Multidimensional nonlinear pseudo-differential evolution equation with p-adic spatial variables
Alexandra V. Antoniouk, Andrei Yu. Khrennikov, Anatoly N. Kochubei
aa r X i v : . [ m a t h . A P ] S e p Multidimensional nonlinear pseudo-differential evolutionequation with p -adic spatial variables Alexandra V. Antoniouk
Institute of Mathematics,National Academy of Sciences of Ukraine,Tereshchenkivska 3, Kiev, 01004 Ukraine,E-mail: [email protected]
Andrei Yu. Khrennikov
International Center for Mathematical Modelingin Physics and Cognitive Sciences,Mathematical Institute, Linnaeus University,V¨axj¨o, SE-35195 Sweden,E-mail: [email protected]
Anatoly N. Kochubei
Institute of Mathematics,National Academy of Sciences of Ukraine,Tereshchenkivska 3, Kiev, 01004 Ukraine,E-mail: [email protected]
September 17, 2019
Abstract
We study the Cauchy problem for p -adic non-linear evolutionary pseudo-differen-tial equations for complex-valued functions of a real positive time variable and p -adicspatial variables. Among the equations under consideration there is the p -adic analogof the porous medium equation (or more generally, the nonlinear filtration equation)which arise in numerous application in mathematical physics and mathematicalbiology. Our approach is based on the construction of a linear Markov semigroup ona p -adic ball and the proof of m-accretivity of the appropriate nonlinear operator.The latter result is equivalent to the existence and uniqueness of a mild solution ofthe Cauchy problem of a nonlinear equation of the porous medium type. Key words: p -adic numbers; porous medium equation; Markov process; m-accretiveoperator MSC 2010 . Primary: 35S10; 47J35. Secondary: 11S80; 60J25; 76S05.1
Introduction and Preliminaries
Over a period of several hundred years theoretical physics has been developed on thebasis of real and complex analysis. However, for the past half a century the field ofp-adic numbers Q p (as well as its algebraic extensions) has been intensively used intheoretical and mathematical physics (see [25, 15, 16, 17, 20] and the references therein).Since in p -adic analysis associated with the mappings from Q p to C , the operation ofdifferentiation is not defined, many p -adic models instead of differential equations usepseudo-differential equations generated by so called Vladimirov operator D α .A wide class of p -adic pseudo-differential equations was intensively used in applica-tions, for example to model basin-to-basin kinetics [2, 3, 19, 20], for instance to constructthe simplest p -adic pseudo-differential heat type equation [3]. In [21] and in [11] ultra-metric and p -adic nonlinear equations were used to model turbulence.Some types of p-adic pseudo-differential equations were studied in detail in the books[17, 1] and [26]; see also the recent survey [5].At the same time very little is known about nonlinear p-adic equations. We can men-tion only some semilinear evolution equations solved using p-adic wavelets [1] and a kindof equations of reaction-diffusion type studied in [27]. A nonlinear evolution equation forcomplex-valued functions of a real positive time variable and a p -adic spatial variable,which is a non-Archimedean counterpart of the fractional porous medium equation, thatis the equation: ∂u∂t + D α (cid:0) ϕ ( u ) (cid:1) = 0 , u = u ( t, x ) , t > , x ∈ Q p (1.1)was studied in the recent paper [18]. Here Q p is the field of p -adic numbers, D α , α > ϕ is a strictly monotone increasingcontinuous function. Developing an L -theory of Vladimirov’s p -adic fractional differ-entiation operator, the authors of [18] proved the m -accretivity of the correspondingnonlinear operator and obtained the existence and uniqueness of a mild solution.In this paper we prove a similar result for a more complicated multi-dimensionalnonlinear equation resembling (1.1) but with more general nonlocal operator W (1.8)instead of the Vladimirov operator D α . We follow essentially the strategy of the paper[18] and use the abstract theory for nonlinear m -accretive operators in Banach space L developed in [6] and further in [7].In order to use this method we need to build the L -theory for the weighted Vladimirovoperator W and prove some special properties of its restriction onto p -adic ball. Thismeets several difficulties, which we overcome by reconstruction of the associated Markovprocess in the p -adic ball and by recovering of the associated Levy measure. We proveseveral additional properties of W , which are necessary for our tasks and may haveindependent interest.Let us formulate the main definitions and auxiliary statement, which will be usedfurther. p -Adic numbers. Let p be a prime number. The field of p -adic numbers is thecompletion Q p of the field Q of rational numbers, with respect to the absolute value | x | p | | p = 0, | x | p = p − ν if x = p ν mn , where ν, m, n ∈ Z , and m, n are prime to p . Q p is a locally compact topological field.Note that by Ostrowski’s theorem there are no absolute values on Q , which are notequivalent to the “Euclidean” one, or one of | · | p .The absolute value | x | p , x ∈ Q p , has the following properties: | x | p = 0 if and only if x = 0; | xy | p = | x | p · | y | p ; | x + y | p ≤ max( | x | p , | y | p ) . The latter property called the ultra-metric inequality (or the non-Archimedean prop-erty) implies the total disconnectedness of Q p in the topology determined by the metric | x − y | p , as well as many unusual geometric properties. Note also the following conse-quence of the ultra-metric inequality: | x + y | p = max( | x | p , | y | p ) if | x | p = | y | p . The absolute value | x | p takes the discrete set of non-zero values p N , N ∈ Z . If | x | p = p N , then x admits a (unique) canonical representation x = p − N (cid:0) x + x p + x p + · · · (cid:1) , (1.2)where x , x , x , . . . ∈ { , , . . . , p − } , x = 0. The series converges in the topology of Q p . For example, − p −
1) + ( p − p + ( p − p + · · · , | − | p = 1 . The fractional part of element x ∈ Q p in canonical representation (1.2) is given by: { x } p = (cid:26) , if N ≤ x = 0; p − N (cid:0) x + x p + . . . + x N − p N − (cid:1) , if N > . Q np . The space Q np = Q p × · · · × Q p consists of points x = ( x , . . . , x n ), where x j ∈ Q p , j = 1 , . . . , n , n ≥
2. The p -adic norm on Q np is k x k p = max j =1 ,...,n | x j | p , x ∈ Q np . This norm is also non-Archimedean, since for any x, y ∈ Q np : k x + y k p = max j =1 ,...,n | x j + y j | p ≤ max j =1 ,...,n max (cid:0) | x j | p + | y j | p (cid:1) == max (cid:0) max j =1 ,...,n | x j | p , max j =1 ,...,n | y j | p (cid:1) = max (cid:0) k x k p , k y k p (cid:1) . Q np is a complete metric locally compact and totally disconnected space. Thescalar product of vectors x, y ∈ Q np is defined by x · y = n P j =1 x j y j . In the space Q np the following change of variables formula is valid (see [25, Ch.I, §
