By Lev A. Sakhnovich

In a few well-known works, M. Kac confirmed that a variety of equipment of chance thought should be fruitfully utilized to big difficulties of research. The interconnection among likelihood and research additionally performs a principal position within the current publication. notwithstanding, our method is especially in response to the applying of research tools (the approach to operator identities, indispensable equations idea, twin structures, integrable equations) to chance conception (Levy strategies, M. Kac's difficulties, the main of imperceptibility of the boundary, sign theory). the fundamental a part of the publication is devoted to difficulties of statistical physics (classical and quantum cases). We give some thought to the corresponding statistical difficulties (Gibbs-type formulation, non-extensive statistical mechanics, Boltzmann equation) from the sport standpoint (the video game among strength and entropy). One bankruptcy is devoted to the development of precise examples rather than lifestyles theorems (D. Larson's theorem, Ringrose's speculation, the Kadison-Singer and Gohberg-Krein questions). We additionally examine the Bezoutiant operator. during this context, we are not making the idea that the Bezoutiant operator is generally solvable, permitting us to enquire the precise periods of the whole services.

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In a few recognized works, M. Kac confirmed that a number of equipment of chance thought could be fruitfully utilized to big difficulties of research. The interconnection among chance and research additionally performs a valuable function within the current booklet. even though, our strategy is especially in line with the appliance of study equipment (the approach to operator identities, necessary equations conception, twin structures, integrable equations) to likelihood idea (Levy procedures, M.

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**Example text**

19) where ψ(x, s) = 0, x∈Δ. 20) The probabilistic meaning of ψ(x, s) follows from the equality ∞ 0 e−st p(t, Δ)dt = ψ(x, s)dx. 16) the following assertion. 31. Let the considered Levy process have a continuous density. 22) (sI − LΔ )f, ψ(x, s) Δ = f (0) holds. 32. 22) was deduced by M. Kac [67] and for the non-symmetric stable processes it was deduced in our works [144, 146, 147]. 33. It is known that stable processes, variance damped Levy processes, variance Gamma processes, the normal inverse Gaussian process, and the Meixner process have continuous densities (see [167]).

Donsker [6] for the case when Δ = (−∞, a]. We express the important function ψ(x, s) with the help of the quasi-potential B. 6. 35. Let the considered Levy process have continuous density and let the quasi-potential B be regular. 22). Proof. 4) we have −BLΔ f = f, f ∈ CΔ . 24) imply that (sI − LΔ )f, ψ(x, s) Δ = − (I + sB)LΔ f, ψ Δ = − LΔ f, Φ(0, x) Δ . 22) is valid. 22). 26) is valid. 26) in the form LΔ f, (I + sB ∗ )ϕ Δ = 0. 4) the range of LΔ is dense in Lp (Δ). 27) we have ϕ = 0. The theorem is proved.

13) implies that ϕ(x) ∈ Lp [−2c, 2c]. The proposition is proved. 51 that the operator B is bounded in all the spaces Lp (−c, c), p ≥ 1. We shall prove that the operator B is compact. 52. 50 be fulﬁlled. Then the operator B is compact in all the spaces Lp (−c, c), p ≥ 1. Proof. Let us consider the operator B ∗ in the space Lq (−c, c), 1/p + 1/q = 1. 14) where the functions fn (x) → 0 in the weak sense. 14) can be represented in the form B ∗ fn = c −c c+(y−x−|x−y|)/2 q(t, t − y + x)dtdy. 15) we see that B ∗ fn → 0, that is, the operator B ∗ is compact.