Abstract
In this article given y : [0, η) → H, a continuous map into a Hilbert space H, we study the equation
where c(s, ⋅) is a given “potential” on C([0, η), H). Applying the transformation \(y \rightarrow \hat y\) to the solutions of the SPDE and SDE underlying a diffusion, we study the Feynman–Kac formula.
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References
Chung, K.L., Varadhan, S.R.S. (1980) Kac functionals and Schrodinger equations. In: Bhatia, R., Bhat, A.G., Parthasarathy, K.R. (eds.) Collected Papers of S.R.S. Varadhan, vol. 2, pp. 304–315. Hindustan Book Agency, New Delhi
Da Prato, G., Zabczyk, J.: Stochastic Equations in Infinite Dimensions. Cambridge University Press, Cambridge (1992)
Gawarecki, L., Mandrekar, V.: Stochastic Differential Equations with Applications to Stochastic Partial Differential Equations. Springer, Berlin (2011)
Itô, K.: Foundations of Stochastic Differential Equations in Infinite Dimensional Spaces. CBMS 47. SIAM, Philadelphia (1984)
Kac, M.: On the distribution of certain Wiener functionals. Trans. Am. Math. Soc. 65, 1–13 (1949)
Kallenberg, O.: Foundations of Modern Probability. Springer, New York (2010)
Kallianpur, G., Xiong, J.: Stochastic Differential Equations in Infinite Dimensional Spaces. Lecture Notes, Monograph Series, vol. 26. Institute of Mathematical Statistics, Hayward (1995)
Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus. Springer, New York (1998)
Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications (Universitext). Springer, Berlin (2010)
Rajeev, B.: Translation invariant diffusions in the space of tempered distributions. Indian J. Pure Appl. Math. 44(2), 231–258 (2013)
Rajeev, B.: Translation invariant diffusions and stochastic partial differential equations in \(\mathcal {S}^{\prime }\) (2019). http://arxiv.org/abs/1901.00277
Rajeev, B., Thangavelu, S.: Probabilistic representations of solutions to the forward equation. Potential Anal. 28, 139–162 (2008)
Rajeev, B., Vasudeva Murthy, A.S.: Existence and uniqueness for 2nd order quasi linear parabolic PDE’s. Preprint
Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer, Berlin (1999)
Stroock, D.W.: Partial Differential Equations for Probabilists. Cambridge University Press, Cambridge (2008)
Yosida, K.: Functional Analysis. Springer, Berlin (1979)
Acknowledgements
The author would like to acknowledge the financial support from the SERB (Science and Engineering Research Board, India) through the MATRIX project No. MTR/2017/000750.
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Rajeev, B. (2020). On the Feynman–Kac Formula. In: Joshua, V., Varadhan, S., Vishnevsky, V. (eds) Applied Probability and Stochastic Processes. Infosys Science Foundation Series(). Springer, Singapore. https://doi.org/10.1007/978-981-15-5951-8_29
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DOI: https://doi.org/10.1007/978-981-15-5951-8_29
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