First hitting time and place, monopoles and multipoles for pseudo-processes driven by the equation ∂u/∂t
=± ∂Nu/∂xN
Aimé Lachal, INSA de Lyon
Abstract
Consider the high-order heat-type equation
∂u/∂t
=± ∂Nu/∂xN
for an integer N>2 and introduce the related Markov pseudo-process
(X(t))t≥ 0.
In this paper, we study several functionals related
to (X(t))t≥ 0:
the maximum M(t) and minimum m(t) up to time t;
the hitting times ta+
and ta- of the half lines
(a,+∞) and (-∞,a) respectively.
We provide explicit expressions for the distributions of
the vectors (X(t),M(t)) and (X(t),m(t)), as well as those of the vectors
(ta+,X(ta+))
and (ta-,X(ta-)).
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