L ₁ and L _∞ stability of transition densities of perturbed diffusions

A.1 Proof of Lemma 3.1

Let us consider the case when | ν | = 0 . Let us now identify the transition densities p ~ ⁢ ( t , s , x , y ) and p ~ ε ⁢ ( t , s , x , y ) with ( d + 1 ) × d matrices Ω and Ω ε consisting of the first rows that are the components of the respective mean vectors

and the d × d covariance matrices, namely

where i > 1 .

For ( d + 1 ) × d matrix A, we will denote by A 1 the first row and by A 2 : d + 1 the square matrix composed of the rows from 2 to d + 1 .

We can rewrite p ~ ⁢ ( t , s , x , y ) and p ~ ε ⁢ ( t , s , x , y ) in terms of Ω and Ω ε :

where

So, applying the Taylor expansion and (2.8) combined with the consequence of (A3) gives

We will use a slightly different bound:

Now the upper bound on the difference in the L 1 - L 1 norm easily follows from (A.1) and Jensen’s inequality:

The bounds for | ν | ≥ 1 follow from differentiation of the Taylor expansion and similar bounds (2.8) for the derivatives of the Gaussian densities p ~ and p ~ ε .

A.2 Proof of Lemma 3.2

In order to prove Lemma 3.2, we decompose the difference | H - H ε | ⁢ ( t , s , x , y ) into six parts in the following way:

For II and V , by regularity assumptions and (2.8), we have

To obtain an upper bound for I , we first estimate the first part of the corresponding expression:

Combining the estimate above and Lemma 2.8 with the semigroup property of the flows, we have

and

Finally, using the control Lemma 3.1 for III and VI , we get

Summing up the bounds derived above, we complete the proof.