# American Institute of Mathematical Sciences

January  2021, 26(1): 121-154. doi: 10.3934/dcdsb.2020230

## Bilinear equations in Hilbert space driven by paths of low regularity

 1 Charles University, Faculty of Mathematics and Physics, Sokolovská 83, Prague 8,186 75, Czech Republic 2 Universidad de Sevilla, Dpto. Ecuaciones Diferenciales y Análisis numérico, Avda. Reina Mercedes s/n, 41012-Sevilla, Spain

* Corresponding author: María J. Garrido-Atienza

Received  January 2020 Revised  June 2020 Published  July 2020

Fund Project: The first author is supported by the Czech Science Foundation, project GAČR 19-07140S. The second author is supported by Ministerio de Ciencia, Innovación y Universidades, Grant No. PGC2018-096540-I00

In the article, some bilinear evolution equations in Hilbert space driven by paths of low regularity are considered and solved explicitly. The driving paths are scalar-valued and continuous, and they are assumed to have a finite $p$-th variation along a sequence of partitions in the sense given by Cont and Perkowski [Trans. Amer. Math. Soc. Ser. B, 6 (2019) 161–186] ($p$ being an even positive integer). Typical functions that satisfy this condition are trajectories of the fractional Brownian motion with Hurst parameter $H=1 / p$. A strong solution to the bilinear problem is shown to exist if there is a solution to a certain non–autonomous initial value problem. Subsequently, sufficient conditions for the existence of the solution to this initial value problem are given. The abstract results are applied to several stochastic partial differential equations with multiplicative fractional noise, both of the parabolic and hyperbolic type, that are solved explicitly in a pathwise sense.

Citation: Petr Čoupek, María J. Garrido-Atienza. Bilinear equations in Hilbert space driven by paths of low regularity. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 121-154. doi: 10.3934/dcdsb.2020230
##### References:
 [1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York/London, 1966.   Google Scholar [2] M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer International Publishing, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar [3] E. Alòs and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch. Rep., 75 (2003), 129-152.  doi: 10.1080/1045112031000078917.  Google Scholar [4] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sci., 11 (1984), 593-676.   Google Scholar [5] A. Ananova and R. Cont, Pathwise integration with respect to paths of finite quadratic variation, J. Math. Pures Appl., 107 (2017), 737-757.  doi: 10.1016/j.matpur.2016.10.004.  Google Scholar [6] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag London, 2006.  Google Scholar [7] M. Caruana and P. Friz, Partial differential equations driven by rough paths, J. Differ. Equ., 247 (2009), 140-173.  doi: 10.1016/j.jde.2009.01.026.  Google Scholar [8] M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 27-46.  doi: 10.1016/j.anihpc.2010.11.002.  Google Scholar [9] P. Cheridito and D. Nualart, Stochastic integral of divergence type with respecto to fractional Brownian motion with Hurst parameter ${H}\in\left(0, \frac{1}{2}\right)$, Ann. I. H. Poincaré Probab. Stat., 41 (2005), 1049-1081.  doi: 10.1016/j.anihpb.2004.09.004.  Google Scholar [10] R. Cont and P. Das, Quadratic variation and quadratic roughness, preprint, arXiv: 1907.03115. Google Scholar [11] R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.  Google Scholar [12] R. Cont and N. Perkowski, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 161-186.  doi: 10.1090/btran/34.  Google Scholar [13] G. Da Prato, M. Iannelli and L. Tubaro, Some results on linear stochastic differential equations in Hilbert spaces, Stochastics, 6 (1982), 105-116.  