c Pleiades Publishing, Ltd., 2009. ISSN 2070-0466, p-Adic Numbers, Ultrametric Analysis and Applications, 2009, Vol. 1, No. 2, pp. 105–117. 

RESEARCH ARTICLES

Stochastic Processes in Qp Associated with Systems of Nonlinear PIDEs Sergio Albeverio1** and Yana Belopolskaya2*** 1

2

Chair of Stochastic Analysis, Institute of Applied Mathematics, Bonn University, Bonn, Germany St.Petersburg State University for Architecture and Civil Engineering, 2-ja Krasnoarmejskaja 4, 190005, St.Petersburg, Russia Received July 28, 2008

Abstract—We construct a Markov process and its multiplicative operator functional associated with a system of nonlinear partial integral-differential equations (PIDE). As a result we derive a probabilistic representation of the solution to the Cauchy problem for a system of nonlinear PIDEs in Qp . DOI: 10.1134/S2070046609020022 Key words: stochastic process in Qp , pseudo-differential equations, the Cauchy problem.

1. INTRODUCTION In the Euclidean case there exists a well known connection between second order parabolic equations and integro-differential equations and Markov processes. In particular, we can consider a stochastic differential equation (SDE)  f (ξ(t), z, u(t, ξ(t)))ν(dt, dz), ξ(s) = x ∈ Rd , 0 ≤ s ≤ t ≤ T (1.1) dξ(t) = Rd

with f (y, z, u) ∈ Rd , y, z ∈ Rd , u ∈ R1 driven by a Poisson random measure ν(dt, dz) with Eν(dt, dz) = π(dz)dt and add to (1.1) the relation u(s, x) = E[u0 (ξs,x (T ))],

(1.2)

where ξs,x (t) is a solution to (1.1) and u0 is a bounded scalar function defined on Rd . As a result we obtain a closed system of equations. Then providing that the solution of (1.1), (1.2) exists and ξ(t) is a Markov process in Rd we can prove that the function u(s, x) solves the following Cauchy problem  ∂u + [u(s, f (x, z, u(s, x))) − u(s, x)]π(dz) = 0, u(T, x) = u0 (x). (1.3) ∂s Rd Hence we can construct a solution of a boundary value problem for a nonlinear PIDE as an average over trajectories of the corresponding Markov process which is a process with cadlag paths (continuous from the right and having limits from the left). If one turns to the Qp case instead of Rd one finds that there exists a natural type of stochastic processes in Qp similar to a jump process ξ(t) in Rd . Stochastic equations in Qp driven by such processes were studied by Kochubei in [1]. In particular this author has shown that through a solution of a stochastic differential equation  f (ξ(t), z)ν(dt, dz), ξ(s) = x ∈ Qp , 0 ≤ s ≤ t ≤ T, (1.4) dξ(t) = Qp ∗

The text was submitted by the authors in English. E-mail: [email protected] *** E-mail: [email protected] **

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one can construct a Markov process ξs,x (t) ∈ Qp such that given a real valued function u0 (x) defined on Qp the function u(s, x) = Eu0 (ξs,x (T )) solves the Cauchy problem  pα −1  ∂u −α−1 dz = 0, ∂s + 1−p−α−1 K [u(s, x + f (x, z)) − u(s, x)]zp (1.5) u(T, x) = u0 (x). A new problem arises if one wishes to construct a probabilistic representation for the solution of a nonlinear PIDE. In the Euclidean case as it was mentioned above the theory of stochastic differential equations allows to construct a Markov process associated with a nonlinear PIDE by considering a SDE with coefficients depending on the distribution of its own solution. Unfortunately this approach can not be immediately implemented in the Qp case if one starts with a SDE driven by a standard random Poisson measure as in [1]. The key point is that to solve (1.1) one needs some regularity of the function f (x, z, u) in the argument u ∈ R1 but this is impossible for a function f valued in Qp . A way to overcome this obstacle is to consider SDEs driven by fields of Poisson measures and allow their intensities to depend on distributions of the SDE solutions. This approach was developed in our previous paper [2]. Let us mention as well one more approach to constructing a Markov process associated with a nonlinear PIDE developed by Albeverio and Zhao [5] which is based on a notion of a super Markov process or a measure valued branching process. This approach allows to construct a probabilistic representation to a PIDE of the form ∂u + Hu(t, x) − [u(t, x)]1+β , x ∈ Qp ∂s with 0 < β ≤ 1. Here the linear operator H is the generator of a certain Markov process in Qp . In the Euclidean case constructing of a solution to the Cauchy problem for a system of nonlinear PIDEs of the type (1.3) with respect to a function u valued in Rd1 can be reduced to solving a closed system of stochastic equations  f (ξ(t), z, u(t, ξ(t)))ν(dt, dz), ξ(s) = x ∈ Rd , 0 ≤ s ≤ t ≤ T, (1.6) dξ(t) = Rd

