 51

Pseudodi¤erential operators

Given a smooth symbol a (ξ), we attach to it the following pseudodi¤erential operator: DC(QN

p )

! L2 \ C unif g ! Ag, where (Ag) (x) = F 1

ξ!x

• a (ξ) Fx!ξg
• .

Example

• Lα + m2

Lemma

For any l 2 N, the mapping A : HC (l + 1) ! HC (l) is a well-de…ned continuous mapping between Banach spaces.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 26 / 51

Pseudodi¤erential operators

The above lemma is the non-Archimedean counterpart of the following fact: C r+1(R) ! C r(R) f !

d dx f .

Theorem

(i) The mapping A : HC (∞) ! HC (∞) is a bi-continuous isomorphism

• f locally convex spaces. (ii) HC (∞) L∞ \ C unif \ L1 \ L2.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 27 / 51

Pseudodi¤erential Operators and Green Functions

We take l (ξ) 2 Zp [ξ1, , ξN] to be an elliptic polynomial of degree d, this means that l is homogeneous of degree d and satis…es l (ξ) = 0 , ξ = 0. There are in…nitely many elliptic polynomials.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 28 / 51

Pseudodi¤erential Operators and Green Functions

We take l (ξ) 2 Zp [ξ1, , ξN] to be an elliptic polynomial of degree d, this means that l is homogeneous of degree d and satis…es l (ξ) = 0 , ξ = 0. There are in…nitely many elliptic polynomials. We consider the following elliptic pseudodi¤erential operator: (Lαh) (x) = F 1

ξ!x

• jl (ξ)jα

p Fx!ξh

• ,

where α > 0 and h 2 DC(QN

p ).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 28 / 51

Pseudodi¤erential Operators and Green Functions

We take l (ξ) 2 Zp [ξ1, , ξN] to be an elliptic polynomial of degree d, this means that l is homogeneous of degree d and satis…es l (ξ) = 0 , ξ = 0. There are in…nitely many elliptic polynomials. We consider the following elliptic pseudodi¤erential operator: (Lαh) (x) = F 1

ξ!x

• jl (ξ)jα

p Fx!ξh

• ,

where α > 0 and h 2 DC(QN

p ).

We shall call a fundamental solution G (x; m, α) of the equation

• Lα + m2

u = h, with h 2 DC(QN

p ), m > 0,

(3) a Green function of Lα.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 28 / 51

Pseudodi¤erential Operators and Green Functions

As a distribution on DC(QN

p ), the Green function is given by

G (x; m, α) = F 1

ξ!x

1 jl (ξ)jα

p + m2

! . (4)

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 29 / 51

Pseudodi¤erential Operators and Green Functions

As a distribution on DC(QN

p ), the Green function is given by

G (x; m, α) = F 1

ξ!x

1 jl (ξ)jα

p + m2

! . (4) Notice that since C α

0 kξkαd p jl (ξ)jα C α 1 kξkαd p ,

(5) for some positive constants C0, C1, 1 jl (ξ)jα

p + m2 2 L1

QN

p , dNξ

• for αd > N,

and in this case, G (x; m, α) is an L∞-function.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 29 / 51

Pseudodi¤erential Operators and Green Functions

Proposition

The Green function G (x; m, α) veri…es the following properties: (i) the function G (x; m, α) is continuous on QN

p r f0g;

(ii) if αd > N, then the function G (x; m, α) is continuous; (iii) for 0 < αd N, the function G (x; m, α) is locally constant on QN

p r f0g, and

jG (x; m, α)j 8 > < > : C kxk2αdN

p

for 0 < αd < N C0 C1 ln kxkp for N = αd, for kxkp 1, where C, C0, C1 are positive constants, (iv) jG (x; m, α)j C2 kxkαdN

p

as kxkp ! ∞, where C2 is positive constant; (v) G (x; m, α) 0 on QN

p r f0g.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 30 / 51

Pseudodi¤erential Operators and Green Functions

Kochubei, Anatoly N. Pseudo-di¤erential equations and stochastics

• ver non-Archimedean …elds. Monographs and Textbooks in Pure and

Applied Mathematics, 244. Marcel Dekker, Inc., New York, 2001.

