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Stochastic solutions for time-fractional heat equations with complex spatial variables

We deal with complex spatial diffusion equations with time-fractional derivative and study their stochastic solutions. In particular, we complexify the integral operator solution to the heat-type equation where the time derivative is replaced with the convolution-type generalization of the regularized Caputo derivative. We prove that this operator is solution of a complex time-fractional heat equation with complex spatial variable. This approach leads to a wrapped Brownian motion on a circle time-changed by the inverse of the related subordinator. This time-changed Brownian motion is analyzed and, in particular, some results on its moments, as well as its construction as weak limit of continuous-time random walks, are obtained. The extension of our approach to the higher dimensional case is also provided.


Introduction
The study of the evolution equations with complex spatial variables is a quite recent research topic in the theory of the partial differential equations and complex analysis. In the pioneering works [8]- [9], the authors proposed two different methods to complexify the spatial variable appearing in different evolution equations (and keeping the time variable real). In particular, for the heat equation a possible approach consists in the complexification of the spatial variable in the linear semigroup operator representing the unique solution to the Cauchy problem ∂u ∂t (x, t) = 1 2 ∂ 2 u ∂x 2 (x, t), u(x, 0) = f (x), where t > 0, x ∈ R and f ∈ BUC(R). This approach is interesting because, by exploiting the theory of semigroups of linear operators, it is possible to obtain a new complex version of the heat equation (see (2.2) below) and study the properties of its analytic solution (see [8]- [9]). Furthermore, it is worth to observe that in this framework a suitable probabilistic interpretation of the solution to (2.2) leads to a wrapped up Brownian motion on a circle. The stochastic analysis of complex diffusion equations still seems to be an unexplored research topic.
The above mentioned theory was developed starting from partial differential equations involving the standard time derivative. The aim of this paper is to study complex versions of the time-fractional heat equations obtained by complexifying the spatial variable only (and keeping the time variable real). The main idea is to complexify the spatial variable in the corresponding integral operator arising in the study of time-fractional evolution equations. In particular, we study the stochastic solution of the complex heat equation, when the time-derivative is replaced by a convolution-type operator, which generalizes the Caputo fractional derivative. We will adopt the definition given in [6], i.e.
where w(·) is the tail Lèvy measure of a subordinator H g := {H g (t)} t≥0 and g(·) is its Laplace exponent, i.e. Ee −θHg(t) = e −g(θ)t , where t, θ ≥ 0 (see Section 3 for details on this definition). The so-called generalized fractional calculus has been developed in recent years, starting from Kochubei in [13], by many authors (see, among the others, [23], [10], [14]). They extend the traditional construct of fractional derivatives and integrals in order to allow a wider class of kernels. Indeed, it is immediate to check that, by choosing w(t) = t −α /Γ(1 − α), for α ∈ (0, 1), which coincides with the tail Lévy measure of an α-stable subordinator H α , the fractional derivative in (1.1) reduces to the so-called "regularized Caputo derivative". Thus we will define the latter as It has been proved in [17] that the solution to The previous result has been extended to the case of a strong Markov process X := {X(t)} t≥0 (on a separable locally compact Hausdorff space E) whose transition semigroup is a uniformly bounded strong continuous semigroup in some Banach space and has infinitesimal generator A. In this case, the solution to ))], where E g := {E g (t)} t≥0 , is the inverse of the general subordinator H g , with drift b, and is independent of X (see [6], for details). If we denote the unit disk as D := {z ∈ C; |z| < 1} and we consider the space A(D) = {f : D →: f is analytic on D, continuous on D}, endowed with the uniform norm, then we study here the solution to the following Cauchy problem The paper is organized as follows. Section 2 contains the probabilistic interpretation of the solution to the complex Cauchy problem introduced in [8] and a discussion on the properties of the circular Brownian motion. The generalized fractional setting and the complex time-fractional heat equation are introduced in Section 3, where the stochastic solution related to the related complex Cauchy problem is obtained. Section 4 is devoted to the analysis of the time-changed Brownian motion emerging in the previous section. Some results on the moments of the process are provided, as well as the construction of the time-changed process based on the convergence of time-continuous random walks. Furthermore, some special cases involving stable and tempered stable subordinators are examined. The last section contains the analysis of the complex timefractional heat equation in higher dimensions. In the interesting paper [8], it was proved (see Theorem 2.1 in [8]) that the singular (at t = 0) complex integral (that is known as a Gauss-Weierstrass integral) is a C 0 -contraction semigroup of linear operators on A(D). Furthermore, u(z, t) = W t f (z), is the unique solution (with u(z, t) ∈ A(D) for a fixed t) for the Cauchy problem where f ∈ A(D).
