SUPPORT THEOREM FOR PINNED DIFFUSION PROCESSES

[1] S. Aida, Vanishing of one-dimensional L2 -cohomologies of loop groups, J. Funct. Anal. 261 (2011), no.

8, 2164–2213.

[2] S. Aida, Rough differential equations containing path-dependent bounded variation terms, Preprint

(2016), arXiv:1608.03083.

[3] S. Aida, S. Kusuoka and D. Stroock, On the support of Wiener functionals, Asymptotic problems in

probability theory: Wiener functionals and asymptotics (Sanda/Kyoto, 1990), 3–34, Pitman Res. Notes

Math. Ser. 284, Longman Sci. Tech., Harlow, 1993.

[4] V. Bally, A. Millet and M. Sanz-Sol´e, Approximation and support theorem in H¨

older norm for parabolic

stochastic partial differential equations, Ann. Probab. 23 (1995), no. 1, 178–222.

[5] G. Ben Arous, M. Grˇ

adinaru and M. Ledoux, H¨

older norms and the support theorem for diffusions,

Ann. Inst. H. Poincar´e Probab. Statist. 30 (1994), no. 1, 415–436.

[6] G. Ben Arous and R. L´eandre, H¨

older norms and the support theorem for diffusions, Probab. Theory

Related Fields 90 (1991), no. 3, 377–402.

[7] H. Boedihardjo , X. Geng, X. Liu and Z. Qian , A quasi-sure non-degeneracy property for the Brownian

rough path, Potential Anal. 51 (2019), no. 1, 1–21

[8] H. Boedihardjo, X. Geng and Z. Qian, Quasi-sure existence of Gaussian rough paths and large deviation

principles for capacities, Osaka J. Math. 53 (2016), no. 4, 941–970.

[9] T. Cass, G. dos Reis and W. Salkeld, Rough functional quantization and the support of McKean-Vlasov

equations, Preprint (2019), arXiv: 1911.01992.

[10] K. Chouk and P. K. Friz, Support theorem for a singular SPDE: the case of gPAM , Ann. Inst. Henri

Poincar´e Probab. Stat. 54 (2018), no. 1, 202–219.

[11] R. Cont and A. Kalinin, On the support of solutions to stochastic differential equations with pathdependent coefficients, Stochastic Process. Appl. 130 (2020), no. 5, 2639–2674.

SUPPORT THEOREM FOR PINNED DIFFUSION

23

[12] S. Dereich and G. Dimitroff, A support theorem and a large deviation principle for Kunita flows,

Stoch. Dyn. 12 (2012), no. 3, 1150022, 16 pp.

[13] J. D. Deuschel, P. K. Friz, A. Jacquier, and S. Violante, Marginal density expansions for diffusions

and stochastic volatility I: Theoretical foundations, Comm. Pure Appl. Math. 67 (2014), no. 1, 40–82.

[14] J. D. Deuschel, P. K. Friz, A. Jacquier, and S. Violante, Marginal density expansions for diffusions

and stochastic volatility II: Applications, Comm. Pure Appl. Math. 67 (2014), no. 2, 321–350.

[15] H. Doss and P. Priouret, Support d’un processus de r´eflexion, Z. Wahrsch. Verw. Gebiete 61 (1982),

no. 3, 327–345.

[16] P. Florchinger, Malliavin calculus with time dependent coefficients and application to nonlinear filtering,

Probab. Theory Related Fields 86 (1990), no. 2, 203–223.

[17] P. K. Friz, Continuity of the Itˆ

o-map for H¨

older rough paths with applications to the support theorem in

H¨

older norm, Probability and partial differential equations in modern applied mathematics, 117–135,

IMA Vol. Math. Appl. 140, Springer, New York, 2005.

[18] P. K. Friz and N. Victoir, Differential equations driven by Gaussian signals, Ann. Inst. Henri Poincar´e

Probab. Stat. 46 (2010), no. 2, 369–413.

[19] P. K. Friz and N. Victoir, Multidimensional stochastic processes as rough paths, Cambridge University

Press, Cambridge, 2010.

[20] I. Gy¨

ongy, D. Nualart and M. Sanz-Sol´e, Approximation and support theorems in modulus spaces,

Probab. Theory Related Fields 101 (1995), no. 4, 495–509.

[21] I. Gy¨

ongy and T. Pr¨

ohle, On the approximation of stochastic differential equation and on StroockVaradhan’s support theorem, Comput. Math. Appl. 19 (1990), no. 1, 65–70.

[22] M. Hairer and P. Sch¨

onbauer, The support of singular stochastic partial differential equations, Forum

Math. Pi 10 (2022), Paper No. e1, 127 pp.

[23] Y. Hu, Analysis on Gaussian spaces, World Scientific, 2017.

[24] N. Ikeda and S. Watanabe, Stochastic differential equations and diffusion processes, Second edition.

North-Holland Publishing Co., Amsterdam; Kodansha, Ltd., Tokyo, 1989.

[25] Y. Inahama, Quasi-sure existence of Brownian rough paths and a construction of Brownian pants,

Infin. Dimens. Anal. Quantum Probab. Relat. Top. 9 (2006), no. 4, 513–528.

[26] Y. Inahama, Large deviation principle of Freidlin-Wentzell type for pinned diffusion processes, Trans.

Amer. Math. Soc. 367 (2015), 8017–8137.

[27] Y. Inahama, Malliavin differentiability of solutions of rough differential equations, J. Funct. Anal. 267

(2014), no. 5, 1566–1584.

[28] Y. Inahama, Large deviations for rough path lifts of Watanabe’s pullbacks of delta functions, Int. Math.