4, Sect. 4, p. 68]).If F : x = x ( y ) (i.e. x i = x i ( y , . . . , y n ) , i = 1 , . . . , n ) is a homeomorphic map ofthe open compact set K onto the (open) compact set F ( K ), moreover the functions x i ( y ) , i = 1 , . . . , n are analytic in K and det F ′ ( y ) = det h ∂x j /∂y j i ( y ) = 0 , y ∈ K , thenfor any f ∈ L ( T ( K )): Z F ( K ) f ( x ) d n x = Z K f ( F ( y )) k det F ′ k p d n y. (1.3) Q np . The function χ ( x ) = exp (cid:0) πi { x } p (cid:1) is an additive character of the field Q p , i.e. it is a character of its additivegroup, the continuous complex valued function on Q p satisfying the conditions: | χ ( x ) | = 1 , χ ( x + y ) = χ ( x ) + χ ( y ) . (1.4)Moreover χ ( x ) = 1 if and only if | x | p ≤ . Denote by dx the Haar measure on the additivegroup Q p . Then the integral on the p -adic ball B N = (cid:8) x ∈ Q np : k x k p ≤ p N (cid:9) , N ∈ Z , equals ([24, (7.14), p. 25]): Z B N χ ( ξ · x ) d n x = (cid:26) p nN , if k ξ k p ≤ p − N ;0 , if k ξ k p > p − N , (1.5)Lemma 4.1 in [23, Ch.III, §
4, p. 137] implies that for z such that k z k p = 1: Z S j χ ( z · x ) d n x = p jn (1 − p − n ) , j ≤ − p n p − n , j = 1;0 , j > , (1.6)where S j = (cid:8) x ∈ Q np : k x k p = p j (cid:9) , j ∈ Z .The Fourier transform of a test function ϕ ∈ D ( Q np ) is defined by the formula b ϕ ( ξ ) ≡ (cid:0) F x → ξ ϕ (cid:1) ( ξ ) = Z Q np χ ( ξ · x ) ϕ ( x ) d n x, ξ ∈ Q np , where χ ( ξ · x ) = χ ( ξ x ) · · · χ ( ξ n x n ) = exp(2 πi P nj =1 { ξ j x j } p ).Remark that the additive group of Q p is self-dual, so that the Fourier transform of acomplex-valued function ϕ ∈ Q np is again a function on Q np and if F x → ξ ϕ ∈ L ( Q np ) thenit is true the inversion formula ϕ ( x ) = R Q np χ ( − x · ξ ) b ϕ ( ξ ) d n ξ. Let us also remark that,in contrast to the Archimedean situation, the Fourier transform ϕ → F x → ξ ϕ is a linearand continuous automorphism of the space D ( Q np ) (cf. [1, Lemma 4.8.2], see also [13,Ch. II,2.4.], [23, III,(3.2)], [25, VII.2.], i.e. ϕ ( x ) = F − ξ → x (cid:16) F x → ξ ϕ (cid:17) . Here D ( Q np ) denotesthe vector space of test functions , i.e. of all locally constant functions with compact4upport. Recall that a function ϕ : Q np → C is locally constant if there exist such aninteger ℓ ≥ x ∈ Q np ϕ ( x + y ) = ϕ ( x ) , if k y k p ≤ p − ℓ , ( ℓ is independent on x ).The smallest number ℓ with this property is called the exponent of constancy of thefunction ϕ . Note that D ( Q np ) is dense in L q ( Q np ) for each q ∈ [1 , ∞ ).Let us also introduce the subspace D ℓN ⊂ D ( Q np ) consisting of functions with supportsin a ball B N and with the exponents of local constancy less than ℓ . Then the topologyin D ( Q np ) is defined as the double inductive limit topology, so that D ( Q np ) = lim −→ N →∞ lim −→ ℓ →∞ D ℓN . If V ⊂ Q np is an open set, the space D ( V ) of test functions on V is defined as asubspace of D ( Q np ) consisting of functions with supports in V . For a ball V = B N , wecan identify D ( B N ) with the set of all locally constant functions on B N .The space D ′ ( Q np ) of Bruhat-Schwartz distributions on Q np is defined as a strongconjugate space to D ( Q np ). By duality, the Fourier transform is extended to a linear (andtherefore continuous) automorphism of D ′ ( Q np ). For a detailed theory of convolutionsand direct product of distributions on Q np closely connected with the theory of theirFourier transforms see [1, 17, 25] Q np . In this paper we consider the class of non-local operatorsintroduced in [26]. Definition 1.1.
Let us fix a function w : Q np → R + , which satisfies the followingproperties:(i) w is radial, i.e. depending on k y k p , w = w (cid:0) k y k p (cid:1) , continuous and increasingfunction of k y k p ;(ii) w (0) = 0, if y = 0;(iii) there exist such constants C > α > n that C k ξ k αp ≤ w ( k ξ k p ) ≤ C k ξ k αp , for any ξ ∈ Q np . (1.7)Remark that condition (iii) implies that for some M ∈ Z Z k y k p ≥ p M d n yw (cid:0) k y k p (cid:1) < ∞ . The nonlocal operator W is defined by( W ϕ )( x ) = κ Z Q np ϕ ( x − y ) − ϕ ( x ) w ( k y k p ) d n y, for ϕ ∈ D ( Q np ) , (1.8)5here κ is some positive constant. From [26, (2.5), p.15] it follows that for ϕ ∈ D ( Q np )and some constant M = M ( ϕ ) operator W has the following representation:( W ϕ )( x ) = κ Q np \ B M w ( k x k p ) ∗ ϕ ( x ) − ϕ ( x ) Z k y k p >p M d n yw ( k y k p ) . (1.9)Moreover from Lemma 4 and Proposition 7, Ch.2 in [26] it follows that this operator,acting from D ( Q np ) to L q ( Q np ), is linear bounded operator for each q ∈ [1 , ∞ ) and hasthe representation:( W ϕ )( x ) = − κ F − ξ → x (cid:0) A w ( ξ ) F x → ξ ϕ (cid:1) , for ϕ ∈ D ( Q np ) , (1.10)where A w ( ξ ) := Z Q np − χ ( y · ξ ) w ( k y k p ) d n y. (1.11)From [26, (2.9), p.16] for any z ∈ Q np and k z k p = p − γ it follows that A w ( z ) =(1 − p − n ) ∞ X j =2 p γn + jn w ( p γ + j ) + p γn + n w ( p γ +1 ) = (1.12)= (1 − p − n ) ∞ X j = γ +2 p nj w ( p j ) + p n ( γ +1) w ( p γ +1 ) . (1.13)The condition (1.7) on function w implies that there exist positive constants C and C such that C k ξ k α − np ≤ A w ( ξ ) ≤ C k ξ k α − np , (1.14)(see Lemma 8, Ch.2 in [26]).Remark also that W ϕ ∈ C ( Q np ) ∩ L q ( Q np ) for each q ∈ [1 , ∞ ) and ϕ ∈ D ( Q np ) andoperator W may be extended to a densely defined operator in L ( Q np ) with the domain Dom ( W ) = (cid:8) ϕ ∈ L ( Q np ) : A w ( ξ ) F x → ξ ϕ ∈ L ( Q np ) (cid:9) . The operator (cid:0) − W, Dom ( W ) (cid:1) is essentially self-adjoint and positive. In particular, itgenerates a C -semigroup of contractions T ( t ) in the space L ( Q np ) (Proposition 20 in[26]): T ( t ) u = Z t ∗ u = Z Q np Z ( t, x − y ) u ( y ) d n y, t > T (0) u = u. (1.15)Here Z t ( x ) = Z ( t, x ) Z ( t, x ) = Z Q np e − κ tA w ( ξ ) χ ( − x · ξ ) d n ξ, for t > . (1.16)6s the heat kernel or fundamental solution of the corresponding Cauchy problem. Laterwe need the following properties of the fundamental solution Z ( t, x ) (Lemma 10 and 11and Theorem 13, Ch. 2 in [26]):1) Z ( t, x ) ≥ Z t ( x ) ∈ L ( Q np ) , for t > Z Q np Z ( t, x ) d n x = 1; (1.18)3) Z ( t + s, x ) = Z Q np Z ( t, x − y ) Z ( s, y ) d n y, t, s > , x ∈ Q np ; (1.19)4) Z ( t, x ) = F − ξ → x (cid:2) e − κ tA w ( ξ ) (cid:3) ∈ C ( Q np ; R ) ∩ L ( Q np ) ∩ L ( Q np ); (1.20)5) Z ( t, x ) ≤ max { α C , α C } t (cid:16) k x k p + t α − n (cid:17) − α , for t > x ∈ Q np ; (1.21)6) D t Z ( t, x ) = − κ Z Q np A w ( ξ ) e − κ tA w ( ξ ) χ ( x · ξ ) d n ξ, for t > , x ∈ Q np ; (1.22)7) Z ( t, x ) = k x k − np " (1 − p − n ) ∞ X j =0 p − nj e − κ tA w ( p − ( β + j ) ) − e − κ tA w ( p − β +1 ) , if k x k p = p β . (1.23)Since Z t ( x ) ∈ L ( Q np ) for t >