 C(ξ(t), z, u(t, ξ(t)))η(t)ν(dt, dz),

dη = Rd

η(s) = h ∈ Rd1 ,

h, u(s, x) = E[η(T ), u0 (ξs,x (T ))],

(1.7) (1.8)

where C is a function on Rd × Rd × Rd1 valued in the space of d1 × d1 -matrices and ·, · ≡ ·, ·d1 is the inner product in Rd1 . By  · d1 ≡  ·  we denote the norm in Rd1 . Then we can prove that the function u defined by (1.8) solves the Cauchy problem for a system of nonlinear PIDEs (see [3] for details). In the present paper we consider a Markov process ξ(t) ∈ Qp with a generator A which is a particular case of the operator H from [5] and construct this process via a stochastic differential equation. Then we give a construction of a multiplicative operator functional (MOF) of this process. To derive a probabilistic representation of the equation ∂u + Au + M (u)u = 0 ∂s with a nonlinear functional M (u), we consider a MOF of the processes ξ(t) constructed via the solution of a linear SDE with coefficients depending both on the process ξ(t) and its distribution. The paper is organized as follows. In section 2 we recall some results from [1, 2] necessary for our construction. In section 3 we study a system of stochastic equations describing the stochastic process ξ(t) ∈ Qp and its multiplicative functional generated by a solution of a linear SDE with coefficients depending on the process ξ(t) and study the connections between a solution of this system of SDEs and a solution of the Cauchy problem for a nondiagonal linear system of PIDEs. In section 4 we consider the stochastic process ξ(t) ∈ Qp and its multiplicative functional generated by a solution of a linear SDE with coefficients depending both on the process ξ(t) ∈ Qp and its distribution and study the connections between a solution of this system of SDEs and a solution of the Cauchy problem for a nondiagonal nonlinear system of PIDEs. p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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2. STOCHASTIC EQUATIONS IN Qp We start our considerations introducing necessary notions and notations. Consider the field of p-adic numbers Qp which is the completion of the field Q of rational numbers with respect to the absolute value xp defined by setting 0p = 0, m xp = p−k if x = pk , n where k, m, n are integers (k, m, n ∈ Z, where Z denotes the field of integers) and m, n are prime to p. Let (Ω, F, P ) be a complete probability space and ζ(t, ω) ∈ Qp be a symmetric stable process defined for t ∈ [0, T ), ω ∈ Ω. Given a Borel subset G ∈ Qp \{0} denote by ν(t, G) the number of jumps of the process of ζ belonging to G on the interval [0, t). If Γ is contained in the complement Qp \Bn of a ball Bn (a) = {x ∈ Qp : x − ap ≤ pn } with n ∈ Z and a ∈ Qp then ν(t, Γ) < ∞. We denote Bn = Bn (0). The sets Bn (a) are both open and compact and Qp can be uniquely represented as a countable union of disjoint balls of radius pn . Denote by Bp the σ-algebra generated by all balls in Qp . The couple (Qp , Bp ) is a measurable space. It is known that the set of functions defined on balls by χ(Bn (a)) = pn can be uniquely extended to a measure χ(dz) on Bp called the Haar measure on the additive group Qp . The Haar measure χ is a spatially invariant measure finite on any compact set. We denote by dz the normalized Haar measure  such that B0 (a) dz = 1. Let π(z) be a nonnegative measurable function on Qp such that for any a ∈ Qp and n ∈ Z  π(z)dz < ∞. 0< Bn (a)

Then π determines a σ-finite measure π(dz) = π(z)dz on Qp . For G ∈ Bp , f ∈ R1 denote by ν(dt, G) a Poisson measure with parameter dtπ(t, G). Namely ν(dt, G) is a random non-negative countably additive measure on the Borel σ-algebra B((0, T ]) × B(Qp \0) such that the random variables ν(t, G1 ) and ν(t, G2 ) are independent if G1 ∩ G2 = ∅. Given ´ fixed α > 0, α = 1 we assume that Eν(t, G) = tπ(G), where π(dz) is a Levy’s measure π(dz) =

pα − 1 z−α−1 dz = Γα z−α−1 dz. p p 1 − p−α−1

In what follows we assume that

(2.1)

 π(dz) < ∞. Qp

If the set G is inside of the set Bnc = Qp \Bn , where Bn = {x ∈ Qp : xp ≤ pn }, n ∈ Z, then ν(t, G) < ∞. Given (Ω, F, P ) we define a set Lγ (Ω, Qp ), γ ≥ 1 of all F-measurable functions f : Ω → Qp such that  1 f γ = { f (ω)γp P (dω)} γ < ∞. Ω

Let Mt denote the set of Ft -measurable functions f : [0, T ] × Qp → Qp such that f (t, z) = 0, if z ∈ Bk where the integer k depends on f , and let NT denote the set of all Ft - measurable functions defined on the complement of the set Mt with respect to the norm  T f (t, z)p πα (t, z)dzdt. f 1 = 0

Qp

Let C(S, Qp ) denote the space of all continuous functions valued in Qp and defined on a compact set S ⊂ Qp , and let D([0, T ], X) denote the space of cadlag functions valued in the space X and defined on the interval [0, T ]. Note that a function belonging to D([0, T ], Qp ) is constant between jumps since Qp is totally disconnected. p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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Consider the random integer-valued variable ζ with the Poisson distribution having the parameter k a > 0, P {ζ = k} = e−a ak! . Then one can check that for any prime number p and γ ∈ [1, ∞) the estimate γ Eζp ≤ pγp−1 a holds. Given a Poisson measure ν(dt, dz) and a map f : Qp → Qp we denote by I(f ) the stochastic integral  T f (z)ν(dt, dz). (2.2) I(f ) = 0