Res. Not. IMRN 2016, 20, 6378–6414.

[29] Y. Inahama, Large deviations for small noise hypoelliptic diffusion bridges on sub-Riemannian

manifolds., To appear in Publ. Res. Inst. Math. Sci. (2024), arXiv:2109.14841.

[30] Y. Inahama and B. Pei, Positivity of the density for rough differential equations, J. Theoret. Probab.

35 (2022), no. 3, 1863–1877.

[31] A. Kalinin, Support characterization for regular path-dependent stochastic Volterra integral equations,

Electron. J. Probab. 26, (2021), Paper No. 29, 29 pp.

[32] H. Kunita, Stochastic flows and jump-diffusions, Springer, Singapore, 2019.

[33] S. Kusuoka and D. Stroock, Applications of the Malliavin calculus. II, J. Fac. Sci. Univ. Tokyo Sect.

IA Math. 32 (1985), no. 1, 1–76.

[34] T. Lyons, M. Caruana and T. L´evy, Differential equations driven by rough paths, Lecture Notes in

Math. 1908. Springer, Berlin, 2007.

[35] M. Ledoux, Z. Qian and T. Zhang, Large deviations and support theorem for diffusion processes via

rough paths, Stochastic Process. Appl. 102 (2002), no. 2, 265–283.

[36] P. Malliavin, Stochastic analysis, Springer, Berlin, 1997.

[37] T. Matsuda, Characterization of the support for Wick powers of the additive stochastic heat equation,

Preprint (2020), arXiv: 2001.11705.

[38] H. Matsumoto and S. Taniguchi, Stochastic analysis. Itˆ

o and Malliavin calculus in tandem, Cambridge

University Press, Cambridge, 2017.

[39] A. Millet and D. Nualart, Support theorems for a class of anticipating stochastic differential equations,

Stochastics Stochastics Rep. 39 (1992), no. 1, 1–24.

[40] A. Millet and M. Sanz-Sol´e, On the support of a Skorohod anticipating stochastic differential equation,

Barcelona Seminar on Stochastic Analysis (St. Feliu de Gu´ıxols, 1991), 103–131, Progr. Probab. 32,

Birkh¨

auser, Basel, 1993.

24

Y. INAHAMA

[41] A. Millet and M. Sanz-Sol´e, A simple proof of the support theorem for diffusion processes, S´eminaire de

Probabilit´es, XXVIII, 36–48, Lecture Notes in Math. 1583, Springer, Berlin, 1994.

[42] A. Millet and M. Sanz-Sol´e, The support of the solution to a hyperbolic SPDE, Probab. Theory Related

Fields 98 (1994), no. 3, 361–387.

[43] A. Millet and M. Sanz-Sol´e, Approximation and support theorem for a wave equation in two space

dimensions, Bernoulli 6 (2000), no. 5, 887–915.

[44] T. Nakayama, Support theorem for mild solutions of SDE’s in Hilbert spaces, J. Math. Sci. Univ. Tokyo

11 (2004), no. 3, 245–311.

[45] D. Nualart, The Malliavin calculus and related topics, Second edition. Springer, Berlin, 2006.

[46] C. Ouyang and W. Roberson-Vickery, Quasi-sure non-self-intersection for rough differential equations

driven by fractional Brownian motion, Electron. Commun. Probab. 27 (2022), Paper No. 15, 12 pp.

[47] J. Ren and J. Wu, On approximate continuity and the support of reflected stochastic differential

equations, Ann. Probab. 44 (2016), no. 3, 2064–2116.

[48] T. Simon, Support theorem for jump processes, Stochastic Process. Appl. 89 (2000), no. 1, 1–30.

[49] I. Shigekawa, Stochastic analysis, Translations of Mathematical Monographs 224, Iwanami Series in

Modern Mathematics, American Mathematical Society, Providence, RI, 2004.

[50] H. Sugita, Positive generalized Wiener functions and potential theory over abstract Wiener spaces,

Osaka J. Math. 25 (1988), no. 3, 665–696.

[51] D. Stroock, Markov processes from K. Itˆ

o’s perspective, Princeton University Press, Princeton, NJ, 2003.

[52] D. Stroock and S. R. S. Varadhan, On the support of diffusion processes with applications to the strong

maximum principle, Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and

Probability (Univ. California, Berkeley, Calif., 1970/1971), Vol. III: Probability theory, 333–359. Univ.

California Press, 1972.

[53] S. Takanobu and S. Watanabe, Asymptotic expansion formulas of the Schilder type for a class of conditional Wiener functional integrations, Asymptotic problems in probability theory: Wiener functionals

and asymptotics (Sanda/Kyoto, 1990), 194–241, Pitman Res. Notes Math. Ser. 284, Longman Sci.

Tech., Harlow, 1993.

[54] S. Taniguchi, Applications of Malliavin’s calculus to time-dependent systems of heat equations, Osaka

J. Math. 22 (1985), no. 2, 307–320.

[55] P. Tsatsoulis and H. Weber, Spectral gap for the stochastic quantization equation on the 2-dimensional

torus, Ann. Inst. Henri Poincar´e Probab. Stat. 54 (2018), no. 3, 1204–1249.

[56] J. Xu and J. Gong, Wong-Zakai approximations and support theorems for stochastic McKean-Vlasov

equations, Forum Math. 34 (2022), no. 6, 1411–1432.

Yuzuru Inahama

Faculty of Mathematics

Kyushu University

744 Motooka, Nishi-ku,

Fukuoka

Japan.

email:inahama@math.kyushu-u.ac.jp

...