Caruana, Michael; Friz, Peter K.; Oberhauser, Harald A (rough) pathwise approach to a class of non-linear stochastic partial differential equations. (English) Zbl 1219.60061 Ann. Inst. Henri Poincaré, Anal. Non Linéaire 28, No. 1, 27-46 (2011). The authors of the present paper study a stochastic parabolic evolution equation of the form \(\partial_tu=F(t,x,Du,D^2u)dt+Du(t,x) V(x)\circ dB_t,\, t\geq 0,\, u(0,x)=u_0(x),\, x\in R^d,\) where \(B\) is a multi-dimensional Brownian motion and the stochastic integral is understood in Stratonovich’s sense. Motivated by an essentially pathwise approach by P.-L. Lions and P. E. Souganidis [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 9, 1085–1092 (1998; Zbl 1002.60552)], the authors apply {T. J. Lyons}’ rough path analysis [Rev. Mat. Iberoam. 14, No. 2, 215–310 (1998; Zbl 0923.34056)].The authors’ core arguments are purely deterministic and allow even to consider instead of the driving Brownian motion \(B\) a function \(z\in C^{0,p-\text{var}}([0,T];G^{[p]}(R^d)).\) A continuous dependence property of the solution \(u\) of the above equation driven by \(z\) on the initial condition \(u_0\) gives the possibility for various probabilistic applications. Reviewer: Rainer Buckdahn (Brest) Cited in 1 ReviewCited in 48 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 34F05 Ordinary differential equations and systems with randomness 35K55 Nonlinear parabolic equations Keywords:rough path theory; stochastic PDE; viscosity solution Citations:Zbl 1002.60552; Zbl 0923.34056 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Barles, Guy, Solutions de viscosité des équations de Hamilton-Jacobi (2004), Springer · Zbl 0819.35002 [2] Barles, Guy; Biton, Samuel; Bourgoing, Mariane; Ley, Olivier, Uniqueness results for quasilinear parabolic equations through viscosity solutions’ methods, Calc. Var. Partial Differential Equations, 18, 2, 159-179 (2003) · Zbl 1036.35001 [3] Barles, Guy; Biton, Samuel; Ley, Olivier, A geometrical approach to the study of unbounded solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 162, 4, 287-325 (2002) · Zbl 1052.35084 [4] Breuillard, Emmanuel; Friz, Peter; Huesmann, Martin, From random walks to rough paths, Proc. Amer. Math. Soc., 137, 3487-3496 (2009) · Zbl 1179.60017 [5] Brzeźniak, Zdzisław; Flandoli, Franco, Almost sure approximation of Wong-Zakai type for stochastic partial differential equations, Stochastic Process. Appl., 55, 2, 329-358 (1995) · Zbl 0842.60062 [6] Buckdahn, Rainer; Ma, Jin, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I, Stochastic Process. Appl., 93, 2, 181-204 (2001) · Zbl 1053.60065 [7] Buckdahn, Rainer; Ma, Jin, Stochastic viscosity solutions for nonlinear stochastic partial differential equations. II, Stochastic Process. Appl., 93, 2, 205-228 (2001) · Zbl 1053.60066 [8] Buckdahn, Rainer; Ma, Jin, Pathwise stochastic Taylor expansions and stochastic viscosity solutions for fully nonlinear stochastic PDEs, Ann. Probab., 30, 3, 1131-1171 (2002) · Zbl 1017.60061 [9] Buckdahn, Rainer; Ma, Jin, Pathwise stochastic control problems and stochastic HJB equations, SIAM J. Control Optim., 45, 6, 2224-2256 (2007), (electronic) · Zbl 1140.60031 [10] Coutin, Laure; Friz, Peter; Victoir, Nicolas, Good rough path sequences and applications to anticipating stochastic calculus, Ann. Probab., 35, 3, 1172-1193 (2007) · Zbl 1132.60053 [11] Coutin, Laure; Qian, Zhongmin, Stochastic analysis, rough path analysis and fractional Brownian motions, Probab. Theory Related Fields, 122, 1, 108-140 (2002) · Zbl 1047.60029 [12] Crandall, Michael G., Viscosity Solutions: A Primer, Lecture Notes in Math., vol. 1660 (1995) · Zbl 0901.49026 [13] Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27, 