Fractional Cauchy problem on random snowflakes
FFRACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES
RAFFAELA CAPITANELLI* AND MIRKO D’OVIDIO
Abstract.
We consider time-changed Brownian motions on random Koch (pre-fractal andfractal) domains where the time change is given by the inverse to a subordinator. In particular,we study the fractional Cauchy problem with Robin condition on the pre-fractal boundaryobtaining asymptotic results for the corresponding fractional diffusions with Robin, Neumannand Dirichlet boundary conditions on the fractal domain.
Keywords:
Time changes; fractional operators; time-fractional equations; asymptotics.
AMS-MSC:
Introduction
Many physical and biological phenomena take place across irregular and wild structures inwhich boundaries are “large”while bulk is “small”. In this framework, domains with fractalboundaries provide a suitable setting to model phenomena in which the surface effects areenhanced like, for example, pulmonary system, root infiltration, tree foliage, etc..In this paper, we consider random Koch domains which are domains whose boundary areconstructed by mixtures of Koch curves with random scales. These domains are obtained aslimit of domains with Lipschitz boundary whereas for the limit object, the fractal given by therandom Koch domain, the boundary has Hausdorff dimension between 1 and 2.Our attention will be focused on fractional Cauchy problems on the random Koch domainswith boundary conditions.Literature on fractional Cauchy problems is extensive both from the probability and theanalysis point of view. Here, our aim is not providing a large list of references. We mention hereonly few works investigating basic and fondamental aspects: [1], [3], [14], [17], [18], [20], [22],[28], [32].The non-local time-operator we deal with is very general and covers a huge class of non-local (convolution type) operators. Such operators have been recently considered in the papers[13; 31]. From the probabilistic point of view, we consider time-changed Brownian motions wherethe time change is given by an inverse to a subordinator characterized by a symbol which is aBernstein function. Thus, with this time-fractional operator at hand, we study the fractionalCauchy problem with Robin condition on the pre-fractal boundary and we obtain asymptoticresults for the corresponding fractional diffusions with Robin, Neumann and Dirichlet boundaryconditions on the fractal domain.The asymptotic problem we deal with can be illustrated, in the simple case, by the followingparabolic Dirichlet-Robin problem on the interval (0 , a ), a >
0. More precisely, we consider theheat equation ∂ t u n = ∂ xx u n , t > , x ∈ (0 , a ) u n ( t,
0) = 0 , t > u n (0 , x ) = f ( x ) , x ∈ (0 , a )with Robin boundary condition ∂ x u n ( t, a ) + c n u n ( t, a ) = 0 , t > Date : December 23, 2020. a r X i v : . [ m a t h . P R ] D ec RAFFAELA CAPITANELLI* AND MIRKO D’OVIDIO where c n > n ∈ N . The solution can be written as follows u n ( t, x ) = (cid:80) k ≥ e − tλ ( n ) k φ ( n ) k ( x ) f k where f ( n ) k = (cid:82) f ( x ) φ ( n ) k ( x ) dx , k ∈ N . Notice that φ ( n ) k ( x ) = sin( x (cid:113) λ ( n ) k ) and (cid:113) λ ( n ) k = z ( n ) k arethe eigenvalues associated to φ ( n ) k where z ( n ) k are solutions to tan( az ( n ) k ) = − z ( n ) k /c n , k ∈ N .Our aim here is to point out the asymptotic behaviour of the solution u n as n → ∞ . Weobtain three different limit problems. If c n →
0, then z ( n ) k → z Nk = (cid:0) π + πk (cid:1) a and therefore u n ( t, x ) → u ( t, x ) = (cid:80) k ≥ e − t ( z Nk ) sin( xz Nk ) f k where f k = (cid:82) f ( x ) sin( xz Nk ) dx and u is thesolution to (1.1) with Neumann condition ∂ x u ( t, a ) = 0 , t > . If c n → ∞ , then z ( n ) k → z Dk = πka and therefore u n → u where the solution u ( t, x ) = (cid:80) k ≥ e − t ( z Dk ) sin( xz Dk ) f k with f k = (cid:82) f ( x ) sin( xz Dk ) dx solves (1.1) with Dirichlet condition u ( t, a ) = 0 , t > . If c n → c ∈ (0 , ∞ ), then z ( n ) k → z Rk > az Rk ) = − z Rk c and therefore u n → u where thesolution u ( t, x ) = (cid:80) k ≥ e − t ( z Rk ) sin( xz Rk ) f k with f k = (cid:82) f ( x ) sin( xz Rk ) dx solves (1.1) with Robinboundary condition ∂ x u ( t, a ) + cu ( t, a ) = 0 , t > . Now we wonder if a similar asymptotic behaviour holds for the analogue time-fractional problem.A simple example is given by the problem ∂ βt u n = ∂ xx u n , t > , x ∈ (0 , a ) ∂ x u n ( t, a ) + c n u n ( t, a ) = 0 , t > u n ( t,