0, for u ∈ D ( Q np ) ⊂ L ∞ ( Q np ) the convolution in (1.15)exists and is a well-defined continuous in x function (See Th, 1.11). . We use the analytic definition of a Markov process,that is a definition of a transition probability. (See for example [8].) Suppose that ( E, E )is a measurable space. A family of real-valued non-negative measurable with respect tovariable x functions P ( s, x ; t, Γ), s < t , x ∈ E, Γ ∈ E , such that P ( s, x ; t, · ) is a measureon E and P ( s, x ; t, Γ) ≤
1, is called a transition probability , if the Kolmogorov-Chapmanequality Z E P ( t, y ; τ, Γ) P ( s, x ; t, dy ) = P ( s, x ; τ, Γ)holds whenever s < t < τ , x ∈ E , Γ ∈ E .We consider E = (cid:16) Q np , k · k p (cid:17) as a complete non-Archimedean metric space. Let E denote the Borel σ -algebra In Chapter 2.2.5 (Lemma 14, p. 22) of [26] is was shown thatthe transition probability P ( t, x, B ) = ( R B p ( t, x, y ) d n y, for t > x, y ∈ Q np , B ∈ E B ( x ) , for t = 0 , with p ( t, x, y ) := Z ( t, x − y ) and Z ( t, x ) defined in (1.16), is normal, i.e. lim t ↓ s P ( s, x ; t, Q np ) =1, for any s > x ∈ Q np .Due to Theorem 16 in [26, § Z ( t, x ) is the transition density of a time and space homogeneous Markov process ξ t §
1, p.46] this process has an independent increments as a space and time homogeneousMarkov process. (See also [12, Vol. I, Ch. III, §
1, p. 188]) L ( Q np ) Let us consider operator T ( t ) defined by (1.15): (cid:0) T ( t ) u (cid:1) ( x ) = Z Q np Z ( t, x − ξ ) u ( ξ ) dξ in Banach space L ( Q np ). From (1.19), (1.21) and Young inequality it follows that T ( t )is a contraction semigroup in L ( Q np ). Lemma 2.1. T ( t ) is a strongly continuous semigroup in L ( Q np ) .Proof. Since the space D ( Q np ) of Bruhat-Schwartz functions is dense in L ( Q np ) [25] it issufficient to prove the convergence of the expression I t := k T ( t ) u − u k L ( Q np ) → , t → u ∈ D ( Q np ).Due to (1.18) we have for u ∈ D ( Q np ): I t := Z Q np (cid:12)(cid:12) T ( t ) u ( x ) − u ( x ) (cid:12)(cid:12) d n x = Z Q np (cid:12)(cid:12)(cid:12) Z Q np Z ( t, x − ξ ) u ( ξ ) d n ξ − u ( x ) (cid:12)(cid:12)(cid:12) d n x == Z Q np Z Q np Z ( t, x − ξ ) (cid:12)(cid:12) u ( ξ ) − u ( x ) (cid:12)(cid:12) d n ξ d n x. Here we also used (1.17). Since function u ∈ D ( Q np ) is locally constant, then such isalso u ( ξ ) − u ( x ) and therefore it exists some m > u ( ξ ) − u ( x ) = 0 for k x − ξ k p ≤ p m . Moreover there exists some N > m such that u ( x ) = 0 for k x k p > p N .Noting this let us represent integral I t as a sum of two integrals: inside and outside the p -adic ball. Since on the ball k x − ξ k p ≤ p m functions u ( x ) and u ( ξ ) coincides, therefore,using (1.21), we have I ,t = Z k x k p ≤ p N d n x Z k x − ξ k p >p m Z ( t, x − ξ ) | u ( ξ ) − u ( x ) | d n ξ ≤≤ C t Z k x k p ≤ p N d n x Z k x − ξ k p >p m (cid:16) t α − n + k x − ξ k p (cid:17) − α d n ξ ≤≤ C t Z k x k p ≤ p N d n x Z k z k p >p m k z k − αp d n z → , t → , for α > n. k x k p > p N , function u vanishes, u ( x ) = 0, we have I ,t = Z k x k p >p N d n x Z Q np Z ( t, x − ξ ) | u ( ξ ) − u ( x ) | d n ξ = Z k x k p >p N d n x Z Q np Z ( t, x − ξ ) | u ( ξ ) | d n ξ == Z k x k p >p N d n x Z k ξ k p ≤ p N Z ( t, x − ξ ) | u ( ξ ) | d n ξ. On the last step we also used that support of function u ( ξ ) is contained in the ball k ξ k p ≤ p N . Finally, using (1.21) we continue: I ,t ≤ C t Z k x k p >p N d n x Z k ξ k p ≤ p N (cid:16) t α − n + k x − ξ k p (cid:17) − α d n ξ == C t Z k x k p >p N d n x Z k ξ k p ≤ p N (cid:16) t α − n + k x k p (cid:17) − α d n ξ → , as t → , where we have used that k x − ξ k p = k x k p for k x k p > p N . Definition 2.2.
Let us define operator A as a generator of semigroup T ( t ) in space L ( Q np ) and let Dom ( A ) be its domain. Lemma 2.3.
Any test function u ∈ D ( Q np ) belongs to the domain of the operator A in L ( Q np ) : D ( Q np ) ⊂ Dom ( A ) . Moreover, on the test functions the operator A coincideswith the representation of operator W of the form (1.10) .Proof. Let us write for u ∈ D ( Q np ):1 t (cid:16) T ( t ) u − u (cid:17) = 1 t (cid:0) Z t ∗ u − u (cid:1) == 1 t (cid:16) F − ξ → x (cid:2) e − κ tA w ( ξ ) (cid:3) ∗ u − F − ξ → x F x → ξ u (cid:3)(cid:17) == 1 t (cid:16) F − ξ → x (cid:2) e − κ tA w ( ξ ) (cid:3) ∗ F − ξ → x F x → ξ u − F − ξ → x F x → ξ u (cid:3)(cid:17) == 1 t (cid:16) F − ξ → x (cid:2) e − κ tA w ( ξ ) · F x → ξ u (cid:3) − F − ξ → x F x → ξ u (cid:3)(cid:17) == F − ξ → x h t (cid:0) e − κ tA w ( ξ ) − (cid:1) F x → ξ u i . Taking into account these calculations, to finish the prove we need to show that F t := Z Q np (cid:12)(cid:12)(cid:12) F − ξ → x (cid:2) t (cid:0) e − κ tA w ( ξ ) − (cid:1) F x → ξ u + κ A w ( ξ ) F x → ξ u (cid:3)(cid:12)(cid:12)(cid:12) d n ξ → , t → . To see this it is sufficient to note that for u ∈ D ( Q np ) such that the support of F x → ξ u iscontained in some ball B N : (cid:12)(cid:12) t (cid:0) e − κ tA w ( ξ ) − (cid:1) + κ A w ( ξ ) (cid:12)(cid:12) · |F x → ξ u | =9 (cid:12)(cid:12)(cid:12) t (cid:16) ∞ X q =0 ( − q (cid:0) κ tA w ( ξ ) (cid:1) q q ! − (cid:17) + κ A w ( ξ ) (cid:12)(cid:12)(cid:12) · |F x → ξ u | == t A w ( ξ ) κ (cid:12)(cid:12) ∞ X m =2 ( − m κ m t m A mw ( ξ )( m + 2)! (cid:12)(cid:12) · |F x → ξ u | ≤≤ C N t A w ( ξ ) |F x → ξ u | since the series converges for any ξ which belongs to the support of u . Therefore F t ≤ Z Q np (cid:12)(cid:12)(cid:12)(cid:16) t (cid:0) e − κ tA w ( ξ ) − (cid:1) + κ A w ( ξ ) (cid:17) F x → ξ u (cid:12)(cid:12) d n ξ ≤≤ C N t Z k ξ k p ≤ p − N A w ( ξ ) |F x → ξ u | d n ξ → , t → . Let ξ t be the Markov process on Q np constructed in 2.4 of Section 1. Like in [17] (Section4.6.1) we construct the Markov process on a ball B N = { x ∈ Q np : k x k ≤ p N } . Supposethat ξ ∈ B N . Denote by ξ ( N ) t the sum of all jumps of the process ξ τ , τ ∈ [0 , t ], whoseabsolute values exceed p N . Since ξ t is right continuous process with left limits, ξ ( N ) t isfinite a.s. Moreover ξ ( N )0 = 0. Let us consider process η t = ξ t − ξ ( N ) t . (3.1)Since the jumps of η t never exceed p N by absolute value, this process remain a.s. in B N (due to the ultra-metric inequality).Below we will identify a function on ball B N with its extension by zero onto thewhole Q np . Let D ( B N ) consist of functions from D ( Q np ) supported in the ball B N . Then D ( B N ) is dense in L ( B N ), and the operator W N in L ( B N ) defined by restricting theoperator W (1.8) to D ( B N ) and considering (cid:0) W u (cid:1) ( x ) only for x ∈ B N . Let us find thegenerator W N of process η t . To do this we need the following bunch of lemmas. Lemma 3.1.
For any z ∈ Q np E χ (cid:0) z · ξ t (cid:1) = exp (cid:0) − t κ A w ( z ) (cid:1) . (3.2) Proof.