doi: 10.1080/17442508208833196.  Google Scholar [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar [15] M. Davis, J. Obłój and V. Raval, Arbitrage boundes for prices of weighted variance swaps, Math. Fin., 24 (2014), 821-854.  doi: 10.1111/mafi.12021.  Google Scholar [16] M. Davis, J. Obłój and P. Siorpaes, Pathwise stochastic calculus with local times, Ann. Inst. H. Poincaré Probab. Statist., 54 (2018), 1-21.  doi: 10.1214/16-AIHP792.  Google Scholar [17] L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.  doi: 10.1023/A:1008634027843.  Google Scholar [18] A. Deya, M. Gubinelli, M. Hofmanová and S. Tindel, One-dimensional reflected rough differential equations, Stoch. Proc. Appl., 129 (2019), 3261-3281.  doi: 10.1016/j.spa.2018.09.007.  Google Scholar [19] A. Deya, M. Gubinelli, M. Hofmanová and S. Tindel, A priori estimates for rough PDEs with application to rough conservation laws, J. Funct. Anal., 276 (2019), 3577-3645.  doi: 10.1016/j.jfa.2019.03.008.  Google Scholar [20] A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations, Probab. Theory Relat. Fields, 153 (2012), 97-147.  doi: 10.1007/s00440-011-0341-z.  Google Scholar [21] J. Dieudonné, Foundations of Modern Analysis, vol. 1, Academic Press, New York/London, 1969.   Google Scholar [22] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stoch. Proc. Appl., 115 (2005), 1357-1383.  doi: 10.1016/j.spa.2005.03.011.  Google Scholar [23] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag New York, 2000.  Google Scholar [24] E. H. Essaky and D. Nualart, On the $\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter ${H}<\frac{1}{2}$, Stoch. Proc. Appl., 125 (2015), 4117-4141.  doi: 10.1016/j.spa.2015.06.001.  Google Scholar [25] H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités XV, vol. 850 of Lecture Notes in Mathematics, Springer, Berlin, 1981,143–150.  Google Scholar [26] H. Föllmer and A. Schied, Probabilistic aspects of finance, Bernoulli, 19 (2013), 1306-1326.  doi: 10.3150/12-BEJSP05.  Google Scholar [27] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.  Google Scholar [28] P. K. Friz and N. B. Victoir., Multidimensional Stochastic Processes as Rough Paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar [29] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.  Google Scholar [30] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar [31] M. J. Garrido-Atienza, B. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.  Google Scholar [32] M. Gitterman, Classical harmonic oscillator with multiplicative noise, Physica A, 352 (2005), 309-334.  doi: 10.1016/j.physa.2005.01.008.  Google Scholar [33] M. Gubinelli, Ramification of rough paths, J. Differ. Equ., 248 (2010), 693-721.  doi: 10.1016/j.jde.2009.11.015.  Google Scholar [34] M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum of Mathematics, Pi, 3 (2015), e6, 75pp. doi: 10.1017/fmp.2015.2.  Google Scholar [35] M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.  Google Scholar [36] M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.  Google Scholar [37] J. M. E. Guerra and D. Nualart, The $1/{H}$-variation of the divergence integral with respect to the fractional Brownian motion for ${H}>1/2$ and fractional Bessel processes, Stoch. Proc. Appl., 115 (2005), 91-115.  doi: 10.1016/j.spa.2004.07.008.  Google Scholar [38] M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley Series in Physics Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962.  Google Scholar [39] R. Hesse and A. Neamţu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.  Google Scholar [40] Y. Hirai, Remarks on Föllmer's pathwise Itô calculus, Osaka J. Math., 56 (2019), 631-660.   Google Scholar [41] A. Hocquet and M. Hofmanová, An energy method for rough partial differential equations, J. Differ. Equ., 265 (2018), 1407-1466.  