Qp

We can define (2.2) on Bn by 

T



In (fm ) =

fm (z)ν(dt, dz) = 0

(m)

where Δi (m) Δi

Bn

Nm   i=1

are disjoint intervals [0, T ] =

Nm

(m) i=1 Δi

(m)

Bn

fm (z)ν(Δi

, dz),

and norm of the oscillations of the function f on

do not exceed the value εm where εm → 0 as m → ∞.

It results from the definition of I(f ) and the ultrametric inequality that the sequence In,m (f ) = on the choice of approximations. In (fm ) is fundamental in Qp almost surely and its limit does not depend  Now we define In by In (f ) = limm→∞ In,m (f ) and finally set I(f ) = ∞ n=−∞ In (f ), where the series converges in probability. Given a random function f (ω, z) ∈ Qp belonging to Mt we define a stochastic integral I(f ) by  T m n   f (z)ν(dt, dz) = f (zj )ν([tk , tk+1 ], Gj ), (2.3) I(f ) = 0

Qp

k=0 j=1

where Gj are disjoint Borel subsets from Qp and ∪m j=1 Gj = Qp . It is easy to verify that there exists a positive constant C such that the estimate  T Ef (z)p πα (dz)dt (2.4) EI(f )p ≤ C 0

Qp

holds. As a result we can prove that f → I(f ) is a linear operator from Mt to Lγ (Ω, Qp ). 1 -valued functions f (z) and a couple Consider a couple of random predictable and Ft -measurable R+ i of stochastic integrals  t fi (z)να (dt, dz), i = 1, 2. Ψi (t) = 0

Qp

Lemma 2.1. Let fi ∈ Nt , i = 1, 2 . Then  T  Ψ1 (t−)dΨ2 (t) + Ψ1 (T )Ψ2 (T ) = 0



T

Ψ2 (t−)dΨ1 (t) + 0

and

T

f1 (z)f2 (z)ν(dt, dz),

(2.5)

0

 2

f 2 (z)ν(dt, dz).

dΨ (t) = 2Ψ(t)dΨ(t) +

(2.6)

Qp

Proof. Let the functions f1 and f2 be equal to 1 outside a ball Bk where k is a fixed integer. In this case we prove that for any subset G ⊂ Qp dν 2 (t, G) = [(ν(t, G) + 1)2 − ν 2 (t, G)]ν(dt, G), due to the properties of the Poisson measure ν(t, dz) on Qp . p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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For step functions fin (t, x, z) belonging to Nt we obtain

109



d[Ψ1n (t)Ψ2n (t)] = Ψ1n (t)dΨ2n (t) + Ψ2n (t)dΨ1n (t) +

f1n (z)f2n (z)ν(dt, dz) Qp

and passing to the limit as n → ∞ we derive (2.5), (2.6). Let Ft be a filtration generated by the Poisson process να ([0, t), dy) and let X denote the Banach space of (classes of stochastically equivalent) Qp -valued random processes ξ(t), t ∈ [0, T ] which are Ft -measurable for each t and continuous as functions on [0, T ] valued in L1 (Ω, Qp ) with the norm 1

ξX = { sup Eξ(t)γp } γ . 0≤t≤T

Consider a stochastic differential equation  f (ξ(t), z)να (dt, dz), dξ(t) =

ξ(s) = x ∈ Qp ,

0 ≤ s ≤ t ≤ T,

(2.7)

Qp

where f is a scalar deterministic function defined on Qp × Qp . Below we assume that the following assumptions hold. C 2.1. f (x, z) ∈ C(Qp × Bn , Lγ (Ω, Qp )) for every n ∈ Z, where C is the space of continuous functions. C 2.2. The mapping z → f (x, z) from Qp to Lγ (Ω, Qp ) is continuous uniformly with respect to x ∈ Qp .  C 2.3. Qp f (x, z)p πα (z)dz ≤ Cf [1 + xp ]. C 2.4. For each integer n, there exists a constant Ln > 0 such that for all t ∈ [0, T ], Bn



x, y ∈ Qp , z ∈





f (x, z)π(dz) −

f (y, z)π(dz)p ≤ Lfn x − yp

Qp

(2.8)

Qp

and ∞ 

Lfn p−n(α+1) ≤ C < ∞.

n=−∞

Let us consider a random process

 t f (z)ν(dθ, dz).

