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Rough evolution equations. (English) Zbl 1193.60070

The authors consider the equation \[ dy_t= Ay_t dt+ f(y_t)\, dx_t,\quad t\in [0,T]\tag{1} \] with initial condition \(y_0\), where \(A\) is the infinitesimal generator of an analytical semigroup \(\{S_t: t\geq 0\}\) on a separable Banach space and \(f\) is function defined on this space.
The equation (1) is considered in the mild sense, that \(y+t\) satisfies \[ y_t= S_t y_0+ \int^t_0 S_{t-u} f(y_u)\,dx_u. \] The authors discuss a class of linear and nonlinear equations. The results are used in case of the heat equation.
The analysis in the paper bases on the theory of generalized differentials, called \(k\)-increments.

MSC:

60H05 Stochastic integrals
60H07 Stochastic calculus of variations and the Malliavin calculus
60G15 Gaussian processes
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[1] Adams, R. A. (1975). Sobolev Spaces. Pure and Applied Mathematics 65 . Academic Press, New York. · Zbl 0314.46030
[2] Coutin, L. and Qian, Z. (2002). Stochastic analysis, rough path analysis and fractional Brownian motions. Probab. Theory Related Fields 122 108-140. · Zbl 1047.60029 · doi:10.1007/s004400100158
[3] Dalang, R. C. (1999). Extending the martingale measure stochastic integral with applications to spatially homogeneous s.p.d.e.’s. Electron. J. Probab. 4 no. 6, 29 pp. (electronic). · Zbl 0922.60056
[4] Feyel, D. and de La Pradelle, A. (2006). Curvilinear integrals along enriched paths. Electron. J. Probab. 11 no. 34, 860-892 (electronic). · Zbl 1110.60031
[5] Friz, P. and Victoir, N. Multidimensional dimensional processes seen as rough paths. Cambridge Univ. Press. · Zbl 1193.60053
[6] Gubinelli, M. (2004). Controlling rough paths. J. Funct. Anal. 216 86-140. · Zbl 1058.60037 · doi:10.1016/j.jfa.2004.01.002
[7] Gubinelli, M. (2006). Rough solutions of the periodic Korteweg-de Vries equation. · Zbl 1278.35213
[8] Gubinelli, M. (2006). Rooted trees for 3D Navier-Stokes equation. Dyn. Partial Differ. Equ. 3 161-172. · Zbl 1132.35434 · doi:10.4310/DPDE.2006.v3.n2.a3
[9] Gubinelli, M. (2008). Abstract integration, combinatorics of trees and differential equations. · Zbl 1225.35164
[10] Gubinelli, M. (2009). Ramification of rough paths. J. Differential Equations . · Zbl 1315.60065
[11] Gubinelli, M. (2009). Rough integrals in higher dimensions. Unpublished manuscript.
[12] Gubinelli, M., Lejay, A. and Tindel, S. (2006). Young integrals and SPDEs. Potential Anal. 25 307-326. · Zbl 1103.60062 · doi:10.1007/s11118-006-9013-5
[13] Lejay, A. (2003). An introduction to rough paths. In Séminaire de Probabilités XXXVII. Lecture Notes in Math. 1832 1-59. Springer, Berlin. · Zbl 1041.60051
[14] León, J. A. and San Martín, J. (2007). Linear stochastic differential equations driven by a fractional Brownian motion with Hurst parameter less than 1/2. Stoch. Anal. Appl. 25 105-126. · Zbl 1108.60059 · doi:10.1080/07362990601052052
[15] Lototsky, S. V. and Rozovskii, B. L. (2006). Wiener chaos solutions of linear stochastic evolution equations. Ann. Probab. 34 638-662. · Zbl 1100.60034 · doi:10.1214/009117905000000738
[16] Lyons, T. and Qian, Z. (2002). System Control and Rough Paths . Oxford Univ. Press, Oxford. · Zbl 1029.93001 · doi:10.1093/acprof:oso/9780198506485.001.0001
[17] Lyons, T. J. (1998). Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 215-310. · Zbl 0923.34056 · doi:10.4171/RMI/240
[18] Maslowski, B. and Nualart, D. (2003). Evolution equations driven by a fractional Brownian motion. J. Funct. Anal. 202 277-305. · Zbl 1027.60060 · doi:10.1016/S0022-1236(02)00065-4
[19] Nualart, D. (1995). The Malliavin Calculus and Related Topics . Springer, New York. · Zbl 0837.60050
[20] Pazy, A. (1983). Semigroups of Linear Operators and Applications to Partial Differential Equations. Applied Mathematical Sciences 44 . Springer, New York. · Zbl 0516.47023
[21] Pérez-Abreu, V. and Tudor, C. (2002). Multiple stochastic fractional integrals: A transfer principle for multiple stochastic fractional integrals. Bol. Soc. Mat. Mexicana (3) 8 187-203. · Zbl 1020.60050
[22] Quer-Sardanyons, L. and Tindel, S. (2007). The 1-d stochastic wave equation driven by a fractional Brownian sheet. Stochastic Process. Appl. 117 1448-1472. · Zbl 1123.60049 · doi:10.1016/j.spa.2007.01.009
[23] Tindel, S. (1997). Stochastic parabolic equations with anticipative initial condition. Stochastics Stochastics Rep. 62 1-20. · Zbl 0892.60070
[24] Tindel, S., Tudor, C. A. and Viens, F. (2003). Stochastic evolution equations with fractional Brownian motion. Probab. Theory Related Fields 127 186-204. · Zbl 1036.60056 · doi:10.1007/s00440-003-0282-2
[25] Walsh, J. B. (1986). An introduction to stochastic partial differential equations. In École D’été de Probabilités de Saint-Flour , XIV- 1984. Lecture Notes in Math. 1180 265-439. Springer, Berlin. · Zbl 0608.60060
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