doi: 10.1016/j.jde.2018.04.006.  Google Scholar [42] Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.  Google Scholar [43] D. Kim, Local times for continuous paths of arbitrary regularity, preprint, arXiv: 1904.07327. Google Scholar [44] K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.  Google Scholar [45] M. Lemieux, On the Quadratic Variation of Semi-martingales, Master's thesis, University of British Columbia, 1983. Google Scholar [46] B. M. Levitan and G. L. Litvinov, Generalized displacement operators, in Encyclopaedia of Mathematics (ed. M. Hazewinkel), vol. 4, Springer Netherlands, 1989,224–228. Google Scholar [47] R. M. Łochowski, N. Perkowski and D. J. Prömel, A superhedging approach to stochastic integration, Stoch. Proc. Appl., 128 (2018), 4078-4103.  doi: 10.1016/j.spa.2018.01.009.  Google Scholar [48] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.  Google Scholar [49] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar [50] B. Maslowski and J. Šnupárková, Stochastic affine evolution equations with multiplicative fractional noise, Appl. Math., 63 (2018), 7-35.  doi: 10.21136/AM.2018.0036-17.  Google Scholar [51] Y. Mishura and A. Schied, On (signed) Takagi-Landsberg functions: $p$th variation, maximum, and modulus of continuity, J. Math. Anal. Appl., 473 (2019), 258-272.  doi: 10.1016/j.jmaa.2018.12.047.  Google Scholar [52] D. Nualart, The Malliavin Calculus and Related Topics, Springer - Verlag Berlin Heidelberg, 2006.  Google Scholar [53] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [54] N. Perkowski and D. Prömel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520.  doi: 10.3150/15-BEJ735.  Google Scholar [55] V. Pipiras and M. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.  doi: 10.2307/3318624.  Google Scholar [56] M. Pratelli, A remark on the $1/{H}$-variation of the fractional Brownian motion, in Séminaire de Probabilités XLIII (eds. C. Donati-Martin, A. Lejay and A. Rauault), vol. 2006 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2011,215–219. doi: 10.1007/978-3-642-15217-7_8.  Google Scholar [57] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer-Verlag, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar [58] L. C. G. Rogers, Arbitrage with fractional Brownian motion, Math. Fin., 7 (1997), 95-105.  doi: 10.1111/1467-9965.00025.  Google Scholar [59] F. Russo and P. Vallois, Stochastic calculus with respect to continuous finite quadratic variation processes, Stoch. Stoch. Rep., 70 (2000), 1-40.  doi: 10.1080/17442500008834244.  Google Scholar [60] A. Schied and Z. Zhang, On the $p$th variation of a class of fractal functions, preprint, arXiv: 1909.05239. Google Scholar [61] J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Universitatis Carolinae. Mathematica et Physica, 51 (2010), 49-67.   Google Scholar [62] J. Šnupárková, Stochastic Evolution Equations with Multiplicative Fractional Noise, Ph.D. thesis, Charles University in Prague, 2012. Google Scholar [63] H. Tanabe, Equations of Evolution, Pitman, London, 1979.  Google Scholar [64] H. Triebel, Theory of Function Spaces, Birkhäuser Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar [65] C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer International Publishing, 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar [66] M. Würmli, Lokalzeiten Für Martingale, Master's thesis, Universität Bonn, 1980. Google Scholar [67] M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅱ, Math. Nachr., 225 (2001), 145-183.  doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.  Google Scholar

show all references