1, 1-67 (1992) · Zbl 0755.35015 [14] Davis, Mark H. A.; Burstein, Gabriel, A deterministic approach to stochastic optimal control with application to anticipative control, Stoch. Stoch. Rep., 40, 3-4, 203-256 (1992) · Zbl 0774.93085 [15] Fleming, Wendell H.; Soner, H. Mete, Controlled Markov Processes and Viscosity Solutions, Stoch. Model. Appl. Probab., vol. 25 (2006), Springer: Springer New York · Zbl 1105.60005 [16] Friz, P.; Lyons, T.; Stroock, D., Lévy’s area under conditioning, Ann. Inst. H. Poincaré Probab. Statist., 42, 1, 89-101 (2006) · Zbl 1099.60054 [17] Friz, Peter; Oberhauser, Harald, Rough path limits of the Wong-Zakai type with a modified drift term, J. Funct. Anal., 256, 3236-3256 (2009) · Zbl 1169.60011 [18] Friz, Peter; Victoir, Nicolas, Differential equations driven by Gaussian signals, Ann. Inst. H. Poincaré Probab. Statist., 46, 2, 369-413 (2010) · Zbl 1202.60058 [19] Friz, Peter; Victoir, Nicolas, Approximations of the Brownian rough path with applications to stochastic analysis, Ann. Inst. H. Poincaré Probab. Statist., 41, 4, 703-724 (2005) · Zbl 1080.60021 [20] Friz, Peter; Victoir, Nicolas, On uniformly subelliptic operators and stochastic area, Probab. Theory Related Fields, 142, 3-4, 475-523 (2008) · Zbl 1151.31009 [21] Friz, Peter K.; Victoir, Nicolas B., Multidimensional Stochastic Processes as Rough Paths: Theory and Applications, Cambridge Stud. Adv. Math., vol. 120 (2010), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1193.60053 [22] Giga, Y.; Goto, S.; Ishii, H.; Sato, M.-H., Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40, 2, 443-470 (1991) · Zbl 0836.35009 [23] Gyöngy, I., The stability of stochastic partial differential equations and applications. I, Stoch. Stoch. Rep., 27, 2, 129-150 (1989) · Zbl 0726.60060 [24] Gyöngy, I., The stability of stochastic partial differential equations and applications. Theorems on supports, (Stochastic Partial Differential Equations and Applications, II. Stochastic Partial Differential Equations and Applications, II, Trento, 1988. Stochastic Partial Differential Equations and Applications, II. Stochastic Partial Differential Equations and Applications, II, Trento, 1988, Lecture Notes in Math., vol. 1390 (1989), Springer: Springer Berlin), 91-118 · Zbl 0683.93092 [25] Gyöngy, I., The stability of stochastic partial differential equations. II, Stoch. Stoch. Rep., 27, 3, 189-233 (1989) · Zbl 0726.60061 [26] Gyöngy, I., The approximation of stochastic partial differential equations and applications in nonlinear filtering, Comput. Math. Appl., 19, 1, 47-63 (1990) · Zbl 0711.60053 [27] Gyöngy, István, On stochastic partial differential equations. Results on approximations, (Topics in Stochastic Systems: Modelling, Estimation and Adaptive Control. Topics in Stochastic Systems: Modelling, Estimation and Adaptive Control, Lecture Notes in Control and Inform. Sci., vol. 161 (1991), Springer: Springer Berlin), 116-136 · Zbl 0791.60046 [28] Gyöngy, István; Michaletzky, György, On Wong-Zakai approximations with \(δ\)-martingales, Stochastic Analysis with Applications to Mathematical Finance. Stochastic Analysis with Applications to Mathematical Finance, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 460, 2041, 309-324 (2004) · Zbl 1055.60056 [29] Gyöngy, István; Shmatkov, Anton, Rate of convergence of Wong-Zakai approximations for stochastic partial differential equations, Appl. Math. Optim., 54, 3, 315-341 (2006) · Zbl 1106.60050 [30] Iftimie, Bogdan; Varsan, Constantin, A pathwise solution for nonlinear parabolic equations with stochastic perturbations, Cent. Eur. J. Math., 1, 3, 367-381 (2003), (electronic) · Zbl 1031.35156 [31] Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, Ann. Probab., 28, 2, 558-602 (2000) · Zbl 1044.60045 [32] Kunita, Hiroshi, Stochastic Flows and Stochastic Differential Equations, Cambridge Stud. Adv. Math., vol. 24 (1997), Cambridge University Press: Cambridge University Press Cambridge, reprint of the 1990 original · Zbl 0865.60043 [33] Ledoux, M.; Qian, Z.; Zhang, T., Large deviations and support theorem for diffusion processes via rough paths, Stochastic Process. Appl., 102, 2, 265-283 (2002) · Zbl 1075.60510 [34] Lions, P.