0) = 0 , t > u n (0 , x ) = f ( x ) , x ∈ (0 , a )with c n > n ∈ N where ∂ βt u is the Caputo fractional derivative of u (see formula (4.5) below).The solution can be written as follows u n ( t, x ) = (cid:88) k ≥ E β ( − λ ( n ) k t β ) φ ( n ) k ( x ) f ( n ) k (1.3)where E β ( w ) = (cid:88) k ≥ w β Γ( βk + 1) , w ≥ { φ ( n ) k , λ ( n ) k : k ∈ N } has been introduced before. Bysimple arguments, we get that the solution (1.3) uniformly converges to a function u which turnsout to be analogously related to the boundary problems above (Neumann, Dirichlet, Robin) withthe Caputo time-fractional derivative ∂ βt in place of the ordinary derivative ∂ t . This is due tothe fact that we have explicit representation of the system { φ ( n ) k , λ ( n ) k : k ∈ N } .Following the same spirit, in the present paper, we move on to general domains like therandom snowflakes we have introduced before and we address the same asymptotic problemwith a general time-fractional operator. In this case we do not have the same informationsabout the associated system and the compact representation of the solution. We overcome thisdifficulty by using the theory of Dirichlet forms and Markov processes. An essential tool will begiven by the convergence of forms associated with time-changed processes.We remark that the peculiarity in studying the asymptotic behaviour of these approximatingproblems is that one has to deal with an increasing sequence of Lipschitzian domains whichconverges in the limit to the domain whose boundary is a fractal.The plan of the paper is the following: in Section 2 we introduce the random Koch domains;in Section 3 we recall the definition of Dirichlet forms with associated base processes; in Section4 we introduce time-fractional equations and time changes; in the last section we prove our RACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES 3 main results. More precisely, in Theorem 5.1 we solve the asymptotic problem for the the time-changed processes and in Theorem 5.2 we point out some peculiar aspects arising by passingfrom the ordinary to the fractional Cauchy problem.2.
Random Koch domains (RKD)
We first introduce the Koch (snowflake) domain and then we construct the random Kochdomains. Let (cid:96) a ∈ (2 ,
4) with a ∈ I ⊂ N be the reciprocal of the contraction factor for thefamily Ψ ( a ) of contractive similitudes ψ ( a ) i : C → C given by ψ ( a )1 ( z ) = z(cid:96) a , ψ ( a )2 ( z ) = z(cid:96) a e ıθ ( (cid:96) a ) + 1 (cid:96) a ,ψ ( a )3 ( z ) = z(cid:96) a e ıθ ( (cid:96) a ) + 12 + ı (cid:114) (cid:96) a − , ψ ( a )4 ( z ) = z − (cid:96) a + 1where θ ( (cid:96) a ) = arcsin( (cid:112) (cid:96) a (4 − (cid:96) a ) / I N with I ⊂ N , | I | = N , and let ξ = ( ξ , ξ , . . . ) ∈ Ξ. We call ξ an environment sequence where ξ n says which family of contractive similitudes weare using at level n . Set (cid:96) ( ξ ) (0) = 1 and (cid:96) ( ξ ) ( n ) = n (cid:89) i =1 (cid:96) ξ i . (2.1)We define a left shift S on Ξ such that if ξ = ( ξ , ξ , ξ , . . . ) , then Sξ = ( ξ , ξ , . . . ) . For B ⊂ R set Υ ( a ) ( B ) = (cid:91) i =1 ψ ( a ) i ( B )and Υ ( ξ ) n ( B ) = Υ ( ξ ) ◦ · · · ◦ Υ ( ξ n ) ( B ) . The fractal K ( ξ ) associated with the environment sequence ξ is defined by K ( ξ ) = + ∞ (cid:91) n =1 Υ ( ξ ) n (Γ)where Γ = { P , P } with P = (0 ,
0) and P = (1 , . We remark that these fractals do nothave any exact self-similarity, that is, there is no scaling factor which leaves the set invariant:however, the family { K ( ξ ) , ξ ∈ Ξ } satisfies the following relation K ( ξ ) = Υ ( ξ ) ( K ( Sξ ) ) . (2.2)Moreover, the spatial symmetry is preserved and the set K ( ξ ) is locally spatially homogeneous,that is, the volume measure µ ( ξ ) on K ( ξ ) satisfies the locally spatially homogeneous condition(2.3) below. Before describing this measure, we introduce some notations. For ξ ∈ Ξ , we definethe word space W = W ( ξ ) = { ( w , w , ... ) : 1 (cid:54) w i (cid:54) } and, for w ∈ W, we set w | n = ( w , ..., w n ) and ψ w | n = ψ ( ξ ) w ◦ · · · ◦ ψ ( ξ n ) w n . The volume measure µ ( ξ ) is the unique Radon measure on K ( ξ ) such that µ ( ξ ) ( ψ w | n ( K ( S n ξ ) )) = 14 n (2.3)for all w ∈ W, (see Section 2 in [2]) as, for each a ∈ A, the family Ψ ( a ) has 4 contractivesimilitudes. Let K be the line segment of unit length with P = (0 ,
0) and P = (1 ,
0) asendpoints. We set, for each n ∈ N , K ( ξ ) n = Υ ( ξ ) n ( K )and K ( ξ ) n is the so-called n -th prefractal curve. RAFFAELA CAPITANELLI* AND MIRKO D’OVIDIO
Figure 1.