Indeed, let µ t denote the distribution of the process ξ t with ξ = 0. Then, due to[12, Vol.2, p. 24], the Fourier transform of measure µ t is equal to E χ (cid:0) z · ξ t (cid:1) = b µ t ( z ) = Z Q np χ ( z · y ) µ t ( dy ) = Z Q np χ ( z · y ) p ( t, , y ) d n y =10 Z Q np χ ( z · y ) Z ( t, y ) d n y = Z Q np χ ( z · y ) F − ξ → y h e − κ tA w ( ξ ) i d n y = e − κ tA w ( z ) . Here we also used (1.20) and that the Fourier transform F is a linear continuous auto-morphism of the space D ( Q np ). Lemma 3.2. I B N ( z ) = Z B N − χ ( z · x ) w ( k x k p ) d n x = (cid:26) , if k z k p ≤ p − N ; A w ( z ) − λ N , if k z k p > p − N , (3.3) where λ N = (1 − p − n ) ∞ X j = N +1 p nj w ( p j ) . (3.4) Remark 3.3.
Remark that λ N = Z k y k p >p N d n yw ( k y k p ) . (3.5) Indeed. λ N = Z k y k p > p N d n yw ( k y k p ) = ∞ X j = N +1 Z S j d n yw ( p j ) = (1 − p − n ) ∞ X j = N +1 p nj w ( p j ) . Remark 3.4. I Q np ( z ) = A w ( z ) . Proof.
First of all remark that the character χ equals 1 on ball B . Hence E χ ( z · η t ) = 1if k z k p ≤ p − N . Let us consider z ∈ Q np such that k z k p = p k , k ≥ − N + 1. Any such z belongs to S k , k ≥ − N + 1 and we may be represent it as z = p − k z , where k z k p = 1. Since B N \{ } = G j ≤ N p j S = G j ≤ N S j , we may represent in the following form: I B N ( z ) = Z B N − χ ( z · x ) w ( k x k p ) d n x = N X j = −∞ Z S j − χ ( z · x ) w ( k x k p ) d n x = N X j = −∞ Z S j − χ ( p − k z · x ) w ( k x k p ) d n x. The change of variables formula (1.3) implies I B N ( z ) = N X j = −∞ p − kn Z S j + k − χ ( z · y ) w ( p − k k y k p ) d n y =11 N X j = −∞ p − kn w ( p j ) n p ( j + k ) n (1 − p − n ) − Z S j + k χ ( z · y ) d n y o == k + N X ℓ = −∞ p − kn w ( p ℓ − k ) n p ℓn (1 − p − n ) − Z S ℓ χ ( z · y ) d n y o . Using (1.6) we have for z such that k z k p = p k , k ≥ − N + 1 I B N ( z ) = P ℓ = −∞ p − kn p ℓn w ( p ℓ − k ) n (1 − p − n ) − (1 − p − n ) o = 0 , ℓ ≤ p − kn p n w ( p − k ) n (1 − p − n ) − ( p − n ) o , ℓ = 1; k + N P ℓ =2 p − kn p ℓn w ( p ℓ − k ) (1 − p − n ) , ℓ > , and finally I B N ( z ) = k + N X j =2 p − kn + jn w ( p j − k ) (1 − p − n ) + p − kn + n w ( p − k ) . Comparing this with (1.12) we conclude that I B N ( z ) = A w ( z ) − (1 − p − n ) ∞ X j = k + N +1 p − kn + jn w ( p j − k ) == A w ( z ) − (1 − p − n ) ∞ X ℓ = N +1 p ℓ n w ( p ℓ ) = A w ( z ) − λ N , where λ N is given by (3.4). Lemma 3.5.
For any ψ ∈ D ( Q np ) such that its Fourier transform u = F ψ has a supportin the ball B N we have: Z k z k p ≤ p − N A w ( z ) ψ ( z ) d n z = λ N Z k z k ≤ p − N ψ ( z ) d n z (3.6) Proof.
Since supp ψ ⊂ B − N , we have ψ ( z ) = ψ (0) for k z k p ≤ p − N . Therefore J N := Z k z k p ≤ p − N A w ( z ) ψ ( z ) d n z = ψ (0) Z k z k p ≤ p − N A w ( z ) d n z == ψ (0) Z k z k p ≤ p − N Z Q np − χ ( y · z ) w ( k y k p ) d n y d n z =12 ψ (0) Z Q np Z k z k p ≤ p − N − χ ( y · z ) w ( k y k p ) d n z d n y. (3.7)Since the character χ equals 1 on ball B , hence the integral above is equal zero, when k y · z k p = k y k p · k z k p ≤
1. Therefore, for k z k p ≤ p − N , if k y k p ≤ p N , we have k y · z k p = k y k p · k z k p ≤ J N = ψ (0) Z k y k p >p N Z k z k p ≤ p − N − χ ( y · z ) w ( k y k p ) d n z d n y == ψ (0) Z k y k p >p N d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z == ψ (0) ∞ X k = N +1 Z S k d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z. Let k y k p = p k , k ≥ N + 1, then y = p − k y , where k y k p = 1, and change of variablesformula (1.3) gives for the expression under integral: S N ( y ) = − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z = − N X j = −∞ Z S j (cid:16) − χ ( p − k y · z ) (cid:17) d n z == − N X j = −∞ p − kn Z S j + k (cid:16) − χ ( y · v ) (cid:17) d n v = k − N X ℓ = −∞ p − kn Z S ℓ (cid:16) − χ ( y · v ) (cid:17) d n v. (3.8)Due to (1.6) using that Z S ℓ d n z = (1 − p − n ) p nℓ (3.9)we have S N ( y ) = P ℓ = −∞ p − kn (cid:2) (1 − p − n ) p nℓ − p nℓ (1 − p − n ) (cid:3) = 0 , ℓ ≤ p − kn (cid:2) (1 − p − n ) p n + p n p − n (cid:3) = p − kn p n , ℓ = 1; k − N P ℓ =2 p − kn p nℓ (1 − p − n ) , ℓ > . Finally, for y such that k y k p = p k , k ≥ N + 1 we may continue (3.8): S N ( y ) = − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z = k − N X ℓ =2 p − kn p ℓn (1 − p − n ) + p − kn p n =13 p − kn (cid:16) k − N X ℓ =2 p ℓn − k − N X ℓ =2 p ℓn p − n + p n (cid:17) = p − kn (cid:16) k − N X ℓ =1 p ℓn − k − N X ℓ =2 p ℓn p − n (cid:17) == p − kn (cid:16) k − N X ℓ =1 p ℓn − k − N X ℓ =2 p n ( ℓ − (cid:17) = p − kn (cid:16) k − N X ℓ =1 p nℓ − k − N − X s =1 p ns (cid:17) == p − kn p n ( k − N ) = p − nN . Therefore, again using (3.9) we have J N = ψ (0) ∞ X k = N +1 Z S k d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z == ψ (0) p − nN ∞ X k = N +1 Z S k d n yw ( p k ) = ψ (0) p − nN (1 − p − n ) ∞ X k = N +1 p nk w ( p k ) == ψ (0) p − nN λ N = λ N Z k z k ≤ p − N ψ ( z ) d n z. where λ N is given by (3.4). Theorem 3.6. If η t | t =0 = x and u ∈ D ( B N ) , then ddt E u ( η t ) (cid:12)(cid:12) t =0 = − (cid:0) W N u (cid:1) ( x ) + κ λ N u ( x ) , (3.10) where operator W N is defined by restricting W to the function u supported in the ball B N and the resulting function W u is considered only on the ball B N , i.e. (cid:0) W N u (cid:1) ( x ) = (cid:0) W u (cid:1) ↾ B N , for u ∈ D ( B N ) . Remark that the functions on B N we identify with its extension by zero onto Q np . Remark 3.7.
This theorem actually states that the operator W N − κ λ N is the generatorof the stochastic process η t located in the ball B N . Proof.