Theorem

Let α > 0, m > 0, and let Lα be an elliptic operator. (i) There exists a Green function G (x; m, α) for the operator Lα, which is continuous and non-negative on Qn

p r f0g, and tends to zero at in…nity. Furthermore, if

h 2 DC(QN

p ), then u(x) = G (x; m, α) h(x) is a solution of (3) in

D

• C(QN

p ). (ii) The equation

• Lα + m2

u = g, (6) with g 2 HR (∞), has a unique solution u 2 HR (∞).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 31 / 51

Pseudodi¤erential Operators and Green Functions

Proof.

(ii) By a density argument we can take g 2 DR(QN

p ), then by (i),

u(x) = G (x; m, α) g(x) is a real-valued, locally constant function which is a solution of (6) in D

• C(QN

p ). Now, since b

u (ξ) =

b g(ξ) jl(ξ)jα

p+m2 2 L2,

kuk2

l+d C kgk2 0 +

Z

QN

p rB N

kξk2α(l+d)

p

jb g (ξ)j2 dNξ

• jl (ξ)jα

p + m2

2 C kgk2

0 + 1

C α

Z

QN

p rB N

kξk2αl

p

jb g (ξ)j2 dNξ C kgk2

0 + 1

C α kgk2

l

C 0 kgk2

l , for l 2 N.

Then, by Lemma 4, u 2 HR (m), for m d. In the case, 0 m d 1,

• ne gets kukm C 00 kgk0. Therefore u 2 HR (m), for m 2 N.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 32 / 51

Pseudodi¤erential Operators and Green Functions

Corollary

The mapping HR (∞) ! HR (∞) g (x) ! G (x; m, α) g(x), is continuous. Notice that G (x; m, α) g(x) =

• Lα + m21 g.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 33 / 51

In…nitely divisible probability distributions

An in…nitely divisible probability distribution P is a probability distribution having the property that for each n 2 N there exists a probability distribution Pn such that P = Pn Pn (n-times).

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 34 / 51

In…nitely divisible probability distributions

An in…nitely divisible probability distribution P is a probability distribution having the property that for each n 2 N there exists a probability distribution Pn such that P = Pn Pn (n-times). By the Lévy-Khinchine Theorem, the characteristic function CP of P satis…es CP(t) = R

R

eistdP(s) = eΨ(t), t 2 R, (7) where Ψ : R ! C is a continuous function, called the Lévy characteristic of P, which is uniquely represented as follows: Ψ (t) = iat σ2t2 2 + R

Rrf0g

• eist 1

ist 1 + s2

• dM(s), t 2 R, (8)

where a, σ 2 R, with σ 0, and the measure dM(s) satis…es R

Rrf0g

min

• 1, s2

dM(s) < ∞. (9)

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 34 / 51

In…nitely divisible probability distributions

Remark

From now on, we work with in…nitely divisible probability distributions which are absolutely continuous with all …nite moments. This fact is equivalent to all the moments of the corresponding M’s are …nite, cf. [1, Theorem 2.3]. Albeverio Sergio, Wu Jiang Lun, Euclidean random …elds obtained by convolution from generalized white noise, J. Math. Phys. 36 (1995),

• no. 10, 5217–5245.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 35 / 51

Bochner-Minlos Theorem

Let HR (∞) and H

• R (∞) be the spaces introduced before. We

denote by h, i the dual pairing between HR (∞) and H

• R (∞). Let

B be the σ-algebra generated by cylinder sets of H

• R (∞). Then

H

• R (∞) , B
• is a measurable space.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 36 / 51

Bochner-Minlos Theorem

Let HR (∞) and H

• R (∞) be the spaces introduced before. We

denote by h, i the dual pairing between HR (∞) and H

• R (∞). Let

B be the σ-algebra generated by cylinder sets of H

• R (∞). Then

H

• R (∞) , B
• is a measurable space.

By a characteristic functional on HR (∞), we mean a functional C : HR (∞) ! C satisfying the following properties:

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 36 / 51

Bochner-Minlos Theorem

Let HR (∞) and H

• R (∞) be the spaces introduced before. We

denote by h, i the dual pairing between HR (∞) and H

• R (∞). Let

B be the σ-algebra generated by cylinder sets of H

• R (∞). Then

H

• R (∞) , B
• is a measurable space.