Outward curvesLet us consider the random vector ξ = ( ξ , ξ , . . . ) whose components ξ i take values on I withprobability mass function P : Ξ → [0 , n -th pre-fractalcurve K ( ξ ) n = Υ ( ξ ) n ( K )depends on the realization of ξ with probability P ( ξ i = ξ i ) for its i -th component. We assumethat { ξ i } i =1 ,...,n are identically distributed and ξ i ⊥ ξ j for i (cid:54) = j , that is we obtain the curve K ( ξ ) n with probability P ( ξ | n = ξ | n ) = n (cid:89) i =1 P ( ξ i = ξ i )where ξ | n = ( ξ , . . . , ξ n ) and ξ | n = ( ξ , . . . , ξ n ). Further on we only use the superscript ( ξ | n ) or( ξ | n ) in order to streamline the notation.The fractal K ( ξ ) associated with the random environment sequence ξ is therefore defined by K ( ξ ) = + ∞ (cid:91) n =1 Υ ( ξ ) n (Γ)where Γ = { P , P } with P = (0 ,
0) and P = (1 , . Let Ω ( ξ | n ) be the planar domain obtained from a regular polygon by replacing each side witha pre-fractal curve K ( ξ ) n and Ω ( ξ ) be the planar domain obtained by replacing each side with thecorresponding fractal curve K ( ξ ) . We introduce the random planar domains Ω ( ξ | n ) and Ω ( ξ ) byconsidering the random curves K ( ξ ) n and K ( ξ ) . Examples of (pre-fractal) random Koch domainsare given in figures 1 (outward curves), 2 (inward curves), 3 (inward curves) by choosing asregular polygon the square.Since ξ i law = ξ , ∀ i , we have that the Hausdorff dimension d ( ξ ) of the curve K ( ξ ) can beobtained by considering the strong law of large numbers and the fact thatln 4 n (cid:80) ni =1 (cid:96) ξ i = ln 4 n (cid:80) ni =1 (cid:96) ξ i a.s. → ln 4 E [ln (cid:96) ξ ] , n → ∞ . RACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES 5
Figure 2.
Inward curves
Figure 3.
Inward curvesThen (see [2, Lemma 2.3]), d ( ξ ) = ln 4ln (cid:81) a ∈ I ( (cid:96) a ) P ( ξ = a ) = ln 4 E [ln (cid:96) ξ ] . (2.4)Moreover the measure µ ( ξ ) in (2.3) has the property that there exist two positive constants C , C , such that, C r d ( ξ ) ≤ µ ( ξ ) ( B ( P, r ) ∩ K ( ξ ) ) ≤ C r d ( ξ ) , ∀ P ∈ K ( ξ ) , (2.5)where B ( P, r ) denotes the Euclidean ball with center in P and radius 0 < r ≤ K ( ξ ) is a d -set with respect to theHausdorff measure H d , with d = d ( ξ ) . The sequence σ ( ξ | n ) = (cid:96) ( ξ | n ) n , where (cid:96) ( ξ | n ) = n (cid:89) i =1 (cid:96) ξ i (2.6)is obtained from the realization of ξ | n and therefore, from the realization of the random variable (cid:96) ( ξ | n ) with mean value given by E [ (cid:96) ( ξ | n ) ] = n (cid:89) i =1 E [ (cid:96) ξ i ] = (cid:0) E [ (cid:96) ξ ] (cid:1) n . Thus, for α = E (cid:96) ξ ∈ (2 ,
4) we find the mean value E [ σ ( ξ | n ) ] = α n / n .The realization ξ | n can be regarded as the vector a | n = ( a , . . . , a n ) which is a n -dimensionalvector with N different values of I , that is a | n ∈ I n . We introduce the multinomial distribution p a | n = n ! N (cid:89) i =1 p (cid:93) ( a i ) i (cid:93) ( a i )! , N (cid:88) i =1 (cid:93) ( a i ) = n, N (cid:88) i =1 p i = 1where p i = P ( ξ = a i ) and write p = { p i } i =1 ,...,N . Thus, for the realization of the vector ξ | n wehave that E [ Ω ( ξ | n ) ] = (cid:88) ξ | n p ξ | n Ω ( ξ | n ) with p ξ | n = P ( ξ | n = ξ | n ) . or equivalently E [ Ω ( ξ | n ) ] = (cid:88) a | n p a | n Ω ( a | n ) . RAFFAELA CAPITANELLI* AND MIRKO D’OVIDIO
We notice that E [ Ω ( ξ | n ) ( x )] = 1 = ( p + . . . + p N ) N if x ∈ (cid:92) Ξ n (cid:91) i =1 Ω ( ξ | i ) . (2.7)3. Dirichlet forms and Base processes
