# Anomaly Detection in Random Heterogeneous Media: Feynman-Kac - download pdf or read online

By Martin Simon

This monograph is anxious with the research and numerical answer of a stochastic inverse anomaly detection challenge in electric impedance tomography (EIT). Martin Simon experiences the matter of detecting a parameterized anomaly in an isotropic, desk bound and ergodic conductivity random box whose realizations are swiftly oscillating. For this goal, he derives Feynman-Kac formulae to scrupulously justify stochastic homogenization when it comes to the underlying stochastic boundary worth challenge. the writer combines recommendations from the speculation of partial differential equations and sensible research with probabilistic principles, paving how to new mathematical theorems that could be fruitfully utilized in the remedy of the matter handy. additionally, the writer proposes an effective numerical procedure within the framework of Bayesian inversion for the sensible answer of the stochastic inverse anomaly detection challenge.

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**Additional info for Anomaly Detection in Random Heterogeneous Media: Feynman-Kac Formulae, Stochastic Homogenization and Statistical Inversion**

**Example text**

0 Therefore, we obtain Ntv = U1 (vg·σ) NtG1 φ − Nt t t (φ(Xs ) − G1 φ(Xs )) ds − = 0 U1 (vg · σ)(Xs ) ds 0 t v(Xs )g(Xs ) dLs + 0 t t (φ(Xs ) − v(Xs )) ds + = 0 v(Xs )g(Xs ) dLs . 0 Moreover, notice that X g is related to X by a random time change, namely Xsg = Xs , ∂, s < ζg s ≥ ζg, where the lifetime ζ g is given by t ζ g := inf t : g(Xs ) dLs > Z 0 and Z is an exponentially distributed random variable with parameter 1. Hence, we obtain t Ntv = Ntg,v + v(Xsg )g(Xsg ) dLs for all t < ζ g . 35) 0 This equality may be generalized to the case of an arbitrary v ∈ H 1 (D) not necessarily in the range of the resolvent using an approximation argument.

12). s. e. x ∈ D. Proof. s. e. x ∈ D into a martingale additive functional of ﬁnite energy and a continuous additive functional of zero energy of the non-conservative Hunt process X g associated with (E g , H 1 (D)). We study the relation between the continuous additive functionals N v and N g,v . , v(x) = Gg1 φ(x) = Ex ∞ −t− e t 0 g(Xs ) dLs φ(Xt ) dt 0 for some φ ∈ L2 (D). Then we have the identity −LGg1 φ = φ − v so that for all w ∈ H 1 (D) E g (Gg1 φ, w) = (φ − v)w dx. 29 and the Revuz correspondence, we see that N g,v admits a semimartingale decomposition, namely t Ntg,v = (φ(Xsg ) − v(Xsg )) ds.

The term “symmetric” comes from the fact that in the onedimensional case L0 is the local time deﬁned by the Tanaka formula with the convention sign(0) = 0, which is called the symmetric local time, see [140]. In this case we have 1 ε→0 2ε t L0t = lim [[−ε, ε]](Xs ) ds. 8. 18). , where W is a standard d-dimensional Brownian motion, L0 is the symmetric local time of X at ∂D2 and L is the boundary local time. Proof. s. s. for every x ∈ D, =2 0 implying that t Mtv = 0 where W is a standard d-dimensional Brownian motion.