-L.; Souganidis, P. E., Viscosity solutions of fully nonlinear stochastic partial differential equations, Viscosity Solutions of Differential Equations and Related Topics. Viscosity Solutions of Differential Equations and Related Topics, Kyoto, 2001. Viscosity Solutions of Differential Equations and Related Topics. Viscosity Solutions of Differential Equations and Related Topics, Kyoto, 2001, Sūrikaisekikenkyūsho Kōkyūroku, 1287, 58-65 (2002), (in Japanese) [35] Lions, Pierre-Louis; Souganidis, Panagiotis E., Fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., 326, 9, 1085-1092 (1998) · Zbl 1002.60552 [36] Lions, Pierre-Louis; Souganidis, Panagiotis E., Fully nonlinear stochastic partial differential equations: non-smooth equations and applications, C. R. Acad. Sci. Paris Sér. I Math., 327, 8, 735-741 (1998) · Zbl 0924.35203 [37] Lions, Pierre-Louis; Souganidis, Panagiotis E., Fully nonlinear stochastic PDE with semilinear stochastic dependence, C. R. Acad. Sci. Paris Sér. I Math., 331, 8, 617-624 (2000) · Zbl 0966.60058 [38] Lions, Pierre-Louis; Souganidis, Panagiotis E., Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations, C. R. Acad. Sci. Paris Sér. I Math., 331, 10, 783-790 (2000) · Zbl 0970.60072 [39] Lyons, Terry, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14, 2, 215-310 (1998) · Zbl 0923.34056 [40] Lyons, Terry; Qian, Zhongmin, Flow of diffeomorphisms induced by a geometric multiplicative functional, Probab. Theory Related Fields, 112, 1, 91-119 (1998) · Zbl 0918.60009 [41] Lyons, Terry; Qian, Zhongmin, System Control and Rough Paths, Oxford Math. Monogr. (2002), Oxford University Press · Zbl 1029.93001 [42] Lyons, Terry J.; Caruana, Michael; Lévy, Thierry, Differential equations driven by rough paths, (Lectures from the 34th Summer School on Probability Theory Held in Saint-Flour, July 6-24, 2004, With an Introduction Concerning the Summer School by Jean Picard. Lectures from the 34th Summer School on Probability Theory Held in Saint-Flour, July 6-24, 2004, With an Introduction Concerning the Summer School by Jean Picard, Lecture Notes in Math., vol. 1908 (2007), Springer: Springer Berlin) · Zbl 1176.60002 [43] Nualart, David, The Malliavin Calculus and Related Topics, Probab. Appl. (N. Y.) (2006), Springer-Verlag: Springer-Verlag Berlin · Zbl 1099.60003 [44] Pardoux, E., Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3, 2, 127-167 (1979) · Zbl 0424.60067 [45] Pardoux, Étienne; Peng, Shi Ge., Backward doubly stochastic differential equations and systems of quasilinear SPDEs, Probab. Theory Related Fields, 98, 2, 209-227 (1994) · Zbl 0792.60050 [46] Rozovskiĭ, B. L., Evolyutsionnye stokhasticheskie sistemy, (Lineinaya teoriya i prilozkheniya k statistike sluchainykh protsessov (1983), Nauka: Nauka Moscow) [47] Tubaro, Luciano, Some results on stochastic partial differential equations by the stochastic characteristics method, Stoch. Anal. Appl., 6, 2, 217-230 (1988) · Zbl 0647.60070 [48] Twardowska, Krystyna, An approximation theorem of Wong-Zakai type for nonlinear stochastic partial differential equations, Stoch. Anal. Appl., 13, 5, 601-626 (1995) · Zbl 0839.60059 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.