Let E be a locally compact, separable metric space and E ∂ = E ∪ { ∂ } be the one-pointcompactification of E . Denote by B ( E ) the σ -field of the Borel sets in E ( B ∂ is the σ -field in E ∂ ). Let X = { X t , t ≥ } with infinitesimal generator ( A, D ( A )) be the symmetric Markovprocess on ( E, B ( E )) with transition function p ( t, x, B ) on [0 , ∞ ) × E × B ( E ). The point ∂ isthe cemetery point for X and a function f on E can be extended to E ∂ by setting f ( ∂ ) = 0.The associated semigroup is uniquely defined by P t f ( x ) := (cid:90) E f ( y ) p ( t, x, dy ) = E x [ f ( X t )] , f ∈ C ∞ ( E )with X = x ∈ E where E x denote the mean value with respect to the probability measure P x ( X t ∈ dy ) = p ( t, x, dy )and C ∞ is the set of continuous function C ( E ) on E such that f ( x ) → x → ∂ . Let E ( u, v ) = ( √− Au, √− Av ) with domain D ( E ) = D ( √− A ) be the Dirichlet form associated with(the non-positive definite, self-adjoint operator) A . Then X is equivalent to an m -symmetricHunt process whose Dirichlet form ( E , D ( E )) is on L ( E ) (see the books [15; 23]). Withoutrestrictions we assume that the form is regular ([23, page 143]).We say that X is the base process. Our aim is to consider time changes of the base process X . Such random times will be introduced in the next section.4. Time fractional equations and Time changes
We first introduce the subordinator H = { H t , t ≥ } for which E [exp( − λH t )] = exp( − t Φ( λ ))where Φ is the symbol of H . The symbol Φ may be associated also to the inverse L of H , thatis L = { L t , t ≥ } defined as L t = inf { s ≥ H s > t } , t ≥ . We assume that H = 0, L = 0. By definition, we also have that P ( H t < s ) = P ( L s > t ) , s, t > . (4.1)The symbol Φ we consider hereafter is a Bernstein function with representationΦ( λ ) = (cid:90) ∞ (cid:16) − e − λz (cid:17) Π( dz ) , λ ≥ , ∞ ) with (cid:82) ∞ (1 ∧ z )Π( dz ) < ∞ is the associated L´evy measure . We also recallthat Φ( λ ) λ = (cid:90) ∞ e − λz Π( z ) dz, Π( z ) = Π(( z, ∞ )) (4.3)and Π is the so called tail of the L´evy measure . Both random times H, L are non-decreasing.We do not consider step-processes with Π((0 , ∞ )) < ∞ and therefore we focus only on strictlyincreasing subordinators with infinite measures. Thus, the inverse process L turns out to be acontinuous process. For details, see the books [5; 30].We now introduce the fractional operators and the fractional equations governing the time-changed process X L = { X ◦ L t , t ≥ } , that is the base process X = { X t , t ≥ } with the timechange L characterized by the symbol Φ. RACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES 7
Let
M > w ≥
0. Let M w be the set of (piecewise) continuous function on [0 , ∞ ) ofexponential order w such that | u ( t ) | ≤ M e wt . Denote by (cid:101) u the Laplace transform of u . Then,we define the operator D Φ t : M w (cid:55)→ M w such that (cid:90) ∞ e − λt D Φ t u ( t ) dt = Φ( λ ) (cid:101) u ( λ ) − Φ( λ ) λ u (0) , λ > w where Φ is given in (4.2). Since u is exponentially bounded, the integral (cid:101) u is absolutely con-vergent for λ > w . By Lerch’s theorem the inverse Laplace transforms u and D Φ t u are uniquelydefined. Notice that Φ( λ ) (cid:101) u ( λ ) − Φ( λ ) λ u (0) = ( λ (cid:101) u ( λ ) − u (0)) Φ( λ ) λ . (4.4)Simple arguments say that D Φ t can be written as a convolution involving the ordinary derivativeand the inverse transform of (4.3) iff u ∈ M w ∩ C ([0 , ∞ ) , R + ) and u (cid:48) ∈ M w , that is, D Φ t u ( t ) = (cid:90) t u (cid:48) ( s )Π( t − s ) ds. We notice that when Φ( λ ) = λ (that is, the ordinary derivative) we have that a.s. H t = t and L t = t .We also notice that for Φ( λ ) = λ β , the symbol of a stable subordinator, the operator D Φ t becomes the Caputo fractional derivative D Φ t u ( t ) = 1Γ(1 − β ) (cid:90) t u (cid:48) ( s )( t − s ) β ds (4.5)with u (cid:48) ( s ) = du/ds .For Φ( λ ) = ( λ + η ) β − η β , with η ≥ β ∈ (0 , D Φ t becomes the Caputotempered fractional derivative D Φ t u ( t ) = 1Γ(1 − β ) (cid:90) t u (cid:48) ( s )( t − s ) β e − η ( t − s ) ds with u (cid:48) ( s ) = du/ds .For explicit representation of the operator D Φ t see also the recent works [13; 31].Let X be the process with generator ( A, D ( A )) introduced above. In the present work weconsider the time fractional equation D Φ t u = Au, u = f ∈ D ( A ) . (4.6)The probabilistic representation of the solution to (4.6) is written in terms of the time-changedprocess X L , that is u ( t, x ) = E x [ f ( X Lt )] = (cid:90) ∞ P s f ( x ) P ( L t ∈ ds ) , x ∈ E, t > . (4.7)We notice that (4.7) is not a semigroup, indeed the random time L is not Markovian andtherefore, the composition X L is not a Markov process.The fractional Cauchy problem has been investigated by many authors by considering Caputoderivative and only recently, by taking into account more general operators. The followingtheorem has been obtained in [11] for Feller processes (not necessarily Feller diffusions, see [11])and we mention here such a result for the reader’s convenience. Theorem 4.1.
The function (4.7) is the unique strong solution in L ( E ) to (4.6) in the sensethat: (1) ϕ : t (cid:55)→ u ( t, · ) is such that ϕ ∈ C ([0 , ∞ ) , R + ) and ϕ (cid:48) ∈ M , (2) ϑ : x (cid:55)→ u ( · , x ) is such that ϑ, Aϑ ∈ D ( A ) , (3) ∀ t > , D Φ t u ( t, x ) = Au ( t, x ) holds a.e in E (4) ∀ x ∈ E , u ( t, x ) → f ( x ) as t ↓ . RAFFAELA CAPITANELLI* AND MIRKO D’OVIDIO
In [13] the author proves existence and uniqueness of strong solutions to general time fractionalequations with initial datum f ∈ D ( A ). In [16] the authors establish existence and uniquenessfor weak solutions and initial datum f ∈ L . The result in Theorem 4.1 has been proved in ageneral setting, that is by considering a generator of a Feller process as in [13] but following avery different approach. We notice that the condition on the initial datum f must be betterspecified for the compact representation of the solution, this is the case investigated in [17] forinstance (the domain has no boundary) or the case investigated in [14] (with Dirichlet conditionon the boundary).In the next section we will study continuous base processes time changed by continuousrandom times, thus we do not stress the fact that the previous result holds for Feller process(right-continuous with no discontinuity other than jumps).5. Main results
We consider the prefractal RKD Ω ( ξ | n ) defined in Section 2 and we construct the set Ω ( ξ | n ) \ B where B ⊂ Ω ( ξ | is a ball.Then, we consider Brownian diffusions on the Random Koch Domain Ω ( ξ | n ) \ B . Let X n = { X nt , t ≥ } with X n = x ∈ Ω ( ξ | n ) \ B be a sequence of planar Brownian motions for a given ξ ∈ Ξ. Let ( A n , D ( A n )) be the generator of X n , in particular A n = ∆ and D ( A n ) = { u ∈ H (Ω ( ξ | n ) \ B ) : ∆ u ∈ L (Ω ( ξ | n ) \ B ) , u | ∂B = 0 , ( ∂ n u + c n σ ( ξ | n ) u ) | ∂ Ω ( ξ | n ) = 0 } where c n ≥ n ( x ) denote the inward normal vector at x ∈ ∂ Ω ( ξ | n ) and σ ( ξ | n ) is defined in (2.6).It is well-known that there is one to one correspondence between the infinitesimal generator of X n and the closed symmetric form ( E n , D ( E n ) (see [23, Theorem 1.3.1]).We recall that a form ( E n , D ( E n ) can be defined in the whole of L ( F, m ) by setting E n ( u, u ) =+ ∞ ∀ u ∈ L ( F, m ) \ D ( E n ) . Similarly a forms E , E can be defined in the whole of L ( F, m )by setting E ( u, u ) = + ∞ ∀ u ∈ L ( F, m ) \ D ( E ) . For the convenience of the readers we recall the definition of convergence of forms introducedby Mosco in [27], denoted by M -convergence. Definition 1.