##### References:
 [1] J. Aczél, Lectures on Functional Equations and Their Applications, Academic Press, New York/London, 1966.   Google Scholar [2] M. S. Agranovich, Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains, Springer International Publishing, 2015. doi: 10.1007/978-3-319-14648-5.  Google Scholar [3] E. Alòs and D. Nualart, Stochastic integration with respect to the fractional Brownian motion, Stoch. Stoch. Rep., 75 (2003), 129-152.  doi: 10.1080/1045112031000078917.  Google Scholar [4] H. Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sci., 11 (1984), 593-676.   Google Scholar [5] A. Ananova and R. Cont, Pathwise integration with respect to paths of finite quadratic variation, J. Math. Pures Appl., 107 (2017), 737-757.  doi: 10.1016/j.matpur.2016.10.004.  Google Scholar [6] J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer-Verlag London, 2006.  Google Scholar [7] M. Caruana and P. Friz, Partial differential equations driven by rough paths, J. Differ. Equ., 247 (2009), 140-173.  doi: 10.1016/j.jde.2009.01.026.  Google Scholar [8] M. Caruana, P. Friz and H. Oberhauser, A (rough) pathwise approach to a class of non-linear stochastic partial differential equations, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 28 (2011), 27-46.  doi: 10.1016/j.anihpc.2010.11.002.  Google Scholar [9] P. Cheridito and D. Nualart, Stochastic integral of divergence type with respecto to fractional Brownian motion with Hurst parameter ${H}\in\left(0, \frac{1}{2}\right)$, Ann. I. H. Poincaré Probab. Stat., 41 (2005), 1049-1081.  doi: 10.1016/j.anihpb.2004.09.004.  Google Scholar [10] R. Cont and P. Das, Quadratic variation and quadratic roughness, preprint, arXiv: 1907.03115. Google Scholar [11] R. Cont and D.-A. Fournié, Change of variable formulas for non-anticipative functionals on path space, J. Funct. Anal., 259 (2010), 1043-1072.  doi: 10.1016/j.jfa.2010.04.017.  Google Scholar [12] R. Cont and N. Perkowski, Pathwise integration and change of variable formulas for continuous paths with arbitrary regularity, Trans. Amer. Math. Soc. Ser. B, 6 (2019), 161-186.  doi: 10.1090/btran/34.  Google Scholar [13] G. Da Prato, M. Iannelli and L. Tubaro, Some results on linear stochastic differential equations in Hilbert spaces, Stochastics, 6 (1982), 105-116.  doi: 10.1080/17442508208833196.  Google Scholar [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge University Press, 2014.  doi: 10.1017/CBO9781107295513.  Google Scholar [15] M. Davis, J. Obłój and V. Raval, Arbitrage boundes for prices of weighted variance swaps, Math. Fin., 24 (2014), 821-854.  doi: 10.1111/mafi.12021.  Google Scholar [16] M. Davis, J. Obłój and P. Siorpaes, Pathwise stochastic calculus with local times, Ann. Inst. H. Poincaré Probab. Statist., 54 (2018), 1-21.  doi: 10.1214/16-AIHP792.  Google Scholar [17] L. Decreusefond and A. S. Üstünel, Stochastic analysis of the fractional Brownian motion, Potential Anal., 10 (1999), 177-214.  doi: 10.1023/A:1008634027843.  Google Scholar [18] A. Deya, M. Gubinelli, M. Hofmanová and S. Tindel, One-dimensional reflected rough differential equations, Stoch. Proc. Appl., 129 (2019), 3261-3281.  doi: 10.1016/j.spa.2018.09.007.  Google Scholar [19] A. Deya, M. Gubinelli, M. Hofmanová and S. Tindel, A priori estimates for rough PDEs with application to rough conservation laws, J. Funct. Anal., 276 (2019), 3577-3645.  doi: 10.1016/j.jfa.2019.03.008.  Google Scholar [20] A. Deya, M. Gubinelli and S. Tindel, Non-linear rough heat equations, Probab. Theory Relat. Fields, 153 (2012), 97-147.  doi: 10.1007/s00440-011-0341-z.  Google Scholar [21] J. Dieudonné, Foundations of Modern Analysis, vol. 1, Academic Press, New York/London, 1969.   Google Scholar [22] T. E. Duncan, B. Maslowski and B. Pasik-Duncan, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stoch. Proc. Appl., 115 (2005), 1357-1383.  doi: 10.1016/j.spa.2005.03.011.  