我們考慮下列一維的隨機微分方程：
dXt = (Xt −Xt^3)dt +λXtdBt, λ > 0,
對應到的起始值為正數。這方程的係數函數並不滿足利普希茨連續，所以不能用一般已知的定理得到強解的存在唯一性，但我們可以將係數函數截斷使其滿足利普希茨連續，得到截斷後方程強解的存在與唯一性，再取極限得到原方程強解的存在唯一性，而這解為一強馬氏過程。這篇論文主要想要研究解的長時間行為，而這會與解中的λ有關。利用解的比較定理，我們證明當λ大於根號2的時候，只有一個顯然的不變測度；當λ小於2的時候，存在了一個非顯然的不變測度。利用耦合的方法，我們得到藉由遍歷性可以得到全變差距離收斂。另外，我們也得到強範數為指數型收斂。

We consider the one-dimensional stochastic differential equation (SDE) of the form:
dXt = (Xt −Xt^3)dt +λXtdBt, λ > 0,
subject to a positive initial value. \\
It should be emphasized that because the Lipschitz conditions are not satisfied, the existence of a solution to the SDE is not assured. However, we can establish the existence and uniqueness of a strong solution by truncating the drift term and taking the limit as the truncation is removed. As an application of martingale problems, the solution is a strong Markov process. Our primary goal is to investigate the large-time behavior of this process. The conclusions will depend on the value of λ. By utilizing comparison theorems, we demonstrate that the process has a unique trivial invariant measure when λ is large enough. In addition, we prove that a unique invariant probability measure exists when λ is small, which is why we are more interested in this case. We discover, in particular, that the critical point exists and is equal to the square root of 2. Furthermore, using the coupling method, we establish ergodicity, which ensures convergence in the total variation distance, is established. We also obtain an exponential rate of convergence in the strong norm based on results from the application of "drift criteria" for general state space Markov processes.

[1] Ikeda, N., & Watanabe, S. Stochastic differential equations and diffusion processes. 2nd ed. North-Holland , 1989.
[2] Revuz, D., & Yor, M. Continuous martingales and Brownian Motion. Vol 293, 3rd ed. Springer Science & Business Media, 1999.
[3] Karatzas, Ioannis, et al. Brownian motion and stochastic calculus. Vol 113, 2nd ed. Springer Science & Business Media, 1991.
[4] Ethier, Stewart N, & Kurtz, Thomas G. Markov processes: characterization and convergence. John Wiley & Sons, 1986.
[5] Khoshnevisan, D. Multiparameter resources: An introduction to random fields. Springer, 2002.
[6] Gall, Jean-François Le. Brownian motion, martingales, and stochastic calculus. Springer, 2016.
[7] Durrett,R. Stochastic calculus: a practical introduction. CRC press, 1996.
[8] Stratonovich, R.L. " A new representation for stochastic integrals and equations." SIAM Journal on Control, Vol 4, no 2, SIAM, 1966, pp. 362-371.
[9] Decreusefond, Laurent, et al. "Stochastic Analysis of the Fractional Brownian Motion."Potential Analysis, Vol 10, 1999, pp. 177-214
[10] Nualart, David. The Malliavin calculus and related topics. Vol 1995. Springer, 2006.
[11] Biagini, Francesca, et al. Stochastic calculus for fractional Brownian motion and applications. Springer Science & Business Media, 2008.
[12] Samko, Stefan G, et al. Fractional integrals and derivatives. Vol 1. Gordon and breach science publishers, Yverdon Yverdon-les-Bains, Switzerland, 1993.
[13] Billingsley, Patrick. Probability and measure. 3rd ed. John Wiley & Sons, 1995.
[14] Nualart, David. & Ouknine, Youssef. "Stochastic differential equations with additive fractional noise and locally unbounded drift." Stochastic inequalities and applications, Vol 56, Birkhäuser Verlag, Basel, 2003, pp. 353-365.
[15] Russo, F., & Vallois, P. "Forward, backward and symmetric stochastic integration." Probability theory and related fields", Vol 77, no 3, Springer, 1993, pp. 403-421.
[16] Nourdin, Ivan. "Calcul stochastique généralisé et applications au mouvement brownien fractionnaire; Estimation non-paramétrique de la volatilité et test d’adéquation." Ph.D.Dissertation, Université Henri Poincaré-Nancy 1, 2004.
[17] Gradinaru, Mihai, et al. "m-order integrals and generalized Itô’s formula; the case of a fractional Brownian motion with any Hurst index." Annales de l’IHP Probabilités et statistiques, Vol 41, no 4, 2005, pp. 781-806.