A sequence of forms {E n ( · , · ) } M -converges to a form E ( · , · ) in L ( F ) if (a) For every v n converging weakly to u in L ( F )lim E n ( v n , v n ) ≥ E ( u, u ) , as n → ∞ . (5.1) (b) For every u ∈ L ( F ) there exists v n converging strongly in L ( F ) such that lim E n ( v n , v n ) ≤ E ( u, u ) , as n → ∞ . (5.2)In our framework, we consider the pre-fractal form E n ( · , · ) on L (Ω ( ξ ) \ B ) by defining E n ( u, u ) = (cid:90) Ω ( ξ | n ) \ B |∇ u | dxdy + c n σ ( ξ | n ) (cid:90) ∂ Ω ( ξ | n ) | u | ds for u | Ω ( ξ | n ) \ B ∈ H (Ω ( ξ | n ) \ B ) , u | ∂B = 0+ ∞ otherwise . We now introduce the time-changed process X L,n = X n ◦ L and we study the asymptoticbehaviour of X L,n depending on the asymptotics for c n . The process X L,n can be considered inorder to study the corresponding time-fractional Cauchy problem on Ω ( ξ | n ) \ B D Φ t u = A n u, u = f ∈ D ( A n ) . Let D be the set of continuous functions from [0 , ∞ ) to E ∂ = Ω ( ξ ) ∪ ∂ which are rightcontinuous on [0 , ∞ ) with left limits on (0 , ∞ ). We denote by ∂ the cemetery point, that is E n∂ is the one-point compactification of E n = Ω ( ξ | n ) , n ∈ N . Let D the set of non-decreasingcontinuous function from [0 , ∞ ) to [0 , ∞ ). RACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES 9
Proposition 5.1. (Kurtz, [21]. Random time change theorem). Suppose that X n , X are in D and L n , L are in D . If ( X n , L n ) converges to ( X, L ) in distribution as n → ∞ , then X n ◦ L n converges to X ◦ L in distribution as n → ∞ .Proof. The proof follows from part b) of Theorem 1.1 and part a) of Lemma 2.3 in [21]. Lemma2.3 gives convergence for strictly increasing time changes. Since H is strictly increasing, we usepart c) of Theorem 1.1 and find results for L which is non-decreasing and continuous. Then,part b) holds for the random time changes L n . (cid:3) Theorem 5.1. As n → ∞ , X L,n → X L in distribution in D ξ − a.s. on Ω ( ξ ) . In particular, as c n → c ≥ , i) if c = 0 , then X L is reflected on ∂ Ω ( ξ ) , that is the process driven by D Φ t u = ∆ N u, u = f ∈ D (∆ N ) where D (∆ N ) = { u ∈ H (Ω ( ξ ) \ B ) : ∆ u ∈ L (Ω ( ξ ) \ B ) , u | ∂B = 0 , ( ∂ n u ) | ∂ Ω ( ξ ) = 0 } ;ii) if c ∈ (0 , ∞ ) , then X L is (elastic) partially reflected on ∂ Ω ( ξ ) , that is the process drivenby D Φ t u = ∆ R u, u = f ∈ D (∆ R ) where D (∆ R ) = { u ∈ H (Ω ( ξ ) \ B ) : ∆ u ∈ L (Ω ( ξ ) \ B ) , u | ∂B = 0 , ( ∂ n u + cu ) | ∂ Ω ( ξ ) = 0 } ;ii) if c = ∞ , then X L is killed on ∂ Ω ( ξ ) , that is the process driven by D Φ t u = ∆ D u, u = f ∈ D (∆ D ) where D (∆ D ) = { u ∈ H (Ω ( ξ ) \ B ) : ∆ u ∈ L (Ω ( ξ ) \ B ) , u | ∂B = 0 , u | ∂ Ω ( ξ ) = 0 } . Remark 5.1.