By a characteristic functional on HR (∞), we mean a functional C : HR (∞) ! C satisfying the following properties: (i) C is continuous on HR (∞);

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 36 / 51

Bochner-Minlos Theorem

Let HR (∞) and H

• R (∞) be the spaces introduced before. We

denote by h, i the dual pairing between HR (∞) and H

• R (∞). Let

B be the σ-algebra generated by cylinder sets of H

• R (∞). Then

H

• R (∞) , B
• is a measurable space.

By a characteristic functional on HR (∞), we mean a functional C : HR (∞) ! C satisfying the following properties: (i) C is continuous on HR (∞); (ii) C is positive-de…nite;

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 36 / 51

Bochner-Minlos Theorem

Let HR (∞) and H

• R (∞) be the spaces introduced before. We

denote by h, i the dual pairing between HR (∞) and H

• R (∞). Let

B be the σ-algebra generated by cylinder sets of H

• R (∞). Then

H

• R (∞) , B
• is a measurable space.

By a characteristic functional on HR (∞), we mean a functional C : HR (∞) ! C satisfying the following properties: (i) C is continuous on HR (∞); (ii) C is positive-de…nite; (iii) C(0) = 1.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 36 / 51

Bochner-Minlos Theorem

Since HR (∞) is a nuclear space, cf. Theorem 3, by the Bochner-Minlos Theorem, there exists a one to one correspondence between the characteristic functionals C and probability measures P on H

• R (∞) , B
• given by the following relation

C(f ) = R

H

R(∞)

eihf ,T idP (T) , f 2 HR (∞) .

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 37 / 51

The generalized white noise

Theorem

Let Ψ be a Lévy characteristic de…ned by (7). Then there exists a unique probability measure PΨ on H

• R (∞) , B
• such that the Fourier transform
• f PΨ satis…es

R

H

R(∞)

eihf ,T idPΨ (T) = exp 8 < : R

QN

p

Ψ (f (x)) dNx 9 = ; , f 2 HR (∞) . The proof is based on [1, Theorem 6, p. 283] like in the Archimedean case, cf. [1, Theorem 1.1]. However, in the non-Archimedean case the result does not follow directly from [1]. We need some additional results. Gel’fand I. M., Vilenkin N. Ya, Generalized functions. Vol. 4. Applications of harmonic analysis. Academic Press, New York-London, 1964.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 38 / 51

The generalized white noise

Lemma

R

QN

p Ψ (f (x)) dNx < ∞ for any f 2 HR (∞).

Lemma

The function f ! R

QN

p Ψ (f (x)) dNx is continuous on HR (∞).

Set L(f ) := exp nR

QN

p Ψ (f (x)) dNx

• for f 2 HR (∞). Notice that by

Lemma 13 this function is well-de…ned.

Proposition

The function L(f ) is positive-de…nite if and only if esΨ(t) is positive-de…nite for every s > 0.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 39 / 51

Non-Archimedean generalized white noise measures

De…nition

We call PΨ in Theorem 12 a generalized white noise measure with Lévy characteristic Ψ and H

• R(∞), B, PΨ
• the generalized white noise space

associated with Ψ. The associated coordinate process z : HR (∞)

• H
• R(∞), B, PΨ
• ! R

de…ned by z (f , T) = hf , Ti, f 2 HR (∞), T 2 H

• R(∞), is called

generalized white noise. The generalized white noise z is composed by three independent noises: constant, Gaussian and Poisson (with jumps given by M) noises.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 40 / 51

Euclidean random …elds as convoluted generalized white noise

De…nition

Let (Ω, F, P) be a given probability space. By a generalized random …eld Φ on (Ω, F, P) with parameter space HR (∞), we mean a system fΦ (g, ω) : ω 2 Ωgg2HR(∞) ,

• f random variables on (Ω, F, P) having the following properties:

(i) P fω 2 Ω : Φ (c1g1 + c2g2, ω) = c1Φ (g1, ω) + c2Φ (g2, ω)g = 1, for c1, c2 2 R, g1, g2 2 HR (∞); (ii) if gn ! g in HR (∞), then Φ (gn, ω) ! Φ (g, ω) in law. The coordinate process in De…nition 16 is a random …eld on the generalized white noise space H

• R(∞), B, PΨ
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 41 / 51

Euclidean random …elds as convoluted generalized white noise

We now recall that (Gf ) (x) := G (x; m, α) f (x) gives rise to a continuous mapping from HR(∞) into itself, cf. Corollary 11.

• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 42 / 51

Euclidean random …elds as convoluted generalized white noise

We now recall that (Gf ) (x) := G (x; m, α) f (x) gives rise to a continuous mapping from HR(∞) into itself, cf. Corollary 11. Thus, the conjugate operator e G : H

• R(∞) ! H
• R(∞) is a measurable

mapping from H

• R (∞) , B
• into itself.
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 42 / 51

Euclidean random …elds as convoluted generalized white noise

We now recall that (Gf ) (x) := G (x; m, α) f (x) gives rise to a continuous mapping from HR(∞) into itself, cf. Corollary 11. Thus, the conjugate operator e G : H

• R(∞) ! H
• R(∞) is a measurable

mapping from H

• R (∞) , B
• into itself.

The generalized white noise measure PΨ on H

• R (∞) , B
• associated

with a Lévy characteristic Ψ was introduced in De…nition 16.

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Euclidean random …elds as convoluted generalized white noise

We now recall that (Gf ) (x) := G (x; m, α) f (x) gives rise to a continuous mapping from HR(∞) into itself, cf. Corollary 11. Thus, the conjugate operator e G : H

• R(∞) ! H
• R(∞) is a measurable

mapping from H

• R (∞) , B
• into itself.

The generalized white noise measure PΨ on H

• R (∞) , B
• associated

with a Lévy characteristic Ψ was introduced in De…nition 16. We set PΦ to be the image probability measure of PΨ under e G, i.e. PΦ is the measure on H

• R (∞) , B
• de…ned by

PΦ (A) = PΨ

• e

G1 (A)

• , for A 2 B.

(10)

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Euclidean random …elds as convoluted generalized white noise

Proposition

The Fourier transform of PΦ is given by

Z

H

R(∞)

eihf ,T idPΦ (T) = exp 8 > < > :

Z

QN

p

Ψ 8 > < > :

Z

QN

p

G (x y; m, α) f (y) dNy 9 > = > ; dNx 9 > = > ; , for f 2 HR (∞).

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Euclidean random …elds as convoluted generalized white noise

By Proposition 18, the associated coordinate process Φ : HR (∞)

• H
• R (∞) , B
• ! R

given by Φ (f , T) = hGf , Ti, f 2 HR (∞), T 2 H

• R (∞) , B
• , is a

random …eld on H

• R (∞) , B, PΦ
• .
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 44 / 51

Euclidean random …elds as convoluted generalized white noise

By Proposition 18, the associated coordinate process Φ : HR (∞)

• H
• R (∞) , B
• ! R

given by Φ (f , T) = hGf , Ti, f 2 HR (∞), T 2 H

• R (∞) , B
• , is a

random …eld on H

• R (∞) , B, PΦ
• .

In fact, Φ is nothing but e Gz which is de…ned by e Gz (f , T) = z (Gf , T) , f 2 HR (∞) , T 2 H

• R (∞) .
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 44 / 51

Euclidean random …elds as convoluted generalized white noise

By Proposition 18, the associated coordinate process Φ : HR (∞)

• H
• R (∞) , B
• ! R

given by Φ (f , T) = hGf , Ti, f 2 HR (∞), T 2 H

• R (∞) , B
• , is a

random …eld on H

• R (∞) , B, PΦ
• .

In fact, Φ is nothing but e Gz which is de…ned by e Gz (f , T) = z (Gf , T) , f 2 HR (∞) , T 2 H

• R (∞) .

It is useful to see Φ as the unique solution, in law, of the stochastic equation

• Lα + m2

Φ = z, where

• Lα + m2

Φ (f , T) := Φ

• Lα + m2

f , T

• , for f 2 HR (∞),

T 2 H

• R (∞).
• W. A. Zúñiga-Galindo (CINVESTAV)

p-adic Massive Euclidean Fields p-ADICS.2015 44 / 51

Symmetries

Given a polynomial a (ξ) 2 Qp [ξ1, , ξn] and g 2 GLN (Qp), we say that g preserves a if a (ξ) = a (gξ), for all ξ 2 QN

p . By simplicity, we use

gx to mean [gij] xT , x = (x1, , xN) 2 QN

p , where we identify g with

the matrix [gij].