By general result about processes with independent increments on the locallycompact space [14] the processes ξ ( N ) t and η t are independent. Thus E χ (cid:0) z · ξ t (cid:1) = E χ (cid:0) z · ξ ( N ) t (cid:1) · E χ (cid:0) z · η t (cid:1) (3.11)for any z ∈ Q np . Due to Lemma 3.1 we have E χ (cid:0) z · ξ t (cid:1) = exp (cid:0) − t κ A w ( z ) (cid:1) . (3.12)On the other hand from Theorem 5.6.17 of [14, p. 397] for any locally compact Abeliangroup G = Q np having a countable basis of its topology and the distribution µ t := P X t
14f an additive process { X t } t ∈ [0 , on (Ω , F , P ) with values in Q np for a chosen fixed innerproduct g fo G there exist an element x t ∈ G and a positive quadratic form φ t on thecharacter group G ∧ of G such that b µ t ( χ ) = χ ( x t ) exp ( − φ t + Z G (cid:2) χ ( x ) − − ig ( x, χ ) (cid:3) π ( t, dx ) ) holds for all χ ∈ G ∧ for some x t ∈ Q np . From Example 4 of 5.1.9 of [14, Ch. V, p. 342] itfollows that for the totally disconnected group G the zero function on G × G ∧ is a localinner product for G , moreover on the totally disconnected group there is no nonzeroGaussian measure, it follows that E χ (cid:0) z · ξ t (cid:1) ≡ b µ t ( z ) = χ ( x t · z ) exp n Z Q np (cid:2) χ ( z · x ) − (cid:3) π ( t, dx ) o . (3.13)Comparing this with (3.12), where due to (1.11) we conclude that the Levy measure ofprocess ξ t is equal to π ( t, dx ) = κ tw ( k x k p ) d n x (3.14)and x t in (3.13) may be chosen as zero so that χ ( x t · z ) ≡ E χ (cid:0) z · η t (cid:1) = E χ (cid:0) z · ( ξ t − ξ ( N ) t ) (cid:1) = exp n Z B N (cid:2) χ ( z · x ) − (cid:3) π ( t, dx ) o . (3.15)Noting (3.14) and notation (3.3) we may represent the integral in the exponent of (3.15)in the following form: Z B N (cid:2) χ ( z · x ) − (cid:3) π ( t, dx ) = κ t Z B N χ ( z · x ) − w ( k x k p ) d n x = − κ t I B N ( z ) . Due to Lemma 3.2 I B N ( z ) = (cid:26) , if k z k p ≤ p − N ; A w ( z ) − λ N , if k z k p > p − N , with λ N given by (3.4) therefore from (3.15) we have E χ (cid:0) z · η t (cid:1) = (cid:26) , if k z k p ≤ p − N ;exp (cid:8) κ t (cid:0) λ N − A w ( z ) (cid:1)(cid:9) , if k z k p > p − N . (3.16)If we consider u = F ψ for ψ ∈ D ( Q np ), then from (3.16) we have: E u (cid:0) η t (cid:1) = Z k z k p ≤ p − N ψ ( z ) d n z + e κ λ N t Z k z k p >p − N e − t κ A w ( z ) ψ ( z ) d n z, ddt E u (cid:0) η t (cid:1)(cid:12)(cid:12)(cid:12) t =0 = κ λ N Z k z k p >p − N ψ ( z ) d n z − κ Z k z k p >p − N A w ( z ) ψ ( z ) d n z. (3.17)By the Fourier inversion formula Z Q np ψ ( z ) d n z = u (0) . On the other habd, since supp u ⊂ B N , we find that ψ ( z ) = ψ (0) for k z k p ≤ p − N .Lemma 3.5 implies that (3.17) takes the form ddt E u (cid:0) η t (cid:1)(cid:12)(cid:12)(cid:12) t =0 = κ λ N Z Q np ψ ( z ) d n z − κ Z Q np A w ( z ) ψ ( z ) d n z == κ λ N u (0) − (cid:0) W N u (cid:1) (0)which finished the proof of Theorem 3.6. Lemma 3.8.
Let the support of a function u ∈ L ( Q np ) be contained in Q np \ B N . Thenthe restriction to B N of the distribution W u ∈ D ′ ( Q np ) coincides with the constant: R N = R N ( u ) = κ Z Q np \ B N u ( y ) d n yw ( k y k p ) , (3.18) i.e. for u ∈ L ( Q np ) , supp u ⊂ Q np \ B N : (cid:0) W u (cid:1) ↾ x ∈ B N = R N ( u ) . Proof.
Let ψ ∈ D ( B N ). Then h W u, ψ i = h u, W ψ i . Since ψ ( x ) = 0 for k x k p > p N h u, W ψ i = κ Z k x k p >p N u ( x ) d n x Z Q np ψ ( x − y ) − ψ ( x ) w ( k y k p ) d n y ψ ( x )=0 == κ Z k x k p >p N u ( x ) d n x Z Q np ψ ( x − y ) w ( k y k p ) d n y x − y = z == κ Z k x k p >p N u ( x ) d n x Z Q np ψ ( z ) w ( k x − z k p ) d n z = κ Z k x k p >p N u ( x ) d n x Z k z k p ≤ p N ψ ( z ) w ( k x − z k p ) d n z == κ Z k x k p >p N u ( x ) w ( k x k p ) d n x · Z k z k p ≤ p N ψ ( z ) d n z. On the last step we have used that for k x k p > p N and k z k p ≤ p N it follows that k x − z k p = k x k p . Since ψ ∈ D ( B N ) is arbitrary, this implies the required property.16 Semigroup on the p -adic ball Consider on the ball B N , N ∈ Z the following Cauchy problem ∂u ( t, x ) ∂t + (cid:0) W N − κ λ N (cid:1) u ( t, x ) = 0 , x ∈ B N , t > u (0 , x ) = ψ ( x ) , x ∈ B N , (4.1)where operator is given by Theorem 3.6 and λ N is obtained from (3.4). Recall thatoperator W N is defined by restricting W to the function u N supported in the ball B N and considering the resulting function W u N only on the ball B N . The functions on B N we identify with its extension by zero onto Q np : (cid:0) W N u N (cid:1) ( x ) = (cid:0) W u N (cid:1) ↾ B N , for u N ∈D ( B N ) . Remark that for operator W N considered on L ( B N ) κ λ N is its eigenvalue. Thisfollows from (1.9) and expression for λ N given by (3.5).A maximum principle arguments as in the proof of Theorem 4.5 in [17, p. 82] provedthe uniqueness of the solution of Cauchy problem (4.1). The fundamental solution Z N ( t, x − y ) for the problem (4.1) is the transition density of the process η t (3.1). Thenext result gives a formula for this transition density. Theorem 4.1.
The solution of the problem (4.1) is given by the formula u N ( t, x ) = Z B N Z N ( t, x − y ) ψ ( y ) d n y, t > , x ∈ B N , (4.2) where Z N ( t, x ) = e κ λ N t Z ( t, x ) + c ( t ) , x ∈ B N , (4.3) c ( t ) = 1 m ( B N ) − e κ λ N t m ( B N ) Z B N Z ( t, x ) d n x (4.4) and Z ( t, x ) is from (1.16) . Here m ( B N ) = R B N d n x = p nN . Moreover c ′ ( t ) = − e κ λ N t κ Z Q np \ B N Z ( t, ξ ) w ( k ξ k p ) d n ξ. (4.5) Proof.