We point out that the condition on the boundary ∂ Ω ( ξ ) must be meant in the dualof certain Besov spaces (for details, see [8], [26] and the references therein).Proof. Fix ξ ∈ Ξ. First we prove the M-convergence in L (Ω ( ξ ) \ B ) of the Dirichlet forms E n .The case of finite limit has been addressed in Theorem 5.2 in [9]: in particular, it has beenproved that if c n → c ≥
0, then the sequence of forms E n ( · , · ) M –converges in the space L (Ω ( ξ ) \ B ) to the form E c ( u, u ) = (cid:90) Ω ( ξ ) \ B |∇ u | dxdy + c (cid:90) ∂ Ω ( ξ ) | u | dµ ( ξ ) for u | Ω ( ξ ) \ B ∈ H (Ω ( ξ ) \ B ) , u | ∂B = 0+ ∞ otherwise . The last form E c , for c ∈ (0 , ∞ ), is associated with the semigroup ([6; 15]) E x [ f ( X t )] = E x [ f ( ∗ X t ) M t ] (5.3)where the multiplicative functional M t is associated to the Revuz measure given by the pertur-bation of the form E c . Thus, (5.3) is the solution to ∂ t u = ∆ R u , u = f ∈ D (∆ R ).For c = 0, the form E c is associated with E x [ f ( X t )] = E x [ f ( ∗ X t )]solution to ∂ t u = ∆ N u , u = f ∈ D ( A ).Now we prove that if c n → ∞ , the sequence of forms E n M-converges on L (Ω ( ξ ) ) to the form E ∞ ( u, u ) = (cid:90) Ω ( ξ ) \ B |∇ u | dxdy for u | Ω ( ξ ) \ B ∈ H (Ω ( ξ ) \ B )+ ∞ otherwise. First we prove condition ( a ) of Definition 1. Up to passing to a subsequence, which we stilldenote by v n , we can suppose that v n | Ω ( ξ | n ) \ B ∈ H (Ω ( ξ | n ) \ B ) , (5.4)and, for every n, || v n || H (Ω ( ξ | n ) \ B ) (cid:54) c ∗ , (5.5)with c ∗ independent of n. First we extend v n by Jones extension operator (Theorem 1 in [24])and after we restrict it to the domain Ω ( ξ ) \ B : more precisely, we extend v n to a function v ∗ n = Ext J v n | Ω ( ξ ) \ B , such that || v ∗ n || H (Ω ( ξ ) \ B ) (cid:54) C J || v n || H (Ω ( ξ ) n \ B ) (cid:54) C J c ∗ . (5.6)We point out that the constant C J independent of n (see Theorem 3.4 in [9]) that is the normof extension operator is independent of the (increasing) number of sides.Then, there exists v ∗ such that the sequence v ∗ n weakly converges to v ∗ in H (Ω ( ξ ) \ B ) :for the uniqueness of the limit in the weak topology, we obtain that v ∗ = u and, in particular, u ∈ H (Ω ( ξ ) \ B ) . Since the sequence v ∗ n weakly converges to u in H (Ω ( ξ ) \ B ) , we have thatlim (cid:90) Ω ( ξ | n ) \ B |∇ v n | dxdy ≥ (cid:90) Ω ( ξ ) \ B |∇ u | dxdy. (5.7)From the compact embedding of H (Ω ( ξ ) ) in H α (Ω ( ξ ) ) ( < α < || v ∗ n − u || H α (Ω ( ξ ) \ B ) → σ ( ξ | n ) (cid:90) ∂ Ω ( ξ | n ) | v n | ds → k (cid:90) ∂ Ω ( ξ ) | u | dµ ( ξ ) (5.9)when n → ∞ (see Theorem 2.1 in [9]). We stress the fact that the value of σ ( ξ | n ) play a crucialrole in the previous limit.Now, if c n → ∞ , for any k > n such that, for all n > n , c n ≥ k. Then c n σ ( ξ | n ) (cid:90) ∂ Ω ( ξ | n ) | v n | ds ≥ kσ ( ξ | n ) (cid:90) ∂ Ω ( ξ | n ) | v n | ds → k (cid:90) ∂ Ω ( ξ ) | u | dµ ( ξ ) (5.10)when n → ∞ . Dividing for k and letting k → ∞ we obtain that (cid:90) ∂ Ω ( ξ ) | u | dµ ( ξ ) = 0 (5.11)and so u = 0 on ∂ Ω ( ξ ) . By combining (5.7), (5.9), (5.11) we have proved condition ( a ) ofDefinition 1.In order to prove condition (b) of Definition 1, we can assume that u ∈ H (Ω ( ξ ) \ B ) withoutloss of generality: then, the choice of v n = u suffices to achieve the result. So we have provedthe M-convergence of the forms E n ( · , · ) on L (Ω ( ξ ) \ B ) to the form E ∞ when c n → ∞ .From the M-convergence of the forms E n ( · , · ) on L (Ω ( ξ ) \ B ) , by using the results in the recentpaper [11], we obtain the convergence of the time changed processes.More precisely, from the M -convergence of the forms we have the strong convergence ofsemigroups. From Theorem 17.25 (Trotter, Sova, Kurtz, Mackeviˇcius) in [19] we have that strongconvergence of semigroups (Feller semigroups) is equivalent to weak convergence of measures if X n → X in distribution. Then we obtain that X n d → X in D .From Proposition 5.1, we have that ∀ ξ ∈ Ξ , X n ◦ L =: X L,n → X L := X ◦ L on Ω ( ξ ) in distribution as n → ∞ in D .From the pointwise convergence, we get that ξ − a.s. X L,n → X L on Ω ( ξ ) in distribution as n → ∞ in D . RACTIONAL CAUCHY PROBLEM ON RANDOM SNOWFLAKES 11 (cid:3)
Let us consider now the process X L on E . We point out some peculiar aspects of X L andthe corresponding lifetimes. Theorem 5.2.
Let us consider the Cauchy problems ∂ t w = Aw, w = f ∈ D ( A ) (5.12) and D Φ t u = Au, u = f ∈ D ( A ) (5.13) with Φ such that lim λ → Φ( λ ) λ ∈ (0 , ∞ ) . We have that, ∀ x ∈ E : - if Φ (cid:48) (0) < , then (cid:90) ∞ u ( t, x ) dt < (cid:90) ∞ w ( t, x ) dt, - if Φ (cid:48) (0) > , then (cid:90) ∞ u ( t, x ) dt > (cid:90) ∞ w ( t, x ) dt, - if Φ (cid:48) (0) = 1 , then (cid:90) ∞ u ( t, x ) dt = (cid:90) ∞ w ( t, x ) dt. Proof.
The solution to (5.12) has the following probabilistic representation w ( t, x ) = E x [ f ( X t )] = E x [ f ( ∗ X t ) M t ]where M t = ( t<ζ ) is the multiplicative functional written in terms of the lifetime ζ of the process X on E . Then, we consider the part process X of ∗ X where ∗ X = x ∈ E . It is well knownthat M t characterizes uniquely the associated semigroup ([6]), that is the solution w . We alsohave that E x (cid:20)(cid:90) ζ f ( X s ) ds (cid:21) = (cid:90) ∞ w ( t, x ) dt =: w ( x )is the solution to the elliptic problem on E − Aw = f. From Theorem 4.1 we have that the time-changed process X L can be considered in order tosolve the problem (5.13), that is u ( t, x ) = E x [ f ( X Lt )] = E x [ f ( ∗ X Lt ) ( t<ζ L ) ]where ζ L is the lifetime of X L . As before we introduce the u ( x ) := (cid:90) ∞ u ( t, x ) dt = E x (cid:34)(cid:90) ζ L f ( X Ls ) ds (cid:35) which is the solution to the elliptic problem associated with the fractional Cauchy problem(5.13). We are able to obtain the key relation between w and u by taking into considerationthe following plain calculations. First we recall (4.7) where P s f ( x ) here is given by w ( s, x ) with w ( s, x ) → f ( x ) as s →
0. Moreover (see [12]), (cid:90) ∞ e − λt P ( L t ∈ ds ) dt = Φ( λ ) λ e − s Φ( λ ) ds. (5.14)We have that u ( x ) = lim λ → (cid:90) ∞ e − λt u ( t, x ) dt =[by (4.7)]= lim λ → (cid:90) ∞ e − λt (cid:90) ∞ w ( s, x ) P ( L t ∈ ds ) dt =[by (5.14)]= lim λ → (cid:90) ∞ w ( s, x ) Φ( λ ) λ e − s Φ( λ ) ds = (cid:18) lim λ → Φ( λ ) λ (cid:19) (cid:90) ∞ w ( s, x ) ds. That is u ( x ) = (cid:18) lim λ → Φ( λ ) λ (cid:19) w ( x )and this gives a connection between solutions of elliptic problems introduced above in the proof.Since Φ is a Bernstein function with Φ(0) = 0 we get the result. (cid:3) The characterization given in the previous result admits a probabilistic interpretation in termsof mean lifetime of the base and time-changed processes. The problems (5.12) and (5.13) with f = E are associated with w ( x ) = E x [ ζ ] and u ( x ) = E x [ ζ L ] as described in the previousproof and the mean lifetime says how much the time change L modifies the base process X . Byfollowing the definition given in [12] and the relation between w and u we say that X is delayed orrushed on E by L . An example is given by the tempered fractional derivative ([4; 29]) associatedwith the symbol Φ( λ ) = ( λ + η ) β − η β with η > β ∈ (0 , E x [ ζ L ] = βη β − E x [ ζ ]that is, if βη β − < X is rushed by L , whereas if βη β − > X is delayed by L .The previous discussion on either delayed or rushed processes holds according to specificregularity conditions on the boundary ∂E . We must have that sup E w ( x ) < ∞ which is thecharacterization of trap domains (written here for X with generator A ) given in [7] for theBrownian motion. By applying the result in [7] it follows that the following proposition holdstrue. Proposition 5.2.
For ξ ∈ Ξ , the domains Ω ( ξ | n ) , n ∈ N are non trap for the Brownian motion. Since the previous statement holds pointwise for any contraction factor, we immediately obtainthe following general statement.
Proposition 5.3.
For the Ξ -valued random vector ξ the domains Ω ( ξ | n ) , n ∈ N are ξ − a.s. nontrap for the Brownian motion. Grant.
The authors are members of GNAMPA (INdAM) and are partially supported byGrants Ateneo “Sapienza” 2018.
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