## Illinois Journal of Mathematics

### Weak nonmild solutions to some SPDEs

#### Abstract

We study the nonlinear stochastic heat equation driven by space–time white noise in the case that the initial datum $u_0$ is a (possibly signed) measure. In this case, one cannot obtain a mild random-field solution in the usual sense. We prove instead that it is possible to establish the existence and uniqueness of a weak solution with values in a suitable function space. Our approach is based on a construction of a generalized stochastic convolution via Young-type inequalities.

#### Article information

Source
Illinois J. Math., Volume 54, Number 4 (2010), 1329-1341.

Dates
First available in Project Euclid: 24 September 2012

https://projecteuclid.org/euclid.ijm/1348505531

Digital Object Identifier
doi:10.1215/ijm/1348505531

Mathematical Reviews number (MathSciNet)
MR2981850

Zentralblatt MATH identifier
1259.60067

#### Citation

Conus, Daniel; Khoshnevisan, Davar. Weak nonmild solutions to some SPDEs. Illinois J. Math. 54 (2010), no. 4, 1329--1341. doi:10.1215/ijm/1348505531. https://projecteuclid.org/euclid.ijm/1348505531

#### References

• L. Bertini and N. Cancrini, The stochastic heat equation: Feynman–Kac formula and intermittence, J. Statist. Physics 78 (1994), 1377–1402.
• D. L. Burkholder, Martingale transforms, Ann. Math. Statist. 37 (1966), 1494–1504.
• D. L. Burkholder, B. J. Davis and R. F. Gundy, Integral inequalities for convex functions of operators on martingales, Proceedings of the sixth Berkeley symposium on mathematical statistics and probability, vol. II, University of California Press, Berkeley, CA, 1972, pp. 223–240.
• D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249–304.
• E. M. Cabaña, The vibrating string forced by white noise, Z. Wahrsch. Verw. Gebiete 15 (1970), 111–130.
• E. Carlen and P. Kree, $L^p$ estimates for multiple stochastic integrals, Ann. Probab. 19 (1991), 354–368.
• R. A. Carmona and S. A. Molchanov, Parabolic Anderson problem and intermittency, Mem. Amer. Math. Soc., vol. 108, Amer. Math. Soc., Providence, RI, 1994.
• R. A. Carmona and D. Nualart, Random nonlinear wave equations: Propagation of singularities, Ann. Probab. 16 (1988), 730–751.
• D. Conus and R. C. Dalang, The nonlinear stochastic wave equation in high dimensions, Electron. J. Probab. 13 (2008), 629–670.
• D. Conus and D. Khoshnevisan, On the existence and position of the farthest peaks of a family of parabolic and hyperbolic SPDE's, to appear in Probab. Theory Related Fields (2012).
• R. C. Dalang, Extending martingale measure stochastic integral with applications to spatially homogeneous SPDE's, Electron. J. Probab. 4 (1999), 1–29.
• R. C. Dalang and C. Mueller, Some nonlinear S.P.D.E.'s that are second order in time, Electron. J. Probab. 8 (2003), 1–21 (electronic).
• R. C. Dalang, The stochastic wave equation, A minicourse on stochastic partial differential equations, Lecture Notes in Mathematics, vol. 1962, Springer, Berlin, 2009, pp. 39–71.
• B. Davis, On the $L^p$ norms of stochastic integrals and other martingales, Duke Math. J. 43 (1976), 697–704.
• M. Foondun, D. Khoshnevisan and E. Nualart, A local-time correspondence for stochastic partial differential equations, Trans. Amer. Math. Soc. 363 (2011), 2481–2515.
• I. Gyöngy and D. Nualart, On the stochastic Burgers' equation in the real line, Ann. Probab. 27 (1999), 782–802.
• C. Mueller, Singular initial conditions for the heat equation with noise, Ann. Probab. 24 (1996), 377–398.
• D. Nualart and L. Quer-Sardanyons, Existence and smoothness of the density for spatially homogeneous SPDEs, Potential Anal. 27 (2007), 281–299.
• S. Peszat, The Cauchy problem for a nonlinear stochastic wave equation in any dimension, J. Evol. Equ. 2 (2002), 383–394.
• S. Peszat and J. Zabczyk, Nonlinear stochastic wave and heat equations, Probab. Theory Related Fields 116 (2000), 421–443.
• J. B. Walsh, An Introduction to Stochastic Partial Differential Equations, Ecole d'Etè de Probabilités de St-Flour, XIV, 1984, Lecture Notes in Mathematics, vol. 1180, Springer-Verlag, Berlin, 1986, pp. 265–439.
• J. B. Walsh, On numerical solutions of the stochastic wave equation, Illinois J. Math. 50 (2006), 991–1018.