Here we briefly discuss the interesting probabilistic meaning of representing the solution of a complexified Cauchy problem in this way. Indeed, it is evident from (2.1), that the solution of the latter can be expressed as where B := {B(t)} t≥0 is the R-valued Brownian motion on (Ω, F, P) and B z (t) = ze −iB(t) . This means that the probabilistic representation of the solution for this complexified Cauchy problem is directly related to a circular or wrapped Brownian motion B z := {B z (t)} t≥0 moving on a circle with radius r ∈ (0, 1] (hereafter denoted by S r ) and starting point z. Furthermore, B stands for B 1 .
For the sake of simplicity, we set z = 1. From the properties of the classical R-valued Brownian motion, it is easy to characterized B. Let B be the complex conjugate of B. We observe that the wrapped Brownian motion B, satisfies the following properties: 1) B(0) = 1 a.s.; 2) B(t k )B(t k−1 ) with k = 1, 2, ..., n ∈ N, 0 =: t 0 ≤ t 1 < t 2 < ... < t n < ∞ are independent; 3) B(t)B(s) has the same distribution of where W N (µ, e − σ 2 2 ), µ ∈ R, σ 2 > 0, stands for a wrapped normal random variables with probability density function given by This result follows by standard arguments on the wrapped distributions; i.e. by wrapping the N (0, t − s) onto the circle (see, e.g., [16]).

5)
B is a wrapped Gaussian process; i.e. let 0 =: t 0 ≤ t 1 < t 2 < ... < t n < ∞, the random vector (B(t 1 ), B(t 2 ), ..., B(t n )) is multivariate wrapped normal in the following sense Since B is a Gaussian process, it follows that . Then, as in the previous point ).

Time-fractional diffusive-type equations with a complex spatial variable
Let us introduce a time-fractional version of the complex heat equation (2.2) and study its stochastic solution.
Let g : (0, +∞) → R be a Bernstein function (i.e. a non-negative, C ∞ function such that . Then, it is well-known that the following representation holds (see e.g, [22]) where ν(·) is a non-negative measure on (0, +∞), satisfying the condition i.e. ν is a Lévy measure. Let w(s) = +∞ s ν(dz) be its tail, in this paper we consider the following convolution-type derivative (see [6]) Typically w is a non-negative decreasing function on (0, +∞) that blows up at x = 0 and locally integrable on [0, ∞). We refer to [6] for the functional setting, observing that obviously this definition is a generalization of the Caputo fractional derivative (see, e.g., [12]): the latter is recovered, as a special case, for w(s) = s −α Γ(1−α) , with α ∈ (0, 1). Observe that we used a quite different notation from [6] in order to underline the connection between this generalized fractional derivative and the particular choice of the underlying Bernstein function g. We remark that a similar probabilistic approach to the generalized time-fractional derivatives have been developed in [23]. It is similar but not equivalent. Hereafter, we exclude compound Poisson subordinator, namely we assume that a = 0 and that the tail measure w(·) is infinite in the origin and absolutely continuous on (0, +∞). Let now H g := {H g (t)} t≥0 be the subordinator with Lévy measure ν and Laplace exponent g, i.e.