[18] Russo, Francesco and Vallois, Pierre. " Itô formula for C 1-functions of semimartingales."Probability theory and related fields, Vol 104, Springer, 1996, pp. 27-41.
[19] Russo, Francesco and Vallois, Pierre. " Stochastic calculus with respect to continuous finite quadratic variation processes." Stochastics: An International Journal of Probability and Stochastic Processes, Vol 70, no 1-2, Taylor & Francis, 2000, pp. 1-40.
[20] Russo, Francesco and Vallois, Pierre. Stochastic Calculus Via Regularizations. Springer,2022
[21] Nourdin, Ivan. "Calcul stochastique généralisé et applications au mouvement brownien fractionnaire; Estimation non-paramétrique de la volatilité et test d’adéquation." Ph.D. Dissertation, Université Henri Poincaré-Nancy 1, 2004.
[22] Meyn, Sean P, & Tweedie, Richard L. " Stability of Markovian processes II: Continuoustime processes and sampled chains." Advances in Applied Probability, Vol 25, no 3, Cambridge University Press, 1993, pp. 487-517.
[23] Meyn, Sean P, & Tweedie, Richard L. " Stability of Markovian processes III: Foster–Lyapunov criteria for continuous-time processes." Advances in Applied Probability, Vol 25, no 3, Cambridge University Press, 1993, pp. 518-548.
[24] Azema, J., Kaplan-Duflo, M., & Revuz, D. " Mesure invariante sur les classes récurrentes des processus de Markov." Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete, Vol 8, no 3, Springer, 1967, pp. 157-181.
[25] Getoor, Ronald K. " Transience and recurrence of Markov processes." Séminaire de probabilités de Strasbourg, Vol 14, 1980, pp. 397-409.
[26] Khasminskii, Rafail. Stochastic stability of differential equations. Vol 66, 2nd ed. Springer
Science & Business Media, 2011.
[27] Kushner, Harold J. Stochastic stability and control. Vol 33. Brown Univ Providence RI, 1967.
[28] Meyn, Sean P, & Tweedie, Richard L. " Stability of Markovian processes I: Criteria for discrete-time chains." Advances in Applied Probability, Vol 24, no 3, Cambridge University Press, 1992, pp. 542-574.
[29] Oksendal, Bernt. Stochastic differential equations: an introduction with applications. 6th ed. Springer Science & Business Media, 2013.
[30] Skorohod, A.V. Studies in the theory of random processes. Reading, Mass: Addison-Wistey
[31] Baudoin, Fabrice. Diffusion processes and stochastic calculus. European Mathematical Society, 2014.
[32] Koralov, Leonid, & Sinai, Yakov G. Theory of probability and random processes. Springer Science & Business Media, 2007.
[33] Nourdin, Ivan. "A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one." Séminaire de probabilités XLI, Vol 56, Springer, 2008, pp.
181-197.
[34] Doss, Halim. "Liens entre équations différentielles stochastiques et ordinaires." Annales de l'IHP Probabilités et statistiques, Vol 13, no 2, 1977, pp. 99-125.
[35] Sussmann, Héctor J. "An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point." Bulletin of the American Mathematical Society, Vol 83, no 2, American Mathematical Society, 1977, pp. 296-298.
[36] Carlen, Eric, & Kree, Paul. " Lp estimates on iterated stochastic integrals." The Annals of Probability, Vol 19, no 1, JSTOR, 1991, pp. 354-368.
[37] Khoshnevisan, Davar. Stochastic Calculus Math 7880-1; Spring 2008.
[38] Da Prato, Giuseppe, & Zabczyk, Jerzy. Ergodicity for infinite dimensional systems. Vol 229. Cambridge University Press., 1996.
[39] Pavliotis, Grigorios A. Stochastic processes and applications: diffusion processes, the Fokker-Planck and Langevin equations. Vol 60. Springer., 2014.
[40] Klebaner, Fima C. Introduction to stochastic calculus with applications. World Scientific Publishing Company, 2012.
[41] Woess, W. Random walks on infinite graphs and groups. Vol 138. Cambridge university press, 2000.
[42] Stein, E. M., & Shakarchi, R. Complex analysis. Vol 2. Princeton University Press, 2010.
[43] Khoshnevisan, Davar, et al. " Phase Analysis for a family of Stochastic Reaction-Diffusion Equations." arXiv preprint arXiv:2012.12512, 2020.
[44] Evans, Lawrence C. An introduction to stochastic differential equations. Vol 82. American Mathematical Soc., 2012