On Some Properties of a Class of Fractional Stochastic Heat Equations
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Abstract
We consider nonlinear parabolic stochastic equations of the form \(\partial _t u=\mathcal {L}u + \lambda \sigma (u)\dot{\xi }\) on the ball \(B(0,\,R)\), where \(\dot{\xi }\) denotes some Gaussian noise and \(\sigma \) is Lipschitz continuous. Here \(\mathcal {L}\) corresponds to a symmetric \(\alpha \)stable process killed upon exiting B(0, R). We will consider two types of noises: spacetime white noise and spatially correlated noise. Under a linear growth condition on \(\sigma \), we study growth properties of the second moment of the solutions. Our results are significant extensions of those in Foondun and Joseph (Stoch Process Appl, 2014) and complement those of Khoshnevisan and Kim (Proc AMS, 2013, Ann Probab, 2014).
Keywords
Stochastic partial differential equations Fractional Laplacian Stochastic heat equation Heat kernelMathematics Subject Classification (2010)
Primary 60H15 Secondary 82B441 Introduction and Main Results
Assumption 1.1
The above assumptions on \(\sigma \) are quite natural and have been used in various works; see [10, 11]. The lower bound is essentially a growth condition which is needed for our results. These inequalities also imply that \(\sigma (0)=0\). This is needed for nonnegativity of solutions to stochastic heat equations. Even though we do not need nonnegativity of the solution in this paper, the upper bound makes our computations easier to follow.
Assumption 1.2
\(\mathcal {L}\) is the generator of a symmetric \(\alpha \)stable process killed upon exiting \(B(0,\,R)\) so that (1.1) can be thought of as the Dirichlet problem for fractional Laplacian of order \(\alpha \).
Here is our first result.
Theorem 1.3
Definition 1.4
We then have the following corollary.
Corollary 1.5
The excitation index of the solution to (1.1) is \(\frac{2\alpha }{\alpha 1}\).
Theorem 1.6
Corollary 1.7
The excitation index of the solution to (1.5) is \(\frac{2\alpha }{\alpha \beta }\).

We need to compare the heat kernel estimates for killed stable process with that of “unkilled” one. To do that, we will need sharp estimates of the Dirichlet heat kernel.

We will also need to study some renewaltype inequalities, and by doing so, we come across the MittagLeffler function whose asymptotic properties become crucial.

While the above two ideas are enough for the proof of Theorem 1.3, we will also need to significantly modify the localisation techniques of [13] to complete the proof of Theorem 1.6.
Here is a plan of the article. In Sect. 2, we collect some information about the heat kernel and the renewaltype inequalities. In Sect. 3, we prove the main results concerning (1.1). Section 4 contains analogous proofs for (1.5). In Sect. 5, we extend our study to a much wider class of equations.
Finally, throughout this paper, the letter c with or without subscripts will denote constants whose exact values are not important to us and can vary from line to line.
2 Preliminaries
Proposition 2.1
Proof
We now make a simple remark which will be important in the sequel.
Remark 2.2
Proposition 2.3
Proof
Lemma 2.4
Proof
We now present the renewal inequalities.
Proposition 2.5
Proof
We have the “converse” of the above result.
Proposition 2.6
Proof
The above inequalities are well studied; see for instance [12]. But the novelty here is that, as opposed to what is usually done, instead of t, we take \(\kappa \) to be large.
3 Proofs of Theorem 1.3 and Corollary 1.5
Proposition 3.1
Proof
Proposition 3.2
Proof
Proof of Theorem 1.3
Proof of Corollary 1.5
4 Proofs of Theorem 1.6 and Corollary 1.7
Lemma 4.1
Proof
Proposition 4.2
Proof
We have the following lower bound on the second of the solution. Inspired by the localisation arguments of [13], we have the following.
Proposition 4.3
Proof
Fix \(\epsilon >0\) and for convenience, set \(B:=B(0,\,R)\) and \(B_\epsilon :=B(0,\,R\epsilon )\). We will also use the following notation; \(B^2:=B\times B\) and \(B^2_\epsilon :=B_\epsilon \times B_\epsilon \).
Proposition 4.4
Proof
Proof of Theorem 1.6
Proof of Corollary 1.7
The proof is exactly the same as that of Corollary 1.5 and is omitted.\(\square \)
5 Some Extensions
Theorem 5.1
Proof
Example 5.2
Example 5.3
Example 5.4
Example 5.5
Notes
Acknowledgments
The authors thank a referee and an associate editor for insightful comments which improved the paper.
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