Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in $${\ma
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Nonlinear Differential Equations and Applications NoDEA
Surface measures and integration by parts formula on levels sets induced by functionals of the Brownian motion in Rn Stefano Bonaccorsi, Luciano Tubaro and Margherita Zanella Abstract. On the infinite dimensional space E of continuous paths from [0, 1] to Rn , n ≥ 1, endowed with the Wiener measure μ, we construct a surface measure defined on level sets of the L2 -norm of n-dimensional processes that are solutions to a general class of stochastic differential equations, and provide an integration by parts formula involving this surface measure. We follow the approach to surface measures in Gaussian spaces proposed via techniques of Malliavin calculus in Airault and Malliavin (Bull Sci Math 112:3–52, 1988). Mathematics Subject Classification. 60G15, 28C20. Keywords. Surface measures in infinite dimensional spaces, Gaussian measures, Gradient type systems, Integration by parts formulae.
1. Introduction Let E = C([0, 1]; Rn ) denote the Banach space of continuous functions from [0, 1] to Rn , endowed with the sup-norm f ∞ = sup |f (x)|. We denote by [0,1]
E = B(E) the σ-field of Borel measurable subsets of E. Also, we introduce the Hilbert space H = L2 (0, 1; Rn ) of square integrable measurable functions. Let us fix the notation we shall use in the sequel. The norm in Rn is denoted by |x| and the scalar product as x, xRn . In the infinite dimensional spaces E and H we denote the norm respectively by xH , xE . Finally, the scalar product in H is x, xH . By E ∗ we we denote the dual of E. It is known that given a probability space (Ω, F, P), a process B = {B(t), t ∈ [0, 1]} is a standard n-dimensional Brownian motion if it a centered Gaussian process with covariance function E[B(t), B(s)] = (s ∧ t)I, I being the identity matrix in Rn . This process induces a Gaussian measure μ on the space of trajectories (E, E). This measure is known as the Wiener measure; 0123456789().: V,-vol
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the process B(t)(x) = x(t) on the probability space (E, E, μ) will denote the standard n-dimensional Brownian motion. On the space E we introduce the Malliavin derivative D with domain D1,p , which is the closure in Lp (E, μ) of the class of smooth random variables (see for instance [6,13,14,25]). In Sect. 2 below we explain its construction in more details. The adjoint operator of the Malliavin derivative operator D having domain D1,p is the divergence operator, denoted as usual by δ, having domain Dq (δ), where q is the adjoint exponent of p. δ coincides with the Skorohod integral with respect to the Brownian motion B. In addition to the Sobolev spaces D1,p , we shall consider the spaces U Cb (E) of uniformly continuous and bounded functions1 from E to R and U Cb1 (E), of uniformly continuous and bounded functions which are Fr´echet differentiable, with an uniformly continuous and bounded derivative. In the sequel, we simplify the notation to U Cb , U Cb1 , since no confusion may arise. Let u ∈ Lp (E, μ; H) be a stochastic process, inde
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