De…nition

Let q (ξ) be the elliptic quadratic form used in the de…nition of the Fourier transform, and let l (ξ) be the elliptic polynomial that appears in the symbol of the operator Lα. We de…ne the homogeneous Euclidean group

• f QN

p relative to q (ξ) and l (ξ), denoted as E0

• QN

p ; q, l

• := E0
• QN

p

• , as

the subgroup of GLN (Qp) whose elements preserve q (ξ) and l (ξ)

• simultaneously. We de…ne the inhomogeneous Euclidean group, denoted as

E

• QN

p ; q, l

• := E
• QN

p

• , to be the group of transformations of the form

(a, g) x = a + gx, for a, x 2 QN

p , g 2 E0

• QN

p

• .
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p-adic Massive Euclidean Fields p-ADICS.2015 45 / 51

Symmetries

Let (a, g) be a transformation in E

• QN

p

• , the action of (a, g) on a

function f 2 HR (∞) is de…ned by ((a, g) f ) (x) = f

• (a, g)1 x
• , for x 2 QN

p ,

and on a functional T 2 H

• R (∞), by

hf , (a, g) Ti := D (a, g)1 f , T E , for f 2 HR (∞) . The action on a random …eld Φ is de…ned by ((a, g) Φ) (f , T) = Φ

• (a, g)1 f , T
• , for f 2 HR (∞) , T 2 H
• R (∞) .
• W. A. Zúñiga-Galindo (CINVESTAV)

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Symmetries

Example

In the case N = 4 there is a unique elliptic quadratic form, up to linear equivalence, which is l4 (ξ) = ξ2

1 sξ2 2 pξ2 3 + sξ2 4, where s 2 Zr f0g is

a quadratic non-residue, i.e.

• s

p

• = 1. We take q4 (ξ) = l4 (ξ), i.e.

B4 (x, ξ) = ξ1x1 sξ2x2 pξ3x3 + sξ4x4. In this case, E0

• Q4

p, l4, l4

• equals

O (l4) = 8 > < > : g 2 GL4 (Qp) : gT

2 6 6 4 1 s p s 3 7 7 5 g =

2 6 4

1 s p s

3 7 5 9 > = > ; the orthogonal group of l4.

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Symmetries

De…nition

By Euclidean invariance of the random …eld Φ we mean that the laws of Φ and (a, g) Φ are the same for each (a, g) 2 E

• QN

p

• , i.e. the probability

distributions of fΦ (f , ) : f 2 HR (∞)g and f((a, g) Φ) (f , ) : f 2 HR (∞)g coincide for each (a, g) 2 E

• QN

p

• .

We say that G is (a, g)-invariant for some (a, g) 2 E

• QN

p

• , if

(a, g) G = G (a, g). If G is invariant under all (a, g) 2 E

• QN

p

• , we say

that G is Euclidean invariant.

• W. A. Zúñiga-Galindo (CINVESTAV)

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Symmetries

Proposition

The random …eld Φ = e Gz is Euclidean invariant.

• W. A. Zúñiga-Galindo (CINVESTAV)

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Schwinger Functions

De…nition

Set g1, , gm 2 HR(∞). We de…ne the m th Schwinger function Sm

• f Φ as the m th moment of Φ, i.e.

Sm (g1 gm) =

Z

H

R(∞)

hg1, Ti hgm, Ti dPΦ (T) , m 2 N r f0g , (11) with S0 := 1.

• W. A. Zúñiga-Galindo (CINVESTAV)

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Schwinger Functions

Theorem

The Schwinger functions Sm de…ned above are symmetric and Euclidean invariant functionals in H

• R
• QNm

p

; ∞

• for m 1. Furthermore for

g1, , gm 2 HR(QN

p , ∞) we have

Sm (g1 gm) = ∑

I 2P (m)

∏

fj1, ,jlg2I

cl

Z

QN

p

l

∏

k=1

G (x; m, α) gjk (x) dNx, where c1 := a +

Z

Rrf0g s3 1+s2 dM (s) ,

c2 := σ2 +

Z

Rrf0g

s2dM (s) , cm :=

Z

Rrf0g

smdM (s) , for m 3.

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