By Samuel Karlin

This moment path keeps the advance of the idea and functions of stochastic approaches as promised within the preface of a primary direction. We emphasize a cautious therapy of easy buildings in stochastic approaches in symbiosis with the research of usual sessions of stochastic approaches bobbing up from the organic, actual, and social sciences.

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69) is convergent, so that we have for the measure under consideration e−h < √ (constant) −−−−→ 0. 71) 2 {0,0;1} dW x(t) χ Z h This, in turn, means that {0,0;1} h where Z = ∞ h=1 Z . Step 4. Now consider any discontinuous function x(t). By the definition of discontinuous functions, for any h there exist two points t1 = j/2m and t2 = ( j + 1)/2m for some m and j , such that 1 |x(t2 ) − x(t1 )| > h(t2 − t1 )log2 A = h(t2 − t1 ) 2 −ε 0 < ε < 12 . 72) h Since h(t2 − t1 )log2 A = h/(2m log2 A ) = h/Am , any discontinuous function belongs to the set Z m j with h arbitrary h > 0: x(t) ∈ Z m j ∀ h and hence x(t) ∈ Z .

43) θ (t) = 1 if t ≥ 0 0 if t < 0. 4, page 51). 42) with the δfunctions as an inhomogeneous term. 20)). ♦ Semigroup property of the transition probability: Chapman (ESKC) relation Einstein–Smoluchowski–Kolmogorov– Now let us consider the probability densities at three instants of time w(x 0 , t0 ) w(x , t ) w(x, t) t0 < t < t. Brownian motion: introduction to the concept of path integration 21 The distribution w(x , t ) can be considered as an initial one for w(x, t), while w(x 0 , t0 ) can serve as an initial one for both distributions w(x, t) and w(x , t ).

71) 2 {0,0;1} dW x(t) χ Z h This, in turn, means that {0,0;1} h where Z = ∞ h=1 Z . Step 4. Now consider any discontinuous function x(t). By the definition of discontinuous functions, for any h there exist two points t1 = j/2m and t2 = ( j + 1)/2m for some m and j , such that 1 |x(t2 ) − x(t1 )| > h(t2 − t1 )log2 A = h(t2 − t1 ) 2 −ε 0 < ε < 12 . 72) h Since h(t2 − t1 )log2 A = h/(2m log2 A ) = h/Am , any discontinuous function belongs to the set Z m j with h arbitrary h > 0: x(t) ∈ Z m j ∀ h and hence x(t) ∈ Z .