ξ(t) = x + 0

Qp

The process ξ(t) has almost surely a finite number of jumps on the interval [0, t] that allows to present the increment of ϕ(ξ(t)) as a sum of corresponding jumps. Thus  t [ϕ(ξ(θ−) + f (z)) − ϕ(ξ(θ−))]ν(dθ, dz). ϕ(ξ(t)) = ϕ(x) + 0

Qp

This formula is called the Ito formula for the jump process ξ(t). Consider the stochastic equation  t f (ξ(θ), z)ν(dθ, dz). ξ(t) = x + s

It is known that under the above assumptions there exists a unique solution ξ ∈ X to (2.9). To prove this result one can consider the following recursive relations  t f (ξn (θ), z)ν(dθ, dz), n = 0, 1, 2, . . . ξn+1 (t) = x + s

(2.9)

Qp

Qp

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and prove the convergence of the family ξn (t) to the process V (ξ(t)) of the form  t f (ξ(θ), z)ν(dθ, dz). V (ξ(t)) = x + s

Qp

In other words the proof is reduced to checking that V is a contraction in X . See [1] for details of the proof. When f is nonrandom the solution ξ(t) of (2.9) possesses in addition the Markov property. As a result we can state the following assertion. Theorem 2.1. Assume that conditions C 2.1 –C 2.4 hold. Then there exists a unique solution to (2.9). The solution to (2.9) is a Markov process when f is nonrandom. Proof. In view of the above reasons it remains only to check that the solution to (2.9) possesses the Markov property. Let Ft be the minimal σ-algebra, generated by the Poisson measure ν([0, s] × dz) for s ≤ t and the initial data ξ(0). Denote by F t the σ-algebra, generated by the Poisson measure ν([t, τ ] × dz) for τ > t. Obviously the events from Ft and F t are independent. Let ξs,x (t) be a solution of the equation  t f (ξs,x (θ), z)ν(dθ, dz). (2.11) ξs,x (t) = x + Qp

s

By the uniqueness of a solution to (2.9) we have ξs,ξ(s)(t) = ξ(t) where ξ(t) satisfies  t f (ξ(θ), z)ν(dθ, dz). ξ(t) = ξ(s) + s

Qp

As a result we get ξ(t) = n(ξ(s), ω) where n(x, ω) is a random variable independent of Fs . In general choosing a real-valued bounded measurable function m : Ω × Qp → R independent of Ft and an Ft -measurable random variable ζ ∈ Qp we can prove that (2.12) E[m(ζ, ω)|Ft ] = g(ζ), where g(x) = Em(x, ω).  Assume that m(x, ω) = k ψk (x)λk (ω), where ψk are nonrandom functions. Then for any Ft measurable random variable ζ1 we have   ψk (ζ)λk (ω)ζ1 = E ψk (ζ)ζ1 Eλk (ω) = Eg(ζ)ζ1 E[m(ζ, ω)ζ1 ] = E k

k



since g(x) in this case is equal to k ψk (x)Eλk (ω). Finally since any bounded measurable function admits an approximation of this kind we deduce from (2.10) that E[χG (ξ(t))|Fs ] = E[χG (ξs,x (t))|x=ξ(s) ] = P (s, x, t, G)|x=ξ(s) , where χG is the indicator function for a Borel set G ⊂ Qp and P (s, x, t, G) = P {ξs,x (t) ∈ G} is the transition probability. This proves that ξ(t) is a Markov process. Let Ht (Qp ) be the set of real bounded functions on [0, T ] × Qp satisfying the Lipschitz condition |v(t, x) − v(t, y)| ≤ Cnv (t)x − yp if x, y ∈ Bn , t ∈ [0, T ] with ∞ 

Cnv (t)p−n(α+1) < ∞.

n=−∞

If the Lipschitz constant does not depend on t then we denote the corresponding set by H(Qp ). One can verify that given ϕ ∈ H(Qp ) the relation U (t, s)ϕ(x) = E[ϕ(ξs,x (t))] p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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defines an evolution family in H(Qp ). If ϕ ∈ H(Qp ) then  Eϕ(ξs,x (t)) − ϕ(x) Aϕ(x) = lim = Γα [ϕ(x + f (x, z)) − ϕ(x)]z−α−1 dz p t→s t−s Qp

(2.14)

uniformly with respect to x ∈ Qp . The operator A defined by (2.14) is called the generator of the evolution family U (t, s) or the generator of the Markov process ξs,x (t) that satisfies (2.9). Finally the following statement holds. Theorem 2.2. Under the conditions of theorem 2.1 the function (2.15)

u(s, x) = E[ϕ(ξs,x (T ))] determines the unique solution of the Cauchy problem ∂u(s, x) + Au(s, x) = 0, u(T, x) = ϕ(x), ∂s where A is defined by (2.14).

(2.16)

3. MARKOV PROCESSES IN Qp AND THEIR MULTIPLICATIVE OPERATOR FUNCTIONALS Let C : Qp × Qp → Rd × Rd , c : Qp → Rd × Rd be bounded Lipschitz continuous functions and for h ∈ Rd there exists a positive constant L such that  C(x, z)h − C(y, z)hπ(dz) ≤ Lx − yp h. [c(x) − c(y)]h + Qp

Consider a couple of stochastic equations  f (ξ(t), z)ν(dt, dz), dξ(t) =

ξ(s) = x ∈ Qp ,

(3.1)

Qp

 C(ξ(t), z)η(t)ν(dt, dz),

dη(t) = c(ξ(t))η(t)dt +

η(s) = h ∈ Rd ,

(3.2)