By the assumptions on w, the subordinator t → H g (t) associated to g is strictly increasing a.s. As a consequence, its inverse t → E g (t) is continuous a.s. We recall that the time-Laplace transform of the density of E g (t) denoted by m g (s, t) := P (E g (t) ∈ ds) /ds reads see, for example, Proposition 3.2 in [23]. We now consider the standard R-valued Brownian motion B, time-changed by E g (t), t ≥ 0, (under the assumption that B and E g are mutually independent); i.e. {B(E g (t))} t≥0 . Then, the density of B(E g (t)) for a fixed t > 0, is given by For all t > 0, we can define on D, the following complex integral where g is given in (3.6). The following stochastic interpretation of (3.7) emerges , is the time-changed circular Brownian motion moving on a circle with radius r ∈ (0, 1] with starting point z, obtained from the wrapped up process B z introduced in the previous section. We observe that that is W g t arises by the time-change of the C 0 -semigroup (2.1). We have the following analytic results concerning the convolution operator W g t . Theorem 3.1. (i) If f ∈ A(D), then we have that for any t > 0, we have that is the unique solution, belonging to A(D) for any t ≥ 0, of the Cauchy problem Proof. (i) The representation (3.10) follows by considering that f (z) ∈ A(D) and by taking into account (3.6) together with (3.5).
is absolutely convergent, we can write By using the same arguments in [9], it is possible to prove that if z n , z 0 ∈ D, with lim n→∞ z n = z 0 , we get where ω 1 (f ; δ) D := sup{|f (u) − f (v)|; |u − v| ≤ δ, u, v ∈ D} is the modulus of continuity of f. Then if f is continuous on D, the integral W g t f (z) is continuous on D, as n → ∞. From (3.10) and by exploiting (3.5), we obtain that .
(ii) From Theorem 2.1 in [6], we can observe that: 1) D g t W g t f (z) is well-defined since the integral appearing in the definition of D g t is absolutely convergent in the Banach space (A(D), || · ||); 2) for b > 0, t → W g t f (·) is globally Lipschitz in (A(D), || · ||) and then ∂ ∂t W g t f (·) exits in (A(D), || · ||) for a.s. t ≥ 0. In order to prove that (3.10) coincides with the solution of (3.12) with initial condition (3.13) we take the time-Laplace transform of both sides of (3.12). By applying the result (2.18) in [21], on the Laplace transform of the generalized derivative (3.2), we obtain .
Furthermore, from (3.10), it is easy to prove that The result follows by considering (3.14), (3.15) and (3.16) together and taking into account that Remark 3.1. In Theorem 3.1, we proved that u(t, z) = W g t f (z) is the classical solution with u ∈ C ∞ ([0, ∞); A(D)). Actually, it is also possible to prove that u(t, z) is the unique strong solution of the fractional Cauchy problem (3.12)-(3.13) (see Theorem 2.1 in [6] for the exact statement).

Time-changed wrapped Brownian motion
For simplicity and without loss of generality, we assume throughout this section that z = 1 and we consider the definition B g (t) := e iB(Eg(t)) , t ≥ 0, which is equivalent to (3.9). Then, by exploiting the theory of wrapped distribution, for any t > 0 we can write down its probability density The probability distribution on a circle is characterized by its Fourier coefficients (see [7], Theorem XIX 6.1). In our case, for k ∈ N, we have that Therefore, We now evaluate the Laplace transform of the first moments of B g (t) by applying the results on the joint moments of the inverse subordinators given in [24]. In particular, we recall that, for the Laplace transform of K t1,...,tn (s 1 , ..., s n ) := P (E g (t 1 ) > s 1 , ..., E g (t n ) > s n ), the following formula holds  where θ 1 , ..., θ n > 0, 0 = s j(0) ≤ s j(1) ≤ ... ≤ s j(n) and j(1), ..., j(n) is a permutation of the integers 1, ..., n (by convention j(0) = 0). Theorem 4.1. The Laplace transform of the moments of the process B g is equal to for 0 ≤ t 1 < t 2 and θ 1 , θ 2 > 0.
