Abstract
It has been recognised recently that there is a close connection between existence and regularity results for stochastic partial differential equations and functional calculus techniques in harmonic analysis. The connection is made more explicit in this paper with the notion of a stochastic functional calculus.
In the deterministic setting, suppose that A1,A2 are bounded linear operators acting on a Banach space E. A pair \((\mu_1,\mu_2)\) of continuous probability measures on [0, 1] determines a functional calculus \(f\;\mapsto\;f_{\mu_{1},\mu_{2}}(A_1,A_2)\) for analytic functions f by weighting all possible orderings of operator products of A1 and A2 via the probability measures µ1 and µ2. For example, \(f\;\mapsto\;f_{\mu,\mu}(A_1,A_2)\) is the Weyl functional calculus with equally weighted operator products.
Replacing μ1 by Lebesgue measure λ on [0, t] and μ2 by stochastic integration with respect to a Wiener process W, we show that there exists a functional calculus \(f\;\mapsto\;f_{\lambda,W;t}(A+B)\) for bounded holomorphic functions f if A is a densely defined Hilbert space operator with a bounded holomorphic functional calculus and B is small compared to A relative to a square function norm. By this means, the solution of the stochastic evolution equation \(dX_t\;=\;AX_{t}dt+BX_{t}dW_{t},\;X_0\;=\;x\), is represented as \(t\;\mapsto\;e_{\lambda,W;t}^{A+B}x,t\geq\;0.\). We show how to extend some of our results to \(L^{p}-\mathrm{spaces},\;2\leq p < \infty\) and apply them to the regularity of solutions of the Zakai equation.
Mathematics Subject Classification (2010). Primary 47A60; Secondary 47D06, 60H15.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Z. Brzeźniak, Stochastic partial differential equations in M-type 2 Banach spaces,. Potential Anal. 4 (1995), 1–45.
Z. Brzeźniak, J. van Neerven, M.C. Veraar and L. Weis, Itô’s formula in UMD Banach spaces and regularity of solutions of the Zakai equation. J. Differential Equations 245 (2008), 30–58
M. Cowling, I. Doust, A. McIntosh and A. Yagi, Banach space operators with a bounded H ∞ functional calculus. J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.
G. Da Prato and J. Zabczyk, Stochastic equations in infinite dimensions, Encyclopedia of Mathematics and its Applications 44, Cambridge University Press, Cambridge, 1992.
J. Dettweiler, J. van Neerven and L. Weis, Space-time regularity of solutions of the parabolic stochastic Cauchy problem. Stoch. Anal. Appl. 24 (2006), 843–869.
F. Flandoli, On the semigroup approach to stochastic evolution equations. Stochastic Analysis and Appl. 10 (1992), 181–203.
D.J.H. Garling, Brownian motion and UMD-spaces, in: “Probability and Banach Spaces” (Zaragoza, 1985), 36–49, Lecture Notes in Math. 1221, Springer-Verlag, Berlin, 1986.
H. Heinich, Esperance conditionelle pour les fonctions vectorielles. C.R. Acad. Sci. Paris Ser. A 276 (1973), 935–938.
B. Jefferies, Conditional expectation for operator-valued measures and functions. Bull. Austral. Math. Soc. 30 (1984), 421–429.
B. Jefferies, Feynman’s operational calculus and the stochastic functional calculus in Hilbert space, in “The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis”, Proc. Centre Math. Appl. Austral. Nat. Univ. 44, Austral. Nat. Univ., Canberra, 2010, 183–210.
B. Jefferies and G.W. Johnson, Feynman’s operational calculi for noncommuting operators: Definitions and elementary properties. Russ. J. Math. Phys. 8 (2001), 153–171.
B. Jefferies, Feynman’s operational calculi for noncommuting systems of operators: tensors, ordered supports and disentangling an exponential factor. Math. Notes 70 (2001), 815–838.
N.J. Kalton, J.M.A.M. van Neerven, M.C. Veraar, and L.W.Weis, Embedding vectorvalued Besov spaces into spaces of γ-radonifying operators. Math. Nachr. 281 (2008), 238–252.
P. Kunstmann and L. Weis, L p -regularity for parabolic equations, Fourier multiplier theorems and H ∞ -functional calculus. Functional analytic methods for evolution equations, 65–311, Lecture Notes in Math. 1855, Springer, Berlin, 2004.
H.H. Kuo, Gaussian measures in Banach spaces. Lecture Notes in Math. 463, Springer, Berlin, 1975.
S. Kwapień, Decoupling inequalities for polynomial chaos. Ann. Probab. 15 (1987), 1062–1071.
S. Kwapień and W. Woyczyński, Random series and stochastic integrals: single and multiple. Birkhäuser Boston, Inc., Boston, MA, 1992.
J. Maas, Malliavin calculus and decoupling inequalities in Banach spaces. J. Math. Anal. Appl. 363 (2010), 383–398.
A. McIntosh, Operators which have an H ∞ -functional calculus, in: Miniconference on Operator Theory and Partial Differential Equations 1986, 212–222. Proc. Centre for Mathematical Analysis 14, ANU, Canberra, 1986.
J. van Neerven and L. Weis, Stochastic integration of functions with values in a Banach space. Studia Math. 166 (2005), 131–170.
J. van Neerven, M.C. Veraar and L. Weis, Stochastic evolution equations in UMD Banach spaces. J. Funct. Anal. 255 (2008), 940–993.
J. van Neerven, Stochastic maximal L p -regularity. Ann. Probab. 40 (2012), 788–812.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Applied Mathematical Sciences, Vol. 44, New York/Berlin/Heidelberg/Tokyo, 1983.
G. Pisier, Probabilistic methods in the geometry of Banach spaces. Probability and analysis (Varenna, 1985), 167–241, Lecture Notes in Math. 1206, Springer, Berlin, 1986.
J. Rosiński and Z. Suchanecki, On the space of vector-valued functions integrable with respect to the white noise. Colloq. Math. 43 (1980), 183–201.
G. Samorodnitsky and M. Taqqu, Multiple stable integrals of Banach-valued functions. J. Theoret. Probab. 3 (1990), 267–287
H. Schaefer, Topological Vector Spaces, Springer-Verlag, Berlin/Heidelberg/New York, 1980.
L. Schwartz, Radon Measures in Arbitrary Topological Spaces and Cylindrical Measures, Tata Inst. of Fundamental Research, Oxford Univ. Press, Bombay, 1973.
A.V. Skorohod, Random Linear Operators, Riedel, 1984.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Jefferies, B. (2014). Harmonic Analysis and Stochastic Partial Differential Equations: The Stochastic Functional Calculus. In: Ball, J., Dritschel, M., ter Elst, A., Portal, P., Potapov, D. (eds) Operator Theory in Harmonic and Non-commutative Analysis. Operator Theory: Advances and Applications, vol 240. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-06266-2_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-06266-2_9
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-06265-5
Online ISBN: 978-3-319-06266-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)