Google Scholar [23] K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer-Verlag New York, 2000.  Google Scholar [24] E. H. Essaky and D. Nualart, On the $\frac{1}{H}$-variation of the divergence integral with respect to fractional Brownian motion with Hurst parameter ${H}<\frac{1}{2}$, Stoch. Proc. Appl., 125 (2015), 4117-4141.  doi: 10.1016/j.spa.2015.06.001.  Google Scholar [25] H. Föllmer, Calcul d'Itô sans probabilités, in Séminaire de Probabilités XV, vol. 850 of Lecture Notes in Mathematics, Springer, Berlin, 1981,143–150.  Google Scholar [26] H. Föllmer and A. Schied, Probabilistic aspects of finance, Bernoulli, 19 (2013), 1306-1326.  doi: 10.3150/12-BEJSP05.  Google Scholar [27] P. K. Friz and M. Hairer, A Course on Rough Paths, Universitext, Springer, Cham, 2014. doi: 10.1007/978-3-319-08332-2.  Google Scholar [28] P. K. Friz and N. B. Victoir., Multidimensional Stochastic Processes as Rough Paths, vol. 120 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 2010.  doi: 10.1017/CBO9780511845079.  Google Scholar [29] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Local pathwise solutions to stochastic evolution equations driven by fractional Brownian motions with Hurst parameters $H\in (1/3, 1/2]$, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2553-2581.  doi: 10.3934/dcdsb.2015.20.2553.  Google Scholar [30] M. J. Garrido-Atienza, K. Lu and B. Schmalfuß, Random dynamical systems for stochastic evolution equations driven by multiplicative fractional Brownian noise with Hurst parameters $H\in (1/3, 1/2]$, SIAM J. Appl. Dyn. Syst., 15 (2016), 625-654.  doi: 10.1137/15M1030303.  Google Scholar [31] M. J. Garrido-Atienza, B. Maslowski and J. Šnupárková, Semilinear stochastic equations with bilinear fractional noise, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 3075-3094.  doi: 10.3934/dcdsb.2016088.  Google Scholar [32] M. Gitterman, Classical harmonic oscillator with multiplicative noise, Physica A, 352 (2005), 309-334.  doi: 10.1016/j.physa.2005.01.008.  Google Scholar [33] M. Gubinelli, Ramification of rough paths, J. Differ. Equ., 248 (2010), 693-721.  doi: 10.1016/j.jde.2009.11.015.  Google Scholar [34] M. Gubinelli, P. Imkeller and N. Perkowski, Paracontrolled distributions and singular PDEs, Forum of Mathematics, Pi, 3 (2015), e6, 75pp. doi: 10.1017/fmp.2015.2.  Google Scholar [35] M. Gubinelli, A. Lejay and S. Tindel, Young integrals and SPDEs, Potential Anal., 25 (2006), 307-326.  doi: 10.1007/s11118-006-9013-5.  Google Scholar [36] M. Gubinelli and S. Tindel, Rough evolution equations, Ann. Probab., 38 (2010), 1-75.  doi: 10.1214/08-AOP437.  Google Scholar [37] J. M. E. Guerra and D. Nualart, The $1/{H}$-variation of the divergence integral with respect to the fractional Brownian motion for ${H}>1/2$ and fractional Bessel processes, Stoch. Proc. Appl., 115 (2005), 91-115.  doi: 10.1016/j.spa.2004.07.008.  Google Scholar [38] M. Hamermesh, Group Theory and its Application to Physical Problems, Addison-Wesley Series in Physics Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1962.  Google Scholar [39] R. Hesse and A. Neamţu, Local mild solutions for rough stochastic partial differential equations, J. Differential Equations, 267 (2019), 6480-6538.  doi: 10.1016/j.jde.2019.06.026.  Google Scholar [40] Y. Hirai, Remarks on Föllmer's pathwise Itô calculus, Osaka J. Math., 56 (2019), 631-660.   Google Scholar [41] A. Hocquet and M. Hofmanová, An energy method for rough partial differential equations, J. Differ. Equ., 265 (2018), 1407-1466.  doi: 10.1016/j.jde.2018.04.006.  Google Scholar [42] Y. Hu and D. Nualart, Rough path analysis via fractional calculus, Trans. Amer. Math. Soc., 361 (2009), 2689-2718.  doi: 10.1090/S0002-9947-08-04631-X.  