Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

We prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

We will explain some known results related to (1.1). Consider the case where α = 2 and coefficients of noise are Lipschitz continuous, that is, where a : R → R is Lipschitz continuous. Equation (1.2) was studied by Funaki [7] and Walsh [18]. They showed the unique existence of a solution to (1.2) and spatial regularity of the solution. Later Mueller-Perkins [11] and Shiga [17] studied the SPDE (1.2) without assuming the Lipschitz continuty of coefficients a. They imposed the following growth condition on a(u): for some 0 < θ < 1. They showed the existence of probability space (¯ ,F,P) on which there is a space-time white noiseẆ (t, x) such that (1.2) withẆ (t, x) replaced byẆ (t, x) has a mild solution X (t, x). In addition, they proved that, if f is positive, then any solution X (t, x) to (1.2) is positive for all t ≥ 0 and x ∈ R. Moreover, it was shown that, if f (x) is sufficiently rapidly decreasing as x → ∞, then any solution X (t, x) to (1.2) is also sufficiently rapidly decreasing. Mytnik [13] proved the weak uniqueness of solutions to the following equation which is a special one of (1.2): with 1/2 < γ < 1. The idea of his proof is based on a duality argument developed by Ethier-Kurtz [5]. Mytnik [14] studied the dual process Y described by the following SPDE: whereL is a stable noise on R × R + with nonnegative jumps. He coordinated a probability space (˜ ,F,P) on which there exists a stable noiseL and random field Y (t, x), where Y (t, x) is a mild solution of (1.5).
However, it is still an open problem to show the existence and uniqueness of solution to (1.6) without assuming the Lipschitz continuty of a(u). The purpose of this paper is to establish the existence and weak uniqueness of a mild solution to (1.1) which is a special case of (1.6) where a(u) = |u| γ with 1 2 < γ < 1. We intend to use similar aguments to Mueller-Perkins [11], Shiga [17], and Mytnik [13]. The large difference between the fractional Laplacian and the usual Laplacian can be found in the decay properties of fundamental solutions as x → ∞. Since the fundamental solution of the usual Laplacian has an exponential decay property, the solution X (t, x) to (1.2) has the exponential decay property if the condition (1.3) holds. Similarly the fundamental solution of fractional Laplacian decays polynomially, we can show that the solution to (1.1) has the polinomial decay property if the condition (1.3) holds. To prove the uniqueness of solution to (1.4) Mytnik [13] used the exponential decay property. It is enough to use the polynomial decay property in order to prove the uniqueness of solution to (1.1).
This paper is organized as follows: in Sect. 2 we prepare some tools and lemma to prove our main results. In Sect. 3, we show the existence and uniqueness of solution by applying the Banach fixed point theorem with Lipschitz continuous coefficients a(X ). In addition, we follow the argument of Mueller-Perkins [11] and Shiga [17] to prove positivity and polynomial decay properties of solutions to (1.6). From Sect. 4 we begin with consideration (1.1). We prove the existence of solution to (1.1) by tightness arguments. In fact, we can prove a uniqueness of solution in the distributional sense by applying to duality method. Since the proof is almost same as the paper [5], we omit the detail.

Definition of the solution
Let ( , F, F t , P) be a complete probability space with filtration andẆ (t, x) be an {F t }-space-time Gaussian white noise with covariance given by we can define stochastic integral (cf. Walsh [18]) The Eq. (1.1) makes sense if we integrate the equation in time and space and use the initial condition.
x ∈ R} is said to be a solution in the sense of generalized functions of (1.1) if for any φ ∈ C ∞ 0 (R), the following equality holds: Using the Green function, we can describe a solution of (1.1) in a mild form.

Definition 2.2
An (F t )-adapted random field {X (t, x), t ≥ 0, x ∈ R} is said to be a mild solution of (1.1) with initial function f if the following stochastic integral equation holds: where G(t, x, y) denotes the Green function of (2.1).
We introduce a martingale problem induced by (1.1).

Definition 2.3
Let S be a Banach space. A solution to martingale problem for (1.1) we mean a measurable process X with values in S defined on some probability space is an F X t square integrable martingale with the quadratic variation given by

Definition 2.4
Let S be a Banach space and X 1 and X 2 be S-valued mild solutions to the SPDE (1.1) with the same initial value. We say that the SPDE (1.1) has pathwise uniqueness if holds.
To this end, it is required that all the terms in (2.4) and (2.3) are well defined. Here, a relationship between a solution in the sense of generalized functions and a mild solution is well known (cf. [18]).

Proposition 2.1 A solution in the sense of generalized functions of (1.1) is equivalent to the mild solution.
A solution to the martigale problem and a mild solution in weak sense are equivalent (cf. [7]).

Existence and uniqueness of mild solutions
In order to prove the existence and uniqueness to SPDE (1.1), we first consider the case where coefficients are Lipschitz continuous: where a : R → R is Lipschitz continuous. Using the Green function, we can write the solution of Eq. (3.1) in a mild form: For any 0 < ρ < (α + 1)/2, define a weighted L 2 -norm defined by 1 Assume that f ∈ L 2 ρ (R) and a : R → R is a Lipschitz function satisfying linear growth condition, that is, there exists C > 0 such that For every 0 < ρ < (α + 1)/2, define a function space The regularity of solution to (3.1) is well known (cf. [4], [15]).

Positivity of solution
We will follow Shiga's arguments [17] to prove the positivity of the solution to (3.1). At first, we need to prepare a boundness of the solution to (3.1).

Lemma 3.1
Assume that for every T > 0 there exists C T > 0 and such that Then for p ≥ 1 can be written by Proof Assume that p = 1. Then for every 0 < ρ < (α + 1)/2 and t ≥ 0, we have We will apply an induction argument. For p ≥ 2, using the Burkhorder's inequality, the Hölder's inequality shows that Multiply both sides by λ ρ (x) and integrate with variable x, we obtain that Setting p = 2, (3.6) and Grownwall's lemma imply that From an induction argument, we complete the proof for p = 2 m with m ∈ N ∪ {0}.

Remark 3.1
We can get in a similarly way to the proof of Lemma 3.1 and Hölder's inequality together, For any 0 < ρ < (α + 1)/2 define the function space C + ρ (R) by

Theorem 3.3 Let X (t, x) be the mild solution of (3.2)
with the initial function f ∈ C + ρ (R). Assume that a(u) satisfies the condition (3.4). In particular, a(0) = 0. Then, we have Proof For every ε > 0, choose non-negative, symmetric and smooth function ρ ε (x) such that . We consider the following equation Since a(u) is Lipschitz continuous and ε is a bounded operator on L 2 (R), the above equation has the unique strong solution and continuous version(cf. [2]). From now on, we claim that P (X ε (t, ·) ≥ 0, ∀t ≥ 0) = 1. (3.8) Let a n = −2(n 2 + n + 2) −1 be a non-increasing sequence. Immediately, we obtain that a n → 0 as n → ∞ and a n a n−1 x −2 dx = n. Let ψ n (x) be a nonnegative continuous function such that supp(ψ n ) = (a n−1 , a n ), 0 ≤ ψ n (x) ≤ 2 nx 2 and a n a n−1 x 0 ψ n (z)dz ≤ 0 for x < 0 and n (x) = 0 for x ≥ 0. Note that, for x < 0 there exists n 0 ∈ N such that for all n ≥ n 0 we have a n > x. About n we can get the following properties as n → ∞, (X ε (s, x))ds.

By ItO o s formula
From the Lipschitz condition and a(0) = 0, Taking the limit as n → ∞ and by monotone convergence theorem
Therefore, by Grownwall's lemma for sup x∈R E(φ(X ε (t, x))), we obtain for every t > 0 and x ∈ R, which yields (3.8). Let We will prove that To prove this fact, we need the following lemma (cf. Appendix of [1]).

Lemma 3.2 (i)
(ii) For some α > 0 and β > 0 Notice that, X ε (t, x) can be written in the following mild form: where the last term equals to Since f is bounded, it follows that for every T > 0 (see Remark 3.1), Then we have
By the Hölder's inequality, the Lipschitz continuty of function a(x) and Lemm 3.2, we have and similarly According to the inequality the definition of ρ ε gives that By Hölder's inequality and (3.10), In a same way, we have Applying the fact we can get Then there exists some constant C > 0 such that where H (T , ε) = C n=2,5,6 sup 0≤t≤T sup x∈R I n (t, x, ε), Therefore, Grownwall's inequality implies that and thus completes the proof of Theorem 3.3.

Polynomial decay
In this section, we show that the solution of (3.1) has modification in the class C ρ (R).
The following lemma is a variant of a Kolmogorov's continuity criterion theorem.
Hence combining with (3.5) From now on, we claim that Note that, Therefore, to prove (3.11) we need to show that Let us write The first integral is finite, since For the second term one, we use Lemma 2.1 and the change of variable z = (1+v Further, tight. This means that there exists a subsequence {n k } k∈N and a random field X ∈ C [0, T ]; C + ρ (R) such that By Skorohod representation theorem (cf. [5]), we can find some random fields Y n , Y ∈ C [0, T ]; C + ρ (R) on some probability space (¯ ,F , (F t ) 0≤t≤T ,P), such that and X n = Y n in law, X = Y in law.
Then we can get for every φ ∈ D((− ) α/2 ) Hence, (M n φ ) n∈N is a sequence of uniformly integrable martingales, and therefore, there exists a martingale M φ such that Thus we complete the proof of Theorem 4.1.
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