Qp

where ν(dt, dz) is the Poisson measure defined in the previous section with Eν(dt, dz) = π(dz)dt and π(dz) of the form (2.1). The existence, uniqueness and the Markov property of a two component process (ξ(t), η(t)) ∈ Qp × Rd are granted by general results of the stochastic equation theory and the results of the previous section since the above stated properties of c and C ensure that C 2.1-C 2.4 hold. Let Xd be the space of Rd valued random processes with the norm η1 = sup0≤τ ≤T Eη(τ ) and Bd (Qp ) be the space of bounded measurable Rd -valued functions defined on [0, T ] × Qp . We use ξ(t) ∈ Qp , η(t) ∈ Rd to define a random mapping T (t, s; ξ(·)) : Xd → Xd (a multiplicative functional of the process ξ(t)) by (3.3)

η(t) = T (t, s, ξ(·))h.

Lemma 3.1. Let coefficients in (3.1), (3.2) satisfy conditions C 2.1 – C 2.4. Then the solution η(t) to (3.2) gives rise to a random evolution family T (t, s; ξ(·)) defined by (3.3) and acting in Xd . Proof. Since the coefficients of (3.3) satisfy the conditions C 2.1-C 2.4 we are granted the existence and uniqueness of the solution to (3.1), (3.2). We deduce the evolution property of T (t, s; ξ(·)) that is the equality T (t, s, ξ(·))h = T (t, τ, ξ(·))T (τ, s, ξ(·))h from the Markov property of ξ(t) and the uniqueness of the solution to (3.2). p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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Denote by H d (Qp ) the set of all Rd -valued functions defined on Qp satisfying the Lipschitz condition φ(x) − φ(y) ≤ Ln x − yp for x, y ∈ Bn with positive constant Ln such that ∞ 

Ln p−n(α+1) ≤ C < ∞.

−∞

Let C([0, T ], H d (Qp )) denote the set of continuous bounded Rd -valued functions mapping [0, T ] to H d (Qp ) . By the Ito formula we obtain that for v ∈ H d (Qp ) dη(t), v(t, ξ(t)) = dη(t), v(t, ξ(t)) + η(t), dv(t, ξ(t)) + dη(t), dv(t, ξ(t)) and taking into account (3.1), (3.2) we deduce that  t  t c(ξ(τ ))η(τ ), v(s, x)dτ + η(t), v(t, ξ(t)) = h, v(s, x) + s

 t  t

Qp

C(ξ(τ ), z)η(τ ), [v(τ, ξ(τ + z) − v(τ, ξ(τ ))]ν(dτ, dz).

+ s

C(ξ(τ ), z)η(τ ), v(s, x)ν(dτ, dz) Qp

h, v(τ, ξ(τ ) + f (ξ(τ ), z) − v(τ, ξ(τ ))]ν(dτ, dz)

+ s

s

Qp

Now we can prove the following result. Theorem 3.1. Let conditions of Lemma 3.1 hold, (ξ(t), η(t)) solves the Cauchy problem (3.1),(3.2) and v0 ∈ H d (Qp ). Then the function v : [0, T ] × Qp → Rd determined by h, v(s, x) = Eη(T ), v0 (ξ(T )),

(3.4)

satisfies the Cauchy problem  d  ∂vi = [vi (s, x + f (x, z)) − vi (s, x)]π(dz) + cij (x)vj (s, x) − ∂s Qp

(3.5)

j=1



d 

+

Cij (x, z)vj (s, x + f (x, z))π(dz) = 0,

vi (T, x) = v0i (x),

i = 1, . . . d.

Qp j=1

Proof. Let T (t, s, ξ(·)) be a linear random mapping acting in Rd and defined by η(t) = T (t, s, ξ(·))h, where η(t) is a solution to (3.2) and h ∈ Rd . Consider an operator family V (t, s) : H d (Qp ) → H d (Qp ) given by (3.6)

V (t, s)v(x) = E[T (t, s; ξ(·))v(ξ(t))]. We apply Ito’s formula and take into account (3.2) to compute the increment h, [V (t, s) − V (t, s + Δs)]v(x) = E[T (t, s + Δs; ξ(·)) − T (t, s; ξ(·))]h, v(ξ(s + Δs))+ ET (t, s; ξ(·))h, [v(ξ(s + Δs)) − v(ξ(s))] = s+Δs 

 h,

∗



E[C (ξ(τ, z), z)v(ξ(s))]π(dz)dτ + s

s

E[c∗ (ξ(τ ))v(ξ(s))]dτ +

s

Qp



s+Δs

(3.7)

s+Δs 

E[C ∗ (ξ(τ ), z)[v(ξ(τ ) + f (ξ(τ ), z)) − v(ξ(τ ))]]π(dz)dτ +

Qp

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s+Δs 

s

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E[v(ξ(τ ) + f (ξ(τ, z)) − v(ξ(τ ))]π(dz)dτ .