Google Scholar [43] D. Kim, Local times for continuous paths of arbitrary regularity, preprint, arXiv: 1904.07327. Google Scholar [44] K. Kobayasi, On a theorem for linear evolution equations of hyperbolic type, J. Math. Soc. Japan, 31 (1979), 647-654.  doi: 10.2969/jmsj/03140647.  Google Scholar [45] M. Lemieux, On the Quadratic Variation of Semi-martingales, Master's thesis, University of British Columbia, 1983. Google Scholar [46] B. M. Levitan and G. L. Litvinov, Generalized displacement operators, in Encyclopaedia of Mathematics (ed. M. Hazewinkel), vol. 4, Springer Netherlands, 1989,224–228. Google Scholar [47] R. M. Łochowski, N. Perkowski and D. J. Prömel, A superhedging approach to stochastic integration, Stoch. Proc. Appl., 128 (2018), 4078-4103.  doi: 10.1016/j.spa.2018.01.009.  Google Scholar [48] T. J. Lyons, Differential equations driven by rough signals, Rev. Mat. Iberoamericana, 14 (1998), 215-310.  doi: 10.4171/RMI/240.  Google Scholar [49] B. Maslowski and D. Nualart, Evolution equations driven by a fractional Brownian motion, J. Funct. Anal., 202 (2003), 277-305.  doi: 10.1016/S0022-1236(02)00065-4.  Google Scholar [50] B. Maslowski and J. Šnupárková, Stochastic affine evolution equations with multiplicative fractional noise, Appl. Math., 63 (2018), 7-35.  doi: 10.21136/AM.2018.0036-17.  Google Scholar [51] Y. Mishura and A. Schied, On (signed) Takagi-Landsberg functions: $p$th variation, maximum, and modulus of continuity, J. Math. Anal. Appl., 473 (2019), 258-272.  doi: 10.1016/j.jmaa.2018.12.047.  Google Scholar [52] D. Nualart, The Malliavin Calculus and Related Topics, Springer - Verlag Berlin Heidelberg, 2006.  Google Scholar [53] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied mathematical sciences, Springer, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [54] N. Perkowski and D. Prömel, Pathwise stochastic integrals for model free finance, Bernoulli, 22 (2016), 2486-2520.  doi: 10.3150/15-BEJ735.  Google Scholar [55] V. Pipiras and M. Taqqu, Are classes of deterministic integrands for fractional Brownian motion on an interval complete?, Bernoulli, 7 (2001), 873-897.  doi: 10.2307/3318624.  Google Scholar [56] M. Pratelli, A remark on the $1/{H}$-variation of the fractional Brownian motion, in Séminaire de Probabilités XLIII (eds. C. Donati-Martin, A. Lejay and A. Rauault), vol. 2006 of Lecture Notes in Mathematics, Springer, Berlin, Heidelberg, 2011,215–219. doi: 10.1007/978-3-642-15217-7_8.  Google Scholar [57] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion, 3rd edition, Springer-Verlag, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-662-06400-9.  Google Scholar [58] L. C. G. Rogers, Arbitrage with fractional Brownian motion, Math. Fin., 7 (1997), 95-105.  doi: 10.1111/1467-9965.00025.  Google Scholar [59] F. Russo and P. Vallois, Stochastic calculus with respect to continuous finite quadratic variation processes, Stoch. Stoch. Rep., 70 (2000), 1-40.  doi: 10.1080/17442500008834244.  Google Scholar [60] A. Schied and Z. Zhang, On the $p$th variation of a class of fractal functions, preprint, arXiv: 1909.05239. Google Scholar [61] J. Šnupárková, Stochastic bilinear equations with fractional Gaussian noise in Hilbert space, Acta Universitatis Carolinae. Mathematica et Physica, 51 (2010), 49-67.   Google Scholar [62] J. Šnupárková, Stochastic Evolution Equations with Multiplicative Fractional Noise, Ph.D. thesis, Charles University in Prague, 2012. Google Scholar [63] H. Tanabe, Equations of Evolution, Pitman, London, 1979.  Google Scholar [64] H. Triebel, Theory of Function Spaces, Birkhäuser Basel, 1983. doi: 10.1007/978-3-0346-0416-1.  Google Scholar [65] C. A. Tudor, Analysis of Variations for Self-similar Processes, Springer International Publishing, 2013. doi: 10.1007/978-3-319-00936-0.  Google Scholar [66] M. Würmli, Lokalzeiten Für Martingale, Master's thesis, Universität Bonn, 1980. Google Scholar [67] M. Zähle, Integration with respect to fractal functions and stochastic calculus Ⅱ, Math. Nachr., 225 (2001), 145-183.  doi: 10.1002/1522-2616(200105)225:1<145::AID-MANA145>3.0.CO;2-0.  Google Scholar
