Let $Y_j$ be a sequence of independent Gaussian random variables with mean zero and unit variance ($\mathbb{E} Y_j = 0$ and $\mathbb{E} Y_j^2 = 1$) and let $\sigma:\mathbb{R}\to [1,2]$.

We define the random variable $X_k$ so that $X_0=0$ and $$ X_N = \sum_{j=1}^N \sigma(X_{j-1}) Y_j. $$ It is not so hard to show that $N \leq \mathbb{E} X_N^2 \leq 4N$. I would like to know whether one can show the anticoncentration bound $$ \mathbb{P}( X_N \in [-1,1]) \leq C N^{-1/2}, $$ with a constant $C$ independent of the function $\sigma$. More generally, I'd like to know if the probability density function of $X_N$ is bounded by $CN^{-1/2}$ (or $CN^{-d/2}$ in a $d$-dimensional generalization).

As an aside, this seems related to the question of $L^1\to L^\infty$ bounds for the variable coefficient parabolic PDE $$ \partial_t u = \nabla \cdot (a(x) \nabla u), $$ which can be done elegantly with Nash's argument. In this case however (a naive translation of) Nash's argument fails because (1) there isn't an integration by parts formula and (2) the operator mapping the density of $X_{N-1}$ to the density of $X_N$ is not self-adjoint.

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