Let and let X be a solution to (39.1), where all the bi’s are equal to 0, a = σσT, and for x = 0, where δi j is equal to 1 if i = j and 0 otherwise. Let a(0) be the identity matrix. (1) Prove...

Let

and let X be a solution to (39.1), where all the bi’s are equal to 0, a = σσ^T, and

for x = 0, where δi j is equal to 1 if i = j and 0 otherwise. Let a(0) be the identity matrix.

(1) Prove that the matrices a(x) are uniformly elliptic.

(2) Show that |Xt| has the same law as a Bessel process of order

Conclude that if A is sufficiently close to −1, then X is transient, i.e,

while if A is sufficiently large, there exist arbitrarily large times t such that X_t
= 0.