We are also able to give an integral representation for the mixed moment E[B g (t)B g (s)], for t, s ≥ 0.
for t, s ≥ 0 and t = s, while E[B g (t)B g (t)] = 1.
Proof. Let t > s and denoting U g (s, t; k, j) := E[(E g (s)) k (E g (t)) j ], s, t ≥ 0, k, j ∈ N, we can write down where we have singled out the term j = k in the second summation, since it must be treated separately (in view of its different behavior for s = 0). Now we use the recursive representation of the moments given by Theorem 4.2, [24], so that (4.9) can be rewritten as follows =: The first term in (4.10) can be treated as follows while the second one reads The result follows by inserting I 1 and I 2 into (4.10) and treating the case s > t analogously.
We now prove that the wrapped Brownian motion B g can be obtained as a scaling limit of a transformed continuous-time random walk on a circle. Let us denote by J1 =⇒ the convergence in the J 1 topology and by M1 =⇒ the convergence in the M 1 topology in the Skorohod space D([0, T ), R d ), for T > 0 and d = 1, 2, ... (see [25] and [19] for details on J 1 and M 1 topologies).   4.11) {e Proof. The convergence in (4.11) follows by the application of Theorem 2.1 and Corollary 2.4 in [18], in the special case where A(t) = B(t), t ≥ 0 : indeed, let Disc(x) be the set of discontinuities of x, the assumption that Disc({A(t)} t≥0 )∩ Disc({H g (t)} t≥0 ) = ∅ a.s. is automatically satisfied because, as well-known, it is always possible to choose a version of Brownian motion such that its trajectories are continuous with probability one. Moreover, the Lévy measure of H g is infinite on [0, +∞) by assumption. By the independence of J j , for any j = 1, 2... and by the functional central limit theorem, we have that in the J 1 topology on D([0, +∞), R × R + ). Therefore, by the above mentioned Theorem 2.1 in [18], the following convergence holds in the M 1 topology on D([0, +∞), R). The result finally follows by applying the continuous mapping theorem to the function φ(·) : R → C defined as φ(x) = e ix , x ∈ R.
We refer to [18] for the description of some relevant situations where this kind of convergence can be appropriately applied. 4.0.1. The stable case. For g(θ) = θ α , α ∈ (0, 1), formula (3.1) holds for a = 0 and b = lim θ→+∞ g(θ)/θ = 0. Moreover, the process E g (t) reduces to the inverse of the α-stable subordinator (see e.g. [20]) and the operator D g t coincides with the Caputo time-fractional derivative of order α, namely ∂ α /∂t α . In this case, if we denote W g t as W α t , we have that is the Wright function, defined for β, γ, x ∈ C. The representation (4.12) follows by the fact that the fundamental solution of the time-fractional diffusion equation involving Caputo time-fractional derivatives of order α ∈ (0, 1) is given by (4.14) u(x, t) = 1 t α/2 W −α/2,1−α/2 − |x| t α/2 , see for example [15].
Then, in this case, as a consequence of Theorem 3.1, we have the following result.
, α ∈ (0, 1), E α (t) is the inverse of the α-stable subordinator and α is the probability density of the time-changed Brownian motion B α (t) := B(E α (t)). Moreover, W α t f (z) is the unique solution u(z, t) for the fractional Cauchy problem ∂ α u ∂t α (z, t) = 1 2 Remark 4.1. Observing that, for α = 1/2 the following equality in distribution holds (4.17) B where B 1 and B 2 are independent, we have that coincides with the solution to (4.16), for α = 1/2. Therefore, in this special case, we have an iterated Brownian motion on the circle.