 [1] Guo-Niu Han, Huan Xiong. Skew doubled shifted plane partitions: Calculus and asymptotics. Electronic Research Archive, 2021, 29 (1) : 1841-1857. doi: 10.3934/era.2020094 [2] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1749-1762. doi: 10.3934/dcdsb.2020318 [3] Soonki Hong, Seonhee Lim. Martin boundary of brownian motion on Gromov hyperbolic metric graphs. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021014 [4] Kaixuan Zhu, Ji Li, Yongqin Xie, Mingji Zhang. Dynamics of non-autonomous fractional reaction-diffusion equations on $\mathbb{R}^{N}$ driven by multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020376 [5] Hongwei Liu, Jingge Liu. On $\sigma$-self-orthogonal constacyclic codes over $\mathbb F_{p^m}+u\mathbb F_{p^m}$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020127 [6] Yuanfen Xiao. Mean Li-Yorke chaotic set along polynomial sequence with full Hausdorff dimension for $\beta$-transformation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 525-536. doi: 10.3934/dcds.2020267 [7] Hai Q. Dinh, Bac T. Nguyen, Paravee Maneejuk. Constacyclic codes of length $8p^s$ over $\mathbb F_{p^m} + u\mathbb F_{p^m}$. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020123 [8] Nicolas Dirr, Hubertus Grillmeier, Günther Grün. On stochastic porous-medium equations with critical-growth conservative multiplicative noise. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020388 [9] Wenxiong Chen, Congming Li, Shijie Qi. A Hopf lemma and regularity for fractional $p$-Laplacians. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3235-3252. doi: 10.3934/dcds.2020034 [10] Vaibhav Mehandiratta, Mani Mehra, Günter Leugering. Existence results and stability analysis for a nonlinear fractional boundary value problem on a circular ring with an attached edge : A study of fractional calculus on metric graph. Networks & Heterogeneous Media, 2021  doi: 10.3934/nhm.2021003 [11] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020445 [12] Zaizheng Li, Qidi Zhang. Sub-solutions and a point-wise Hopf's lemma for fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (2) : 835-865. doi: 10.3934/cpaa.2020293 [13] Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465 [14] Tommi Brander, Joonas Ilmavirta, Petteri Piiroinen, Teemu Tyni. Optimal recovery of a radiating source with multiple frequencies along one line. Inverse Problems & Imaging, 2020, 14 (6) : 967-983. doi: 10.3934/ipi.2020044 [15] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang. A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, , () : -. doi: 10.3934/era.2020120 [16] Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (2) : 697-735. doi: 10.3934/cpaa.2020286 [17] Bao Wang, Alex Lin, Penghang Yin, Wei Zhu, Andrea L. Bertozzi, Stanley J. Osher. Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training. Inverse Problems & Imaging, 2021, 15 (1) : 129-145. doi: 10.3934/ipi.2020046 [18] Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020366 [19] Baoli Yin, Yang Liu, Hong Li, Zhimin Zhang. Approximation methods for the distributed order calculus using the convolution quadrature. Discrete & Continuous Dynamical Systems - B, 2021, 26 (3) : 1447-1468. doi: 10.3934/dcdsb.2020168 [20] Chiun-Chuan Chen, Yuan Lou, Hirokazu Ninomiya, Peter Polacik, Xuefeng Wang. Preface: DCDS-A special issue to honor Wei-Ming Ni's 70th birthday. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : ⅰ-ⅱ. doi: 10.3934/dcds.2020171

2019 Impact Factor: 1.27

Article outline