Abstract
We consider the stochastic heat equation on Lie groups, that is, equations of the form \({{\partial }_{t}}u = {{\Delta }_{x}}u + \dot{W}\) on ℝ+ × G where G is a compact Lie group, Δ is the Laplace-Beltrami operator on G, and where \(\dot{W}\) is a Gaussian space-correlated noise, which is white-noise in time. We find necessary and sufficient conditions on the space correlation of \(\dot{W}\) such that u exists or is a Hölder-continuous function in the spatial variable x, using some basic tools of stochastic analysis and harmonic analysis on the Lie group G. We study the significance of these conditions on some basic examples. A sufficient condition for non-Hölder continuity is also given.
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References
R. Adler, An introduction to continuity,extrema, and related topics for general Gaussian processes, Inst. Math. Stat., Hayward, CA, 1990.
P. L. Chow, On solutions of the stochastic wave equation, Communication at the Fourth Joint AMS-SMM meeting. UNT, May 1999.
T. Coulhon, L. Salof-Coste, N. Varopoulos, Analysis and Geometry on Groups, Cambridge University Press, 1992.
R. C. Dalang and C. Robert, Extending the martingale measure stochastic integral with applications to spatially homogeneous, s.p.d.e.’s. Electron. J. Probab., 6 (1999).
R. C. Dalang and N. Frangos, The stochastic wave equation in two spatial dimensions, Ann. Probab., 26 (1998), 187–212.
G. DaPrato and J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge Univ. Press, 1992.
H. Fegan, Introduction to compact Lie groups, Prentice Hall, 1991.
G. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.
W. Fulton and J. Harris, Representation Theory, Springer, 1991.
I. Karatzas and S. Shreve, Brownian motion and stochastic calculus, 2nd ed., Springer V., 1991.
A. Karczewska and J. Zabczyk, A note on stochastic wave equation,Preprint.
P. Kotelenez, A class of quasilinear stochastic partial differential equations of Mc Kean-Vlasov type with mass conservation, Probab. Theory Rel. Fields, 102 (1995), 159–188.
N. V. Krylov, An analytic approach to SPDEs, Stochastic partial differential equations: six perspectives, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, (1999), 185–242.
H. Kunita, Stochastic flows and stochastic differential equations, Cambridge Univ. Press, 1990.
T. Kurtz and P. Protter, Weak convergence of stochastic integrals and differential equations II: Infinite dimensional case, Lecture Notes, 1996 CIME school in probab.
M. Marcus and G. Pisier, Random Fourier series with applications to harmonic analysis, Princeton UP, 1981.
A. Millet and M. Sanz-Sole, A stochastic wave equation in two space dimensions: smoothness of the law, Ann. Probab., 27 (1999), 803–844.
S. Peszat and J. Zabczyk, Stochastic evolution equations with a spatially homogeneous Wiener process, Stochastic Processes Appl., 72 (2) (1997), 187–204.
S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equation, Probab. Theory Related Fields., 116 (2000), 421–443.
M. Sanz and M. Sarra, Hölder continuity for the stochastic heat equation with spatially correlated noise, This volume.
M. Sugiura, Fourier series of smooth functions on compact Lie groups, Osaka J. Math., 8 (1971), 33–47.
S. Tindel and F. Viens, On space-time regularity for the stochastic heat equation on Lie groups, J. Func. Analy., 169 (2) (1999), 559–603.
J. B. Walsh, An introduction to stochastic partial differential equations, In: cole d’té de Probabilités de Saint Flour XIV, Lecture Notes in Math., 1180 (1986), 265–438.
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Tindel, S., Viens, F. (2002). Regularity Conditions for Parabolic SPDEs on Lie Groups. In: Dalang, R.C., Dozzi, M., Russo, F. (eds) Seminar on Stochastic Analysis, Random Fields and Applications III. Progress in Probability, vol 52. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8209-5_19
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DOI: https://doi.org/10.1007/978-3-0348-8209-5_19
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9474-6
Online ISBN: 978-3-0348-8209-5
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