For any ψ ∈ D ( Q np ) such that supp ψ ⊂ B N the solution to the Cauchy problem(4.1) for t > u N ( t, x ) = θ N ( x ) Z B N Z N ( t, x − y ) ψ ( y ) d n y == θ N ( x ) e κ λ N t Z B N Z ( t, x − y ) ψ ( y ) d n y + θ N ( x ) c ( t ) Z B N ψ ( y ) d n y = u ( t, x ) + u ( t, x ) , θ N ( x ), x ∈ Q np is an indicator of the set B N and u ( t, x ) = θ N ( x ) e κ λ N t Z B N Z ( t, x − y ) ψ ( y ) d n y ; u ( t, x ) = θ N ( x ) c ( t ) Z B N ψ ( y ) d n y. (4.6)Let us check that (cid:0) D t + W N − κ λ N (cid:1) u N ( t, x ) = 0 (4.7)for ψ ∈ D ( B N ) and x ∈ B N . We may write for ψ ∈ D ( B N ) and x ∈ B N : (cid:0) D t + W N (cid:1) u N ( t, x ) − κ λ N u N ( t, x ) == (cid:0) D t + W N (cid:1)h θ N ( x ) e κ λ N t Z B N Z ( t, x − y ) ψ ( y ) d n y + θ N ( x ) c ( t ) Z B N ψ ( y ) d n y i == (cid:0) D t + W N (cid:1)(cid:2) u ( t, x ) + u ( t, x ) (cid:3) − κ λ N (cid:2) u ( t, x ) + u ( t, x ) (cid:3) . (4.8)Let us introduce functions h ( t, x ) = θ N ( x ) Z B N Z ( t, x − y ) ψ ( y ) d n y = e − κ λ N t u ( t, x ); h ( t, x ) = (cid:0) − θ N ( x ) (cid:1) Z B N Z ( t, x − y ) ψ ( y ) d n y (4.9)and remark that (cid:0) D t + W (cid:1) h = − (cid:0) D t + W (cid:1) h or (cid:0) D t + W (cid:1) h = − W h . Since (cid:0) D t + W (cid:1) h = (cid:0) D t + W (cid:1)(cid:0) e − κ λ N t u (cid:1) = e − κ λ N t (cid:0) D t + W (cid:1) u − κ λ N e − κ λ N t u we have for x ∈ B N that (cid:0) D t + W (cid:1) u − κ λ N u ( t, x ) = − e κ λ N t W h ( t, x ) . Therefore we may continue:(4.8) = (cid:0) D t + W N (cid:1) u ( t, x ) − κ λ N u ( t, x ) − e κ λ N t W h ( t, x ) , (4.10)where c ( t ) is given by (4.4).From (1.9) it follows that function θ N ( x ) is an eigenfunction of the operator W N corresponding to the eigenvalue κ λ N with λ N defined in (3.5). Therefore, taking intoaccount the representation (4.6) for u ( t, x ), we have (cid:0) D t + W N (cid:1) u ( t, x ) − κ λ N u ( t, x ) = c ′ ( t ) θ N ( x ) Z B N ψ ( y ) d n y, (4.11)18nd we continue (4.10) = c ′ ( t ) θ N ( x ) Z B N ψ ( y ) d n y − e κ λ N t W h ( t, x ) . (4.12)To finish the proof it remains to show that r.h.s. of (4.12) equals zero for x ∈ B N .Substituting definition (4.9) of h into the expression for W , making calculations for x ∈ B N and noting that ψ ∈ D ( Q np ), we have: (cid:0) W h (cid:1) ( t, x ) = κ Z Q np h ( x − y ) − h ( x ) w ( k y k p ) d n y = κ Z Q np h ( x − y ) w ( k y k p ) d n y == κ Z Q np \ B N h ( x − y ) w ( k y k p ) d n y = κ Z Q np \ B N h ( z ) w ( k x − z k p ) d n z == κ Z Q np \ B N h ( z ) w ( k z k p ) d n z. Applying further Lemma 3.8 and changing the variable on the last step, we have: (cid:0)
W h (cid:1) ( t, x ) = κ Z Q np \ B N d n yw ( k y k p ) Z B N Z ( t, y − η ) ψ ( η ) d n η == κ Z B N ψ ( η ) d n η Z Q np \ B N Z ( t, y − η ) d n yw ( k y k p ) == κ Z B N ψ ( η ) d n η · Z Q np \ B N Z ( t, ζ ) d n ζw ( k ζ k p ) . Thus for x ∈ B N (4.12) looks as follows:(4.12) = c ′ ( t ) Z B N ψ ( y ) d n y + e κ λ N t κ Z B N ψ ( η ) d n η · Z Q np \ B N Z ( t, ζ ) d n ζw ( k ζ k p ) , therefore it remains to show that c ′ ( t ) = − e κ λ N t κ Z Q np \ B N Z ( t, ζ ) w ( k ζ k p ) d n ζ. (4.13)Let us remark that due to (1.20) Z Q np \ B N Z ( t, ζ ) d n ζw ( k ζ k p ) = Z Q np \ B N Z Q np e − κ tA w ( ξ ) χ ( − ζ · ξ ) d n ξ d n ζw ( k ζ k p ) == Z Q np e − κ tA w ( ξ ) Z Q np \ B N χ ( − ζ · ξ ) d n ζw ( k ζ k p ) d n ξ =19 Z k ξ k≤ p − N e − κ tA w ( ξ ) h − A w ( ξ ) + λ N i d n ξ. (4.14)On the last step we used that due to the expression for λ N (3.5), representation of A w ( ξ )(1.11) and Lemma 3.2 we have Z Q np \ B N χ ( − ζ · ξ ) d n ζw ( k ζ k p ) = Z Q np \ B N (cid:2) χ ( − ζ · ξ ) − (cid:3) d n ζw ( k ζ k p ) + λ N == − A w ( ξ ) + I B N + λ N = (cid:26) − A w ( ξ ) + λ N , if k ξ k p ≤ p − N ;0 , if k ξ k p > p − N . From the other side, due to representation (1.20) for Z ( t, x ), we have the followingrepresentation for c ( t ): c ( t ) = p − nN − e κ λ N t p − nN Z B N Z ( t, x ) d n x == p − nN − e κ λ N t p − nN Z B N Z Q np e − κ tA w ( ξ ) χ ( − x · ξ ) d n ξ d n x == p − nN − e κ λ N t Z k ξ k p ≤ p − N e − κ tA w ( ξ ) d n ξ. (4.15)On the last step we used (1.5), see also [24, (7.14), p. 25], i.e. Z B N χ ( ξ · x ) d n x = p Nn (cid:26) , if k ξ k p ≤ p − N ;0 , otherwise . From (4.15) it follows that c ′ ( t ) = − κ λ N e κ λ N t Z k ξ k p ≤ p − N e − κ tA w ( ξ ) d n ξ + e κ λ N t κ Z k ξ k p ≤ p − N A w ( ξ ) e − κ tA w ( ξ ) d n ξ (4.16)Noting (4.16) and (4.14) we receive the identity (4.13), which proves the statement ofthe theorem. Remark 4.2.
Let us note that from (4.4) it follows the following representation for c ′ ( t ): c ′ ( t ) = − κ λ N m ( B N ) e κ λ N t Z B N Z ( t, x ) d n x − e κ λ N t m ( B N ) Z B N D t Z ( t, x ) d n x. (4.17) Lemma 4.3. c ′ ( t ) = κ e κ λ N t Z B N e − κ tA w ( ξ ) (cid:2) A w ( ξ ) − λ N (cid:3) d n ξ. (4.18) Moreover c ′ (0) = 0 . roof. Representation (4.18) follows from the formula (4.16). To prove second statementof the theorem, it is sufficient to show that Z B N A w ( ξ ) d n ξ = Z B N λ N d n ξ = p − nN λ N . Similar to Lemma 3.5 we have Z k z k p ≤ p − N A w ( z ) d n = Z k y k p >p N d n yw ( k y k p ) Z k z k p ≤ p − N (cid:16) − χ ( y · z ) (cid:17) d n z == Z k y k p >p N d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z == ∞ X k = N +1 Z S k d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z. Let k y k p = p k , k ≥ N + 1, then y = p − k y , where k y k p = 1. Using change of variablesformula (1.3) and (1.6) we have − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z = − N X j = −∞ Z S j (cid:16) − χ ( p − k y · z ) (cid:17) d n z == − N X j = −∞ p − kn Z S j + k (cid:16) − χ ( y · v ) (cid:17) d n v == k − N X ℓ = −∞ p − kn Z S ℓ (cid:16) − χ ( y · v ) (cid:17) d n v == P ℓ = −∞ p − kn (cid:2) (1 − p − n ) p nℓ − p nℓ (1 − p − n ) (cid:3) = 0 , ℓ ≤ p − kn (cid:2) (1 − p − n ) p n + p n p − n (cid:3) = p − kn p n , ℓ = 1; k − N P ℓ =2 p − kn p nℓ (1 − p − n ) , ℓ > . (4.19)Finally, for y such that k y k p = p k , k ≥ N + 1, from (4.19) we have − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z = k − N X ℓ =2 p − kn p ℓn (1 − p − n ) + p − kn p n == p − kn (cid:16) k − N X ℓ =1 p ℓn − k − N X ℓ =2 p ℓn p − n (cid:17) = p − kn (cid:16) k − N X ℓ =1 p nℓ − k − N − X s =1 p ns (cid:17) == p − kn p n ( k − N ) = p − nN . Z k z k p ≤ p − N A w ( z ) d n z = ∞ X k = N +1 Z S k d n yw ( k y k p ) − N X j = −∞ Z S j (cid:16) − χ ( y · z ) (cid:17) d n z == p − nN ∞ X k = N +1 Z S k d n yw ( p k ) = p − nN (1 − p − n ) ∞ X k = N +1 p nk w ( p k ) = p − nN λ N . where λ N is given by (3.4). Lemma 4.4.
The function Z N ( t.x ) is non-negative, and Z B N Z N ( t, x ) d n x = 1 . (4.20) Proof.
From (4.3) and (4.4) we have for x ∈ B N Z N ( t, x ) = e κ λ N t Z ( t, x ) + c ( t ) == e κ λ N t Z ( t, x ) + 1 m ( B N ) − e κ λ N t m ( B N ) Z B N Z ( t, x ) d n x == e κ λ N t h Z ( t, x ) − p − nN Z B N Z ( t, x ) d n x i + p − nN = 1 . The positivity of the function Z N ( t, x ) follows from its probabilistic meaning as thetransition density of the process η t (3.1).On a ball B N , N ∈ Z let us consider the Cauchy problem (4.1). Its fundamentalsolution Z N ( t, x ) (4.3) defines a contraction semigroup( T N ( t ) u )( x ) = Z B N Z N ( t, x − ξ ) u ( ξ ) d n ξ on L ( B N ). Lemma 4.5.
