We announce a result on the existence of a unique local solution to a stochastic geometric wave equation on the one dimensional Minkowski space with values in an arbitrary compact Riemannian manifold. We consider a rough initial data in the sense that its regularity is lower than the energy critical.
Revised:
Accepted:
Published online:
Zdzisław Brzeźniak 1; Nimit Rana 1
@article{CRMATH_2020__358_6_633_0, author = {Zdzis{\l}aw Brze\'zniak and Nimit Rana}, title = {Low regularity solutions to the stochastic geometric wave equation driven by a fractional {Brownian} sheet}, journal = {Comptes Rendus. Math\'ematique}, pages = {633--639}, publisher = {Acad\'emie des sciences, Paris}, volume = {358}, number = {6}, year = {2020}, doi = {10.5802/crmath.38}, language = {en}, }
TY - JOUR AU - Zdzisław Brzeźniak AU - Nimit Rana TI - Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet JO - Comptes Rendus. Mathématique PY - 2020 SP - 633 EP - 639 VL - 358 IS - 6 PB - Académie des sciences, Paris DO - 10.5802/crmath.38 LA - en ID - CRMATH_2020__358_6_633_0 ER -
%0 Journal Article %A Zdzisław Brzeźniak %A Nimit Rana %T Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet %J Comptes Rendus. Mathématique %D 2020 %P 633-639 %V 358 %N 6 %I Académie des sciences, Paris %R 10.5802/crmath.38 %G en %F CRMATH_2020__358_6_633_0
Zdzisław Brzeźniak; Nimit Rana. Low regularity solutions to the stochastic geometric wave equation driven by a fractional Brownian sheet. Comptes Rendus. Mathématique, Volume 358 (2020) no. 6, pp. 633-639. doi : 10.5802/crmath.38. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.38/
[1] Strong solutions to stochastic wave equations with values in Riemannian manifolds, J. Funct. Anal., Volume 253 (2007) no. 2, pp. 449-481 | DOI | MR | Zbl
[2] Weak solutions to stochastic wave equations with values in Riemannian manifolds, Commun. Partial Differ. Equations, Volume 36 (2011) no. 9, pp. 1624-1653 | DOI | MR
[3] Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces, Ann. Probab., Volume 41 (2013) no. 3B, pp. 1938-1977 | DOI | MR | Zbl
[4] Tensor Gaussian measures on , J. Funct. Anal., Volume 33 (1979) no. 3, pp. 297-310 | DOI | MR | Zbl
[5] On fractional Brownian processes, Potential Anal., Volume 10 (1999) no. 3, pp. 273-288 | DOI | MR | Zbl
[6] The Cauchy problem for periodic semilinear PDE in space variables, Séminaire Bourbaki, Vol. 1994/95 (Astérisque), Volume 237, Société Mathématique de France, 1996, pp. 163-187 (Exp. no. 796) | Numdam | Zbl
[7] Rough evolution equations, Ann. Probab., Volume 38 (2010) no. 1, pp. 1-75 | DOI | MR | Zbl
[8] Local and global well-posedness of wave maps on for rough data, Int. Math. Res. Not., Volume 2998 (1998) no. 21, pp. 1117-1156 | DOI | MR | Zbl
[9] Well-posedness for nonlinear Dirac equations in one dimension, Kyoto J. Math., Volume 50 (2010) no. 2, pp. 403-451 | DOI | MR | Zbl
[10] Rough differential equations driven by signals in Besov spaces, J. Differ. Equations, Volume 260 (2016) no. 6, pp. 5202-5249 | DOI | MR | Zbl
[11] The -d stochastic wave equation driven by a fractional Brownian sheet, Stochastic Processes Appl., Volume 117 (2007) no. 10, pp. 1448-1472 | DOI | MR | Zbl
[12] Theory of function spaces, Modern Birkhäuser Classics, Birkhäuser, 2010 (reprint of the 1983 original) | Zbl
[13] An introduction to stochastic partial differential equations, École d’été de probabilités de Saint-Flour XIV-1984 (Lecture Notes in Mathematics), Volume 1180, Springer, 1984, pp. 265-439 | DOI | Zbl
Cited by Sources:
Comments - Policy