Qp

Finally, letting Δs → 0 we compute ∂v(s, x) h, [V (t, s) − V (t, s + Δs)]v(x) = −h,  = h, Av(s, x), Δs→0 Δs ∂s lim

where A denotes the operator in the right hand side of (3.5). Since the last relation holds for all h ∈ Rd this yields immediately (3.5). 4. STOCHASTIC EQUATIONS AND NONLINEAR SYSTEMS OF PIDES Let us turn to stochastic equations associated with nonlinear systems of PIDEs. Let C(x, z, v) ∈ Rd × Rd and c(x, v) ∈ Rd × Rd be bounded matrix-valued functions defined on Qp × Qp × Rd and Qp × Rd , respectively. We consider the system of SDEs with respect to a process ξ(t) in Qp and a process η(t) ∈ Rd  dξ(t) = f (ξ(t), z)ν(dt, dz), ξ(s) = x ∈ Qp , (4.1) Qp

 C(ξ(t), z, u(t, ξ(t)))η(t)ν(dt, dz),

dη(t) = c(ξ(t), u(t, ξ(t)))η(t)dt +

(4.2)

Qp

h, u(s, x) = Eη(T ), u0 (ξ(T )),

η(s) = h ∈ Rd .

(4.3)

First we state conditions to ensure the existence and uniqueness of the solution to the system (4.1)(4.3). C 4.1. u0 ∈ Cb (Bn ) for every n ∈ Z, where Cb is the space of bounded continuous Rd -valued functions with sup norm. C 4.2. The mapping z → C(x, z, u) from Qp to Lγ (Ω, Rd × Rd ) is continuous in x, v uniformly with respect to z ∈ Qp .  C 4.3. c(x, v)h, h + Qp C(x, z, v)h, hπ(dz) < h2 ρ0 [1 + vk ], where ρ0 is a constant, k is an integer and h ∈ Rd . C 4.4. For each m ∈ Z, there exists a constant Lm > 0 such that for all t ∈ [0, T ], x, y ∈ Qp , z ∈ Bm  C(x, z, u)h − C(y, z, v)hπ(dz) ≤ ρ1m (u, v)u − v + Mm x − yp , (4.4) Bm

where ρ1m (u, v) depends on the max(u, v), ∞ 

Mn ≤ ρ < ∞,

n=−∞

and

∞ 

ρ1n (u, v) ≤ ρ1 (u, v) < ∞.

n=−∞

In addition c(x, u)h − c(y, v)h ≤ [ρx − yp + ρ1 u − v]h. We determine successive approximations ξn (t), ηn (t), un (s, x) to the solution ξ(t), η(t) of the system (4.1)-(4.3) as follows  f (ξn (t), z)ν(dt, dz), ξn (s) = x ∈ Qp , (4.5) dξn (t) = Qp

dηn (t) = c(ξn (t), un (t, ξn (t)))ηn (t)dt p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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(4.6)

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 +

ηn (s) = h ∈ Rd .

C(ξn (t), z, un (t, ξn (t)))ηn (t)ν(dt, dz), Qp

h, un+1 (s, x) = Eηn (T ), u0 (ξn (T )).

(4.7)

To prove that there exists a limit ξ(t) ∈ Qp , η(t) ∈ Rd and u(t, x) ∈ Rd we need some auxiliary estimates. Consider a couple of SDEs  f (ξ(t), z))ν(dt, dz), ξ(s) = x ∈ Qp , (4.8) dξ(t) = Qp

 C(ξ(t), z, v(t, ξ(t)))η(t)ν(dt, dz),

dη(t) = c(ξ(t), v(t, ξ(t)))η(t)dt +

η(s) = h ∈ Rd .

(4.9)

Qp

We say that v satisfies condition C 4.5 if v is a spatially Lipschitz continuous Rd -valued function defined m on Qp and there exist real valued positive functions Lm v (t), Kv (t) such that v(t, x) − v(t, y) ≤ Lm v (t)x − yp for x, y ∈ Bm , ∞

∞

supx∈Bm v(t, x) ≤ Kvm (t)

and m=−∞ Kvm (t) = Kv (t), m=−∞ Lm v (t) = Lv (t), where Kv (t), Lv (t) are bounded for t ∈ [0, T1 ] for some T1 > 0. Lemma 4.1. Assume that conditions C 4.1-C 4.4 and C 4.5 hold. Then the solutions ξ(t) and η(t) of (4.8), (4.9) satisfy the estimates Eξx (t) − ξy (t) ≤ Cf x − yp ,  t 2 2 Eη(t) ≤ h exp{ 2ρ0 [1 + [Kv ]k (τ )]dτ },

(4.10) (4.11)

s

 t Eηx (t) − ηy (t) ≤ exp{ [L1 + ρ1 Lv (τ ) + CKvk (τ )]dτ }x − yp .

(4.12)

s

Proof. The proof of (4.10) immediately follows from the properties of stochastic integral and the ultra-metric inequality. Since the other two estimates are based on coefficient estimates, properties of stochastic integrals and the Gronwall lemma we give the proof of (4.12) only. It follows from (4.8), (4.9) that  t Ec(ξx (τ ), v(τ, ξx (τ )))ηx (τ ) − c(ξy (τ ), v(τ, ξy (τ )))ηy (τ )dτ Eηx (t) − ηy (t) ≤ s

 t

EC(ξx (τ ), z, v(τ, ξx (τ )))ηx (τ ) − C(ξy (τ ), z, v(τ, ξy (τ )))ηy (τ )π(dz)dτ

+ s

(4.13)

Qp



t

≤

 t [ρ + ρ1 Lv (τ )]Eξx (τ ) − ξy (τ )p dτ exp{ C[1 + Kvk (τ )]dτ },

s

s

that easily yields (4.12) with L1 = ρ + C + Cf due to (4.10). In a similar way we check (4.11). Let us derive estimates for the function u(s, x) given by h, u(s, x) = Eη(T ), u0 (ξ(T )),

(4.14)

where ξ(t), η(t) satisfy (4.8), (4.9). Lemma 4.2. Let C4.1- C 4.4 hold and sup u0 (x) ≤ K0 . Then there exists a function γ(s) bounded on a certain interval Δ1 = [s1 , T ] such that the function u(s, x) defined by (4.14) satisfies the estimate supx u(s, x) ≤ γ(s) if v(s, x) satisfy the estimate supx v(s, x) ≤ γ(s). p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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Proof. Applying the estimates of the process η(t) from Lemma 4.1 we obtain  t u(s, x) ≤ ET ∗ (θ, s, ξ(·))u0 (ξ(θ)) ≤ K0 exp{ ρ0 [1 + [Kv ]k (τ )]dτ } s

and for Ku (s) = supx∈Qp u(s, x) we get



Ku (s) ≤ K0

T

ρ0 [1 + [Kv ]k (τ )]dτ }.

s

It is easy to check that the function γ(s) that solves the equation  T ρ0 [1 + γ k (τ )]dτ } γ(s) = K0 exp

(4.15)

s

possesses the required property. Let us verify that there exists an interval [s1 , T ] such that γ(s) is bounded for s ∈ [s1 , T ]. Assume for simplicity that k = 1 in the condition 4.3. In this case the function γ(s) satisfies the ODE dγ(s) = −ρ0 γ(s)[1 + γ(s)] ds and the boundary condition γ(T ) = K0 . Hence γ(s) =

K0 eρ0 (T −s) 1 + K0 − K0 eρ0 (T −s)

(4.16)

and γ(s) is bounded on the interval of the length κ = T − s with κ<

1 1 ln[1 + ]. ρ0 K0

(4.17)

The similar results we can get for k = 2, 3, 4. Namely for k = 2, 3, 4 we obtain ODEs for the function γ which still we can solve explicitly and show that their solutions are bounded over correspondent time intervals. In general case we prove the assertion of the lemma on an interval where a solution of a correspondent ODE resulted from an inequality of the type (4.15) does exist. Finally we prove that if v(s, x) is Lipschitz continuous then u(s, x) possesses this property. Lemma 4.3. Assume that u0 (x) is Lipschitz continuous u0 (x) − u0 (y) ≤ L0 x − yp and C 4.1- C 4.4 hold. Then there exists a function β(t) bounded on a certain interval Δ2 = [s2 , T ] such that the function u(s, x) defined by (4.14) satisfies the estimate u(s, x) − u(s, y) ≤ β(s)x − yp ) if v(s, x) satisfy the estimate v(s, x) − v(s, y) ≤ β(s)x − yp . Proof. It results from (4.14) and the estimates of lemma 4.1 that Eu(s, x) − u(s, y) ≤ K0 Eηx (T ) − ηy (t) + E[ηy (T )L0 ξx (T ) − ξy (T )]. Applying the estimates of lemma 4.1 we get

 t Eu(s, x) − u(s, y) ≤ K0 exp{ [L1 + ρ1 Lv (τ ) + CKvk (τ )]dτ }x − yp + s

 t Cf exp{ ρ0 [1 + [Kv ]k (τ )]dτ }x − yp . s

Let Lu (t) be a minimal factor such that u(s, x) − u(s, y) ≤ Lu (s)x − y, then we have   T T ρ0 [1+[Kv ]k (τ )]dτ [L1 +ρ1 Lv (τ )+CKvk (τ )]dτ s s 1 + L0 e Cf . Lu (s) ≤ K0 e p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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T Let Q0 = L1 + C supτ ∈[s1 ,T ] Kv (τ ) and D0 = K0 + L0 Cf φ(s1 ) where φ(s1 ) = exp{ s1 ρ0 [1 + [Kv ]k (τ )]dτ } and s1 is defined in the previous lemma. Repeating the arguments used in the proof of lemma 4.2 we check that the function β(s) that solves the integral equation  T Q0 [1 + β(τ )]dτ } β(s) = D0 exp{ s

or the Cauchy problem for the ODE dβ(s) = Q0 [1 + β(s)]β(s), ds As a result we obtain that the function β(s) of the form −

β(s) =

β((T ) = D0 .

D0 eQ0 (T −s) 1 + D0 − D0 eD0 (T −s)

possesses the required property on the interval [s2 , T ] where κ1 ≡ T − s2 satisfies the estimate κ1 ≤

1 1 ln(1 + ). D0 Q0

Finally the estimates of lemma 4.3 and 4.2 allow to prove the convergence of successive approximations (4.5)- (4.7). As a result we can state the following assertion. Theorem 4.1. Let C 4.1-C4.4 and C 4.5 hold. Then there exists an interval [s1 , T ] such that for all s ∈ Δ with Δ = Δ1 ∩ Δ2 there exists a unique solution (ξ(t), η(t), u(s, x)) of the system (4.1)-(4.3). The process ξ(t) ∈ Qp is a Markov process and u(s, x) ∈ Rd is a Lipschitz continuous bounded function on [s1 , T ] × Qp . Proof. Due to estimates of lemma 4.2 and 4.3 we know that the successive approximations un (s, x) of the form (4.7) are uniformly bounded and equicontinuous. Hence by Arzela-Ascoli theorem we get that the family un (s, ·) converges to a Lipschitz continuous function u(s, ·) uniformly on the interval Δ. Given a Lipschitz continuous function u(s, x) we can prove the convergence of ξn (t) and ηn (t) to limiting processes ξ(t) and η(t) that satisfy (4.1), (4.2). To prove that the solution of (4.1)-(4.3) is unique we recall that we have constructed the function u(s, x) as a fixed point of the nonlinear map u(s, x) = Φ(x, u(s, x)) where Φ(x, u) was defined by the right hand side of (4.3). Since this map was proved to be a contraction, the function u(s, x) is unique. In addition it was proved that this function is Lipschitz continuous, that yields the uniqueness of the solution to (4.2) due to results stated in section 2. The uniqueness of the solution to (4.1) under the condition C 2.1 - C 2.4 was proved in [1]. It remains to derive the PIDE that governs the function u(s, x). But this is an immediate consequence of theorem 3.1 since given a bounded Lipschitz continuous function u we are in the framework of section 3 and hence we can check that function v(s, x) defined by the right hand side of (4.3), that is v defined by h, v(s, x) = E[η(T ), u0 (ξ(T ))] solves the system of PIDEs  d  ∂vi + [vi (s, x + f (x, z)) − vi (s, x)]π(dz) + cij (x, u(s, x))vj (s, x)+ ∂s Qp

(4.18)

j=1



d 

Cij (x, z, u(s, x))vj (s, x + f (x, z))π(dz) = 0,

ui (T, x) = u0i (x).

Qp j=1

At the other hand the uniqueness of the solution to (4.1)- (4.3) yields that v(s, x) = u(s, x). Now we can state the final result. p-ADIC NUMBERS, ULTRAMETRIC ANALYSIS AND APPLICATIONS

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Theorem 4.2. Let C 4.1-C 4.4 and C 4.5 hold. Then there exists a unique solution of the Cauchy problem  d  ∂ui + [ui (s, x + f (x, z)) − ui (s, x)]π(dz) + cij (x, u(s, x))uj (s, x) (4.19) ∂s Qp j=1

 +

d 

Cij (x, z, u(s, x))uj (s, x + f (x, z))π(dz) = 0,

ui (T, x) = u0i (x).

Qp j=1

This solution admits a probabilistic representation of the form (4.3), where the processes ξ(t) and η(t) satisfy (4.1), (4.2). Proof. We have already stated based on theorem 3.1 that u(s, x) given by (4.3) satisfies (4.18) and now we can immediately check due to (4.3) that u(T, x) = u0 (x). To prove that the solution given by (4.3) is unique we consider two Lipschitz continuous solutions u1 (s, x) and u(s, x) of (4.18) such that u1 (T, x) = u(T, x) = u0 (x). Then we can check that u(s, x) − u1 (s, x) ≤ Eηu1 (T ), u0 (ξu1 (T ) − Eηu (T ), u0 (ξu (T ) ≤ K0 Eηu1 (T ) − ηu (T ) + L0 Eηu (T )ξu1 (T ) − ξu (T ). Finally due to estimates of lemma 4.1 we get u(s, x) − u1 (s, x) = 0,

sup s∈Δ,x∈Qp

hence u and u1 must coincide. ACKNOWLEDGEMENTS The support of the Grant DFG 436 RUS 113/809, Grant RFBR 05-01-04002-NNIO-a and DFG 436 RUS 113/823 is gratefully acknowledged. REFERENCES 1. A. Kochubei, Pseudo-Differential Equations and Stochastics over non-Archimedian Fields (Marcel Dekker Inc, 2001). 2. S. Albeverio and Ya. Belopolskaya, “Qp -valued jump processes associated with linear and nonlinear pseudodifferential equations,” Zapiski Nauchnyh Seminarov POMI Probability and Statistics, Part 12 351, 5–37 (2007). 3. Ya. Belopolskaya, “On stochastic equations with unbounded coefficients for jump processes,” Lecture Notes in Control and Information Theory 25, 243–254 (1980). 4. Ya. Belopolskaya, “Systems of quasilinear integro-differential equations and Markov processes associated with them,” in Probabilistic Distributions in Infinite Dimensional Spaces, pp. 5–21 (Nauk. Dumka, 1978). 5. S. Albeverio and X. Zhao, “Measure valued branching processes associated with random walks on p-adics,” Annals of Probability 28 (4), 1680–1710 (2000).

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