The semigroup T N ( t ) is strongly continuous in L ( B N ) .Proof. For u ∈ L ( B N ) we may write k T N ( t ) u − u k L ( B N ) ≤ I ( t ) + I ( t ) , where I ( t ) = Z B N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B N e κ λ N t Z ( t, x − ξ ) u ( ξ ) dξ − u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d n x ; I ( t ) = p nN | c ( t ) | Z B N | u ( ξ ) | d n ξ. Using representation (1.20) and (1.5) from (4.4) we have c ( t ) = p − nN − e κ λ N t p − nN Z Q np e − κ tA w ( ξ ) Z B N χ ( − x · ξ ) d n x d n ξ =22 p − nN − e κ λ N t Z k ξ k p ≤ p − N e − κ tA w ( ξ ) d n ξ → , as t → . (4.21)Therefore I ( t ) → t →
0. For small values of t we write I ( t ) = Z B N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B N Z ( t, x − ξ ) u ( ξ ) d n ξ − u ( x ) + Z B N (cid:0) e κ λ N t − (cid:1) Z ( t, x − ξ ) u ( ξ ) d n ξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d n x ≤≤ Z B N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z B N Z ( t, x − ξ ) u ( ξ ) d n ξ − u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d n x + Ct Z B N Z B N Z ( t, x − ξ ) | u ( ξ ) | d n ξ d n x == J ( t ) + J ( t ) . By the Young inequality using the identity (1.18), extending u by zero to a function e u on Q np , we obtain J ( t ) ≤ Ct Z Q np Z Q np Z ( t, x − ξ ) | e u ( ξ ) | d n ξ d n x ≤ Ct k e u k L ( B N ) → , as t → . Moreover by the C -property of T ( t ) stated in Lemma 2.1 we have J ( t ) = Z B N (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Q np Z ( t, x − ξ ) e u ( ξ ) d n ξ − e u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d n x ≤≤ Z Q np (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z Q np Z ( t, x − ξ ) e u ( ξ ) d n ξ − e u ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d n x = k T ( t ) e u − e u k L ( Q np ) → , as t → A N denote the generator of the contraction semigroup T N ( t ) in L ( B N ). Let usalso introduce operator W N which is understood in the sense of D ′ ( B N ), that is ψ N isextended by zero to a function on Q np , W N is applied to it in the distribution sense, andthe resulting distribution is restricted to B N . Proposition 4.6.
Let operator A be a generator of semigroup T ( t ) in space L ( Q np ) with the domain Dom ( A ) . Then for the restriction ψ N of the function ψ ∈ Dom ( A ) tothe ball B N we have: A ψ = W N ψ N + R N , (4.22) where R N = R N ( ψ − ψ N ) is the constant from Lemma 3.8.Proof. To prove this statement we remark that function ψ ∈ Dom ( A ) may be representas ψ = ψ N + ( ψ − ψ N ) and by Lemma 3.8 on B N we may write A ψ = W N ψ N + R N , where R N = R N ( ψ − ψ N ) . heorem 4.7. If ψ ∈ Dom ( A ) in L ( Q np ) (Definition 2.2), then the restriction ψ N ofthe function ψ to B N belongs to Dom ( A N ) and A N ψ N = (cid:0) W N − κ λ N (cid:1) ψ N , (4.23) where W N ψ N is understood in the sense of D ′ ( B N ) , that is ψ N is extended by zeroto a function on Q np , W N is applied to it in the distribution sense, and the resultingdistribution is restricted to B N .Proof. For ψ ∈ Dom ( A ) we have to check that:1) W N ψ N ∈ L ( B N );2) (cid:13)(cid:13)(cid:13) − t (cid:2) T N ( t ) ψ N − ψ N (cid:3) − (cid:0) W N − κ λ N (cid:1) ψ N (cid:13)(cid:13)(cid:13) L ( B N ) →
0, as t → ψ ∈ Dom ( A ) Proposition 4.6implies that A ψ = W N ψ N + R N thus W N ψ N ∈ L ( B N ).Further, from (4.3), expanding exponent in the Taylor series, we have (cid:0) T N ( t ) ψ N (cid:1) ( x ) = Z B N Z ( t, x − y ) ψ ( y ) d n y + c ( t ) Z B N ψ ( y ) d n y + (4.24)+ κ λ N t Z B N Z ( t, x − y ) ψ ( y ) d n y + d ( t ) Z B N Z ( t, x − y ) ψ ( y ) d n y, where d ( t ) = O ( t ), t →
0. By strong continuity of T N ( t ) in L ( B N ) (see Lemma 4.5)we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) − κ λ N Z B N Z ( t, x − y ) ψ ( y ) d n y + κ λ N ψ N ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( B N ) → , as t → . Moreover from Young inequality it follows that1 t (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d ( t ) Z B N Z ( t, x − y ) ψ ( y ) d n y (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( B N ) → , as t → . From (4.21) it follows that c (0) = 0. Moreover Lemma 4.3 implies that c ′ (0) = 0,thus c ( t ) = O ( t ) as t → V ( t, x ) = Z B N Z ( t, x − y ) ψ ( y ) d n y = v ( t, x ) − v ( t, x ) , x ∈ B r , where v ( t, x ) = Z Q np Z ( t, x − y ) ψ ( y ) d n y, ( t, x ) = Z k y k p >p N Z ( t, x − y ) ψ ( y ) d n y. If we show that 1 t v ( t, x ) − R N → , when t → (cid:13)(cid:13)(cid:13) − t (cid:2) T N ( t ) ψ N − ψ N (cid:3) − (cid:0) W N − κ λ N (cid:1) ψ N (cid:13)(cid:13)(cid:13) L ( B N ) ≤≤ (cid:13)(cid:13)(cid:13) − t (cid:16) Z Q np Z ( t, x − y ) ψ ( y ) d n y − v ( t, x ) − ψ (cid:17) − W N ψ N (cid:13)(cid:13)(cid:13) L ( B N ) + o (1) == (cid:13)(cid:13)(cid:13) − t (cid:16) Z Q np Z ( t, x − y ) ψ ( y ) d n y − ψ (cid:17) − ( W N ψ N + R N ) (cid:13)(cid:13)(cid:13) L ( B N ) + o (1) == (cid:13)(cid:13)(cid:13) − t (cid:0) T ( t ) ψ − ψ (cid:1) − W ψ (cid:13)(cid:13)(cid:13) L ( B N ) + o (1) → , as t → , which completes the proof.Let us show (4.25). To do this we write1 t v ( t, x ) − R N = Z k y k p >p N (cid:16) t Z ( t, y ) − κ w ( k y k p ) (cid:17) ψ ( y ) d n y, (4.26)where ψ ∈ Dom ( A ) in L ( Q np ). If we show that Z ( t, y ) = κ tw ( k y k p ) + o ( t ) , (4.27)then this will prove (4.25).Due to (1.23) for k y k p = p β we have Z ( t, y ) = k y k − np " (1 − p − n ) ∞ X j =0 p − nj e − κ tA w ( p − ( β + j ) ) − e − κ tA w ( p − ( β − ) . Then expanding the exponents in the Taylor series we have e − κ tA w ( p − ( β + j ) ) = 1 − κ tA w ( p − ( β + j ) ) + r ( t ); e − κ tA w ( p − ( β − ) = 1 − κ tA w ( p − ( β − ) + r ( t ) , where, for example, r ( t ) is the result of decomposition of function f ( t ) = e − κ tA w ( k y k − p p ) : r ( t ) = t f ′′ ( θt )2 = (cid:2) A w ( k y k − p p ) (cid:3) e − κ θtA w ( k y k − p p ) , θ ∈ (0 , . | r ( t ) ≤ C t (cid:2) k y k − p p (cid:3) α − n ) , for k y k p = p β . Therefore r = o ( t ) as t → r ( t ), thus Z ( t, y ) = k y k − np " (1 − p − n ) ∞ X j =0 p − nj (cid:0) − κ tA w ( p − ( β + j ) ) (cid:1) − κ tA w ( p − ( β − ) + o ( t ) == k y k − np κ t h A w ( p − ( β − ) − (1 − p − n ) ∞ X j =0 p − nj A w ( p − ( β + j ) ) i + o ( t ) . Using representation (1.13) we may write Z ( t, y ) = k y k − np κ t h A + B − C − D i + o ( t ) , (4.28)where A = (1 − p − n ) ∞ X k = β +1 p nk w ( p k ) ; B = p nβ w ( p β ) ; C = (1 − p − n ) ∞ X j =0 p − nj (1 − p − n ) X k = β + j +2 p nk w ( p k ) ; D = (1 − p − n ) ∞ X j =0 p − nj p n ( β + j +1) w ( p β + j +1 ) . Let us in the term C change the order of summation and calculate the finite sum ofgeometric progression, then we have C = (1 − p − n ) ∞ X k = β +2 k − β − X j =0 p − nj p nk w ( p k ) = (1 − p − n ) ∞ X k = β +2 p nk w ( p k ) (cid:0) − p − n ( k − β − (cid:1) == (1 − p − n ) ∞ X k = β +2 p nk w ( p k ) − (1 − p − n ) p n ( β +1) ∞ X k = β +2 w ( p k ) = C − C ; D = (1 − p n ) p n ( β +1) ∞ X j =0 w ( p β + j +1 ) = (1 − p n ) p n ( β +1) ∞ X k = β +1 w ( p k ) . Looking at (4.28) we see that A − C = (1 − p − n ) p n ( β +1) w ( p β +1 ) ; C − D = − (1 − p − n ) p n ( β +1) w ( p β +1 ) , B , i.e. Z ( t, y ) = k y k − np κ t p nβ w ( p β ) + o ( t ) = κ tw ( k y k p ) + o ( t ) , for k y k p = p β , which proves (4.27) and therefore (4.25). To formulate the main result let us first recall the notation of mild solution of nonlinearequation in some real Banach space X . Consider the Cauchy problem (cid:26) D t u + Au = f ( t ) , t ∈ [0 , T ]; u (0) = u , (5.1)where u ∈ X and f ∈ L ([0 , T ]; X ), A is a m -accretive nonlinear operator.Operator A : X → X is called accretive if for every pair x, y ∈ Dom ( A ) h Ax − Ay, w i ≥ , where w ∈ J ( x − y ) and J : X → X ∗ is the duality mapping of the space X . Correspon-dingly operator A is called m -accretive if the range Ran ( I + A ) = X . Definition 5.1.
