Abstract
Here we shall consider multidimensional diffusion processes {u(t), t ∈ I} in \(\mathbb{R}^{d},\) (\(d \in \mathbb{N}\setminus \{0\}\)) solution on a time interval \(I \subset \mathbb{R}_{+}\) of a d−dimensional system of stochastic differential equations of the form
subject to a suitable initial condition.
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References
Arnold, L.: Stochastic Differential Equations: Theory and Applications. Wiley, New York (1974)
Barra, M., Del Grosso, G., Gerardi, A., Koch, G., Marchetti, F.: Some basic properties of stochastic population models. Lecture Notes in Biomathematics, vol. 32, pp. 155–164. Springer, Heidelberg (1978)
Bhattacharya, R.N.: Criteria for recurrence and existence of invariant measures for multidimensional diffusions. Ann. Prob. 6, 541–553 (1978)
Cai, G.Q., Lin, Y.K.: Stochastic analysis of the Lotka-Volterra model for ecosystems. Phys. Rev. E 70, 041910 (2004)
Friedman, A.: Stochastic Differential Equations and Applications. Academic, London (1975). Two volumes bounded as one, Dover, Mineola, NY (2004)
Gard, T.C.: Introduction to Stochastic Differential Equations. Marcel Dekker, New York (1988)
Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer, Berlin (1972)
Has’minskii, R.Z.: Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, The Netherlands (1980)
Ikeda, N., Watanabe, S.: Stochastic Differential Equations and Diffusion Processes. North-Holland, Kodansha (1989)
Itô, K., McKean, H.P.: Diffusion Processes and Their Sample Paths. Springer, Berlin (1965)
Karlin, S., Taylor H.M.: A Second Course in Stochastic Processes. Academic, New York (1981)
Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Van Nostrand, Princeton (1960)
Ludwig, D.: Stochastic Population Theories. Lecture Notes in Biomathematics, vol. 3. Springer, Heidelberg (1974)
Mao, X., Marion, G., Renshaw, E.: Environmental Brownian noise suppresses explosions in population dynamics. Stoch. Proc. Appl. 97, 95–110 (2002)
Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1998)
Roozen, H.: Equilibrium and extinction in stochastic population dynamics. Bull. Math. Biol. 49, 671–696 (1987)
Schuss, Z.: Theory and Applications of Stochastic Differential Equations. Wiley. New York (1980)
Skorohod, A.V.: Asymptotic Methods in the Theory of Stochastic Differential Equations. AMS, Providence, RI (1989)
Tan, W.Y.: Stochastic Models with Applications to Genetics, Cancers, AIDS and Other Biomedical Systems. World Scientific, Singapore (2002)
Veretennikov, A.Y.: On polynomial mixing bounds for stochastic differential equations. Stoch. Proc. Appl. 70, 115–127 (1997)
Veretennikov, A.Y.: On subexponential mixing rate for Markov processes. Theory Prob. Appl. 49, 110–122 (2005)
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Capasso, V., Bakstein, D. (2015). Stability, Stationarity, Ergodicity. In: An Introduction to Continuous-Time Stochastic Processes. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4939-2757-9_5
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DOI: https://doi.org/10.1007/978-1-4939-2757-9_5
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