Stochastic wave equation in a plane driven by spatial stable noise
Volume 3, Issue 3 (2016), pp. 237–248
Pub. online: 8 November 2016
Type: Research Article
Open Access
Open Access
Received
14 October 2016
14 October 2016
Revised
20 October 2016
20 October 2016
Accepted
20 October 2016
20 October 2016
Published
8 November 2016
8 November 2016
Abstract
The main object of this paper is the planar wave equation
with random source f. The latter is, in certain sense, a symmetric α-stable spatial white noise multiplied by some regular function σ. We define a candidate solution U to the equation via Poisson’s formula and prove that the corresponding expression is well defined at each point almost surely, although the exceptional set may depend on the particular point (x,t). We further show that U is Hölder continuous in time but with probability 1 is unbounded in any neighborhood of each point where σ does not vanish. Finally, we prove that U is a generalized solution to the equation.
References
Balan, R.M., Tudor, C.A.: The stochastic wave equation with fractional noise: A random field approach. Stoch. Process. Appl. 120(12), 2468–2494 (2010) MR2728174. doi:10.1016/j.spa.2010.08.006
Dalang, R.C., Frangos, N.E.: The stochastic wave equation in two spatial dimensions. Ann. Probab. 26(1), 187–212 (1998) MR1617046. doi:10.1214/aop/1022855416
Dalang, R.C., Sanz-Solé, M.: Hölder–Sobolev regularity of the solution to the stochastic wave equation in dimension three. Mem. Am. Math. Soc. 199(931), 70 (2009) MR2512755. doi:10.1090/memo/0931
Kôno, N., Maejima, M.: Hölder continuity of sample paths of some self-similar stable processes. Tokyo J. Math. 14(1), 93–100 (1991) MR1108158. doi:10.3836/tjm/1270130491
Millet, A., Morien, P.-L.: On a stochastic wave equation in two space dimensions: Regularity of the solution and its density. Stoch. Process. Appl. 86(1), 141–162 (2000) MR1741200. doi:10.1016/S0304-4149(99)00090-3
Quer-Sardanyons, L., Tindel, S.: The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stoch. Process. Appl. 117(10), 1448–1472 (2007) MR2353035. doi:10.1016/j.spa.2007.01.009
Samorodnitsky, G., Taqqu, M.S.: Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, New York, NY (1994) MR1280932
Walsh, J.B.: An introduction to stochastic partial differential equations. In: École d’Été de Probabilités de Saint-Flour, XIV–1984. Lect. Notes Math., vol. 1180, pp. 265–439. Springer, Berlin (1986) MR0876085. doi:10.1007/BFb0074920