Let f ∈ L ([0; T ]; X ) and ε > be given. An ε -discretization on [0; T ] of the equation D t y + Ay = f consists of a partition t ≤ t ≤ t ≤ . . . ≤ t N of the interval [0; t N ] and a finite sequence { f i } Ni =1 ⊂ X such that t i − t i − < ε for i =1 , . . . , N, T − ε < t N ≤ T and N X i =1 t i Z t i − k f ( s ) − f i k ds < ε. Definition 5.2.
A piecewise constant function z : [0 , t N ] → X whose values z i on ( t i − , t i ] satisfy the finite difference equation z i − z i − t i − t i − + Az i = f i , i = 1 , . . . , N is called an ε − approximate solution to the Cauchy problem (5.1) if it satisfies k z (0) − u k ≤ ε. Definition 5.3.
A mild solution of the Cauchy problem (5.1) is a function u ∈ C ([0 , T ]; X ) with the property that for each ε > there us an ε − approximate so-lution z of D t u + Au = f on [0 , T ] such that k u ( t ) − z ( t ) k ≤ ε for al t ∈ [0 , T ] and u (0) = u . See [4, Ch. 4] for the details. 27 .2 Solvability of the nonlinear equation
Let us consider in L ( Q np ) the equation D t u + A (cid:0) ϕ ( u ) (cid:1) = 0 , u = u ( t, x ) , t > , x ∈ Q np , (5.2)where A is the generator of the semigroup T ( t ) in L ( Q np ) and ϕ : R → R is a continuousstrictly increasing function, ϕ (0) = 0, such that: | ϕ ( s ) | ≤ C | s | m , m ≥ . Consider the nonlinear operator A ϕ with the domain Dom ( A ϕ ) = { u ∈ L (Ω) : ϕ ( u ) ∈ Dom ( A ) } . From Lemma 2.3 it follows that the operator A ϕ is densely defined and therefore so theoperator f A ϕ has the same property. Theorem 5.4.
The operator A ϕ is m -accretive, i.e. for any initial function u ∈ L ( Q np ) the Cauchy problem for equation (5.2) has a unique mild solution.Proof. The statement of the theorem is a consequence of the Crandall-Liggett theorem[4, Theorem 4.3]. Indeed, from Proposition 1 in [7] it follows that ( A ϕ )( u ) = A ( ϕ ( u ))is an accretive nonlinear operator in L ( Q np ) and for any ε > εI + A ) ϕ is m -accretive in L ( Q np ).Therefore in order to prove the m -accretivity of A ϕ it is sufficient to prove that theoperator I + A ϕ has a dense range in L ( Q np ). In other words it suffices to prove thatequation u + A ϕ ( u ) = f is solvable for a dense subset of functions f ∈ L ( Q np ). Equivalently, setting β = ϕ − (the function inverse to ϕ ), we have to study the equation A v + β ( v ) = f. (5.3)Since the space of test functions D ( Q np ) is dense in L ( Q np ), therefore it is enough toprove the solvability of equation (5.3) for any f ∈ L ( Q np ) ∩ L ∞ ( Q np ).For such a function f we consider the regularized equation to (5.3): εv ε + A v ε + β ( v ε ) = f, ε > , (5.4)possessing, due to Proposition 4 in [6, p. 571] a unique solution v ε , such that w ε = f − A v ε satisfies the inequality: k w ε k L ( Q np ) ≤ k f k L ( Q np ) . (5.5)Moreover, if e v ε and e w ε correspond to equation (5.4) with r.h.s. f and e f , then k w ε − e w ε k L ( Q np ) ≤ k f − e f k L ( Q np ) . (5.6)28n addition, if f ∈ L ( Q np ) ∩ L ∞ ( Q np ), then due to the Proposition 4 in [6] applied inspace L q ( Q np ) = L ∞ ( Q np ) we have k f − ( A + ε ) v ε k L ∞ ( Q np ) = k β ( v ε ) k L ∞ ( Q np ) ≤ k f k L ∞ ( Q np ) . (5.7)Using inequality (5.7) we find that | v ε ( x ) | ≤ β − ( k f k L ∞ ( Q np ) ) (5.8)for almost all x ∈ Q np . This means that for any fixed N the constant R N ( v ε ) from (3.18)satisfy inequality: R N ( v ε ) ≤ C, where C does not depend on ε , so that the set of constant functions { R N ( v ε ) , < ε < } is relatively compact in L ( B N ).On the other hand, it follows from (5.5), (5.6) and the translation invariance of A that the family of functions w ε = f − A v ε satisfies the inequalities: k w ε k L ( Q np ) ≤ k f k L ( Q np ) ; (5.9) Z Q np | w ε ( x + h ) − w ε ( x ) | d n x ≤ Z Q np | f ( x + h ) − f ( x ) | d n x (5.10)for any h ∈ Q np . The conditions (5.9) and (5.10) imply relative compactness of sequence { w ε } and therefore of { A v ε } in L ( Q np ), that is the compactness of the closure of therestriction ( A v ε ) (cid:12)(cid:12) X for any bounded measurable subset X ⊂ Q np . This is a consequenceof the criterion for relative compactness in L ( G ) where G is a compact group (seeTheorem 4.20.1 in [9]) applied to the case G = B N (the additive group of p -adic ball).Denote by v ε,N the restriction of v ε to B N . From Proposition 4.6 it follows that W N ψ N = A ψ − R N therefore the set W N v ε,N is relatively compact in L ( B N ). Since W N = A N + κ λ N ,defined as in (4.23), due to Hille-Yosida theorem has bounded inverse on L loc ( Q np ), thisimplies the relative compactness of { v ε,N } in L ( B N ) for each N . The same is true for { v ε } in L loc ( Q np ). Let v be its limit point. Together with the relative compactness of { A v ε } the above reasoning proves the solvability of (5.3) because by Fatou’s lemma and(5.9), a limit point of { A v ε } belongs to L ( Q np ). Therefore β ( v ) ∈ L ( Q np ). By (5.8), v ∈ L ∞ ( Q np ), so that β ( v ) ∈ L ∞ ( Q np ), v = ϕ ( β ( v )), | v ( x ) | ≤ C | β ( v ) | , and v belongs to L ( Q np ) . Acknowledgments
The work by the first- and third-named authors was funded in part under the budgetprogram of Ukraine No. 6541230 “Support to the development of priority researchtrends”. The third-named author was also supported in part in the framework of theresearch work ”Markov evolutions in real and p-adic spaces” of the Dragomanov NationalPedagogical University of Ukraine. 29 eferences [1] Albeverio, S., Khrennikov, A. Yu. and Shelkovich, V. M.
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