By Marco Bramanti

Hörmander's operators are a big classification of linear elliptic-parabolic degenerate partial differential operators with gentle coefficients, that have been intensively studied because the overdue Nineteen Sixties and are nonetheless an energetic box of analysis. this article offers the reader with a normal evaluate of the sphere, with its motivations and difficulties, a few of its basic effects, and a few fresh traces of development.

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**Extra resources for An Invitation to Hypoelliptic Operators and Hörmander's Vector Fields**

**Example text**

9), the first 3 equations do not contain noise. One can read this fact saying that the probability of finding the system in some phase evolves in time as a diffusion process, diffusing (at least “for short time”, that is infinitesimally) in the directions which involve randomness. If the system were totally deterministic, with delta (that is “certain”) initial conditions, the solution would be a delta distribution at any time, and Kolmogorov equation would be a first order transport equation. Moreover, if the drift term X 0 had constant coefficients, there would be no interaction between the two phenomena of diffusion and transport, and the probability density would keep diffusing in a lower dimensional subspace of R6 .

The setting is similar to that studied by Mumford (see the previous example) but now it is the curvature π (s) to be proportional to a Wiener process. The stochastic system is: dx dy dε dπ = cos ε ds = sin ε ds = πds = φ dW and the corresponding backward Kolmogorov equation is ξt p + cos ε ξx p + sin ε ξ y p + πξε p + We can still check that the vector fields φ2 2 ξ p = 0. 1 First Motivation: Kolmogorov-Fokker-Planck Equations 25 φ X 1 = ⊂ ξπ ; X 0 = sum of first order terms 2 satisfy Hörmander’s condition (this time, at step 4).

The same authors, studying Asian options with arithmetic average floating strike call option, that is t A (t) = S (α ) dα 0 with final condition ⎞ ⎝ A (T ) ,0 V (T, S (T ) , A (T )) = max S (T ) − T find the PDE 1 2 −r V + ξt V + r Sξ S V + φ 2 S 2 ξ SS V + Sξ A V = 0. 11) which is of type X 12 + X 0 where X 1 , X 0 do not satisfy Hörmander’s condition at x = 0. Computer vision, the Mumford equation or “the process of random direction” (See Mumford [18]). One of the basic problems in computer vision is to reconstruct the three-dimensional shape and position of real objects starting from some twodimensional image of